1-Cocycles of Unitary Representations of Infinite–Dimensional Unitary Groups

1-Cocycles of unitary representations of inﬁnite–dimensional unitary groups

Der Naturwissenschaftlichen Fakultät

der Friedrich-Alexander-Universität Erlangen-Nürnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Manuel Herbst

aus Bogotá Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der mündlichen Prüfung: 18.12.2018

Vorsitzender des Promotionsorgans: Prof. Dr. Georg Kreimer

Gutachter: Prof. Dr. Karl-Hermann Neeb, Prof. Dr. Alain Valette I

Acknowledgements

I wish to express my sincere gratitude to the following people that supported me throughout my PhD time:

First and foremost, I thank Prof. Dr. Karl–Hermann Neeb for his very friendly and always available help as supervisor and initiator of my project. His knowledge and all the fruitful discussions with him provided an invaluable source of inspiration for my work. I am grateful for his numerous corrections and proof-readings that helped me to improve my thesis. I am very much obliged to him and to the Friedrich–Alexander university (FAU) for providing me an ofﬁce room for my project.

I owe special thanks to the German National Academic Foundation (Studienstiftung des deutschen Volkes) who granted me a PhD scholarship for three years. In particular, I gratefully acknowledge the helpful advices of Prof. Dr. André Kaup concerning the timing of my PhD project.

I would also like to thank my family and my colleagues for encouragement in times of frustation. In particular, I would like to thank Prof. Dr. Catherine Meusburger and Stefan Wiedenmann. II

Abstract

The main objective of this thesis is to establish a classification theorem on the non- vanishing of the first order cohomology space H1 associated to a unitary highest weight representations (HWR), depending on the respective highest weight λ. This is realized for unitary HWR’s of various types of unitary groups: For the finite-dimensional unitary groups U(n), it is well-known that every continuous irreducible unitary rep- n resentation is a HWR with some highest weight λ Z . But in this case, the space 1 H is always trivial because the group U(n) is compact.∈ The situation is drastically different for the direct limit U( ) := lim U(n): Here, we obtain for every λ ZN ∞ ∈ a unitary HWR as a direct limit of unitary−→ HWR’s of the groups U(n). We determine precisely for which λ the space H1 is non-trivial and find that this is true in most cases, in particular whenever λ = 0 is finitely supported. The proof is not constructive and makes use of the classical6 Branching Laws for HWR’s of the U(n). In a second step, 2 we extend the direct limit group U( ) to its Banach completions Up(` ). Here, we ∞ 2 allow p to take values in [1, ) and a unitary operator g belongs to Up(` ) if and only if the difference g 1 is an∞ operator of pth Schatten class. For finitely supported − 2 λ, the corresponding unitary HWR of U( ) uniquely extends to Up(` ) (and even to 2 the whole unitary group U(` )). The associated∞ unitary HWR has a concrete realiza- 2 2 tion in a finite tensor product of Hilbert spaces that is built from ` and its dual (` )∗. Whenever this tensor product consists of at least two factors, there is a natural way to construct unbounded 1-cocycles as infinite sums of 1-coboundaries and we derive that the spaces H1 are non-trivial (even infinite-dimensional). In the complementary 2 2 cases, which are the identical representation of Up(` ) on ` , its dual representation 2 on (` )∗ and the one-dimensional trivial representation, the first order 1-cohomology spaces are trivial. The proofs rely on automatic continuity of group 1-cocycles under mild assumptions and on their automatic smoothness in a suitable Lie group setting. The latter provides a bridge between first order cohomology of Lie groups and that of Lie algebras. These tools are developed and discussed in this thesis. Finally, if the highest weight λ `∞(N, Z) is bounded but not finitely supported, then the corre- ∈ 2 sponding unitary HWR of U( ) only extends to the trace-class completion U1(` ), 1 ∞ 2 and in this case we have H = 0 for U1(` ) without exceptions. This result is ob- { } 2 tained by restricting the representation to the subgroup of diagonal operators T1(` ) (usually called a “maximal torus”). For finite-dimensional semisimple Lie groups, the non-vanishing of the spaces H1 for irreducible representations is a rare phenomenon. In contrast to that, our classification theorem shows that groups like the direct limit U( ) and its Banach completions 2 ∞ Up(` ) have natural irreducible unitary representations with highly non-trivial first order cohomology spaces. In particular, these groups do not have Property (FH). Here, 1 a topological group G is said to have Property (FH) if H = 0 holds for every continuous unitary representation of G. On the other hand, infinite-dimensional{ } groups 2 like U (` ) (the difference g 1 is a compact operator) and the full unitary group 2 U(` ) do∞ have Property (FH) since− they belong to the class of bounded groups (or more generally groups with Rosendal’s Property (OB)). III

Zusammenfassung

Das Hauptziel dieser Arbeit ist die Herleitung eines Klassiﬁkationssatzes über das Nicht-Verschwinden des Kohomologieraumes erster Ordnung, mit H1 bezeichnet, für eine höchste Gewichts-Darstellung (HGD) in Abhängigkeit vom jeweiligen höchsten Gewicht λ. Dies wird für unitäre HGDen verschiedener Klassen von unitären Gruppen realisiert: Es ist bekannt, dass jede stetige irreduzible unitäre Darstellung der endlich- n dimensionalen unitären Gruppen U(n) eine Darstellung mit höchstem Gewicht λ Z 1 ist. In diesem Fall ist jedoch der Raum H stets trivial, weil die Gruppe U(n) kompakt∈ ist. Ganz anders verhält es sich im Falle des direkten Limes U( ) := lim U(n): Hier ∞ erhält man zu jedem λ ZN eine unitäre HGD als direkten Limes von unitären−→ HGDen 1 der Gruppen U(n). Wir∈ bestimmen exakt diejenigen Gewichte λ, für die der Raum H nicht verschwindet und sehen, dass dies in den meisten Fällen zutrifft, insbesondere immer dann, wenn λ = 0 endlichen Träger hat. Der Beweis ist nicht konstruktiver Art und benutzt die klassischen6 Verzweigungsgesetze (“Branching Laws”) für HGDen der 2 U(n). In einem zweiten Schritt betrachten wir die Banach-Vervollständigungen Up(` ) der Gruppe U( ). Hierbei lassen wir Werte aus [1, ) für p zu und ein unitärer ∞ 2 ∞ 1 Operator g ist aus Up(` ) genau dann wenn die Differenz g ein p-ter Schatten- Klasse Operator ist. Für Gewichte λ mit nur endlich vielen Einträgen− lässt sich die 2 entsprechende unitäre HGD von U( ) eindeutig auf Up(` ) fortsetzen (sogar auf 2 die volle unitäre Gruppe U(` )). Die∞ entsprechende unitäre HGD lässt sich in einem endlichen Hilbertraum-Tensorprodukt realisieren, welches aus `2 und seinem Dual- 2 raum (` )∗ gebildet wird. Besteht dieses Tensorprodukt aus mindestens zwei Faktoren, so lassen sich unbeschränkte 1-Kozykel als unendliche Summen von 1-Korändern auf natürliche Art konstruieren und wir leiten daraus ab, dass die Räume H1 nicht trivial (sogar unendlich-dimensional) sind. In den komplementären Fällen, also bei der 2 2 2 identischen Darstellung von Up(` ) auf ` , deren dualer Darstellung auf (` )∗ sowie der eindimensionalen trivialen Darstellung, verschwinden die zugehörigen Räume H1. 1 2 Die Beweise bezüglich der H -Räume für die Gruppen Up(` ) fußen auf automatischer Stetigkeit von Gruppen-1-Koyzkeln unter milden Zusatzannahmen und auf deren automatischer Glattheit in einem geeigneten Lie Gruppen-Kontext. Letzteres stellt eine Beziehung zwischen der 1-Kohomologie von Lie-Gruppen und jener von Lie Algebren her. Wenn schließlich das höchste Gewicht λ `∞(N, Z) beschränkt aber nicht endlich getragen ist, dann setzt sich die jeweilige HGD∈ von U( ) nur noch auf die Spurklasse- 2 ∞1 Vervollständigung U1(` ) fort und in diesem Fall gilt H = 0 ausnahmslos. Dieses Resultat erhalten wir durch Einschränken der Darstellung auf{ die} Untergruppe der Di- 2 agonaloperatoren T1(` ) (üblicherweise als “Maximaler Torus” bezeichnet). Für endlich-dimensionale halbeinfache Lie-Gruppen ist das Nicht-Verschwinden der H1-Räume ein seltenes Phänomen. Im Gegensatz dazu zeigt unser Klassiﬁkationssatz, dass Gruppen wie der direkt Limes U( ) und seine Banach-Vervollständigungen 2 ∞ Up(` ) natürliche irreduzible unitäre Darstellungen mit hochgradig nichttrivialen Ko- homologieräumen H1 aufweisen. Insbesondere haben diese Gruppen nicht die Eigen- schaft (FH). Wir sagen hier, dass eine topologische Gruppe G die Eigenschaft (FH) hat, 1 wenn H = 0 für jede stetige unitäre Darstellung von G gilt. Auf der anderen Seite { } IV

2 haben unendlich-dimensionale Gruppen wie U (` ) (hier ist die Differenz g 1 ein 2 kompakter Operator) und die volle unitäre Gruppe∞ U(` ) durchaus die Eigenschaft− (FH), weil sie zur Klasse der beschränkten Gruppen (oder allgemeiner der Gruppen mit Rosendals Eigenschaft (OB)) zählen. Contents V

Contents

1 Introduction1

2 Preliminary facts on 1-cocycles 20 2.1 Basic definitions...... 20 2.2 1-cocycles, affine isometric actions and negative definite functions... 26 2.3 A generalized version of the Delorme–Guichardet Theorem...... 31

3 Topological Aspects 39 3.1 Baire groups and automatic continuity...... 39 3.2 Boundedness of topological groups and Rosendal’s Property (OB)... 45 3.3 A short remark on bounded orbits and ﬁxed points in Banach spaces. 55

4 Lie Theoretic Aspects 61 4.1 The class of locally exponential Lie groups...... 61 4.2 Integration and differentiation of 1-cocycles...... 73

2 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles 84 5.1 Schatten class operators...... 84

5.2 The Banach–Lie group Up( ) ...... 88 H 5.3 Derivations of Banach algebras and 1-cocycles...... 94

6 Unitary Highest Weight Representations (HWR) of Lie algebras 102 6.1 General facts on unitary highest weight representations...... 102

6.2 Integration of unitary highest weight representations of gl( ) ..... 113 ∞ 6.3 Direct limit structures of unitary highest weight representations.... 120

7 Direct limits and HWR of U( ) 126 ∞ 7.1 Conditional 1-cocycles...... 126

7.2 1-Cocycles of unitary HWR of U( ) ...... 132 ∞ 2 8 Tensor products and HWR of Up(` ) with finitely supported weights 137 8.1 Realization of unitary HWR with finitely supported weight...... 137 8.2 Construction of unbounded 1-cocycles in finite tensor products..... 140 2 8.3 1-Cocycles of unitary HWR of Up(` ) ...... 144 VI Contents

2 9 Maximal tori and general norm-continuous representations of Up(` ) 146 9.1 Restriction of 1-cocycles to a (maximal) torus...... 146 9.2 1-Cocycles of maximal tori of Banach–Lie algebras...... 149 2 9.3 Norm-continuous unitary representations of Up(` ) ...... 153 2 9.4 Unitary HWR of U1(` ) with inﬁnitely supported weights...... 156

Appendix 159

A Automatic continuity of 1-cocycles 159

B Coarsely bounded subsets of topological groups 164

C Inﬁnitesimal weight representations and their globalization 174

References 190 1

1 Introduction

Let G be a topological group, with neutral element e, acting on a real (or complex) Hilbert space via an orthogonal (or a unitary) representation π for which all orbit maps H G , g π(g)v, for v , → H 7→ ∈ H are continuous. Such representations are called continuous. A continuous function β : G is called a (continuous) 1-cocycle if it satisﬁes the equation → H β(gh) = β(g) + π(g)β(h) for g, h G. (1) ∈ For a given continuous representation π, there always exist continuous 1-cocycles of the form ∂v(g) := v π(g)v, where v . Such 1-cocycles are called 1-coboundaries or (synonymously) −trivial 1-cocycles. The∈ H fundamental question we ask in this thesis is the following: Which representations admit nontrivial 1-cocycles, i.e. which representations admit 1-cocycles which are not 1-coboundaries?

In the contemporary mathematical literature, this question is usually formulated using the concept of the 1-cohomology spaces H1. To this end, one introduces the vector 1 space Z (G, π, ) of 1-cocycles associated to a continuous representation (π, ) of 1 G, its subspaceH of 1-coboundaries B (G, π, ) and the resulting quotient vectorH space H 1 1 1 H (G, π, ) := Z (G, π, )/B (G, π, ), H H H which is referred to as the ﬁrst-order cohomology space. Now, the question reads as follows:

Problem 1.1. Which representations (π, ) of G have a non-vanishing ﬁrst-order co- 1 homology space H (G, π)? H

Over the last decades, this problem has been gaining more and more attention since it is related to mathematical problems occurring in a wide range of mathematical disci- plines such as geometric group theory, unitary representations, ergodic theory, stochas- tic processes and theoretical physics (cf. [BHV08, Del77, PS72, PS72b, Ar69]). Of particular importance is the case when (π, ) is an irreducible representation of a Lie group G. For finite-dimensional Lie groupsH it is known that the non-triviality of the first order cohomology spaces H1 for irreducible representations is rather rare and is often considered as an exceptional case. If G is, for example, a connected finite- dimensional semisimple real Lie group, then, up to equivalence, there exists at most a finite number of (topologically complete) irreducible continuous unitary represen- 1 tations (π, ), for which H (G, π, ) = 0 (cf. [PS75, Prop. 11]). Moreover, the 1 space H (GH, π, ) is always finite-dimensionalH 6 { } (cf. [PS75, Prop. 9 and Prop.12]). Much less is knownH about the 1-cohomology spaces of infinite-dimensional Lie groups, which is the main theme of this PhD thesis. We shall see that there exist irreducible representations of infinite-dimensional Lie groups for which the nonvanishing of the 2 1 Introduction spaces H1 is, roughly speaking, the rule and not an exceptional case. The involved Lie groups are concrete subgroups of the unitary group U( ), where is a complex, separable, infinite-dimensional Hilbert space. On the otherH hand, it isH remarkable that the full unitary group U( ) itself does not admit continuous nontrivial 1-cocycles. H

Motivation and ﬁelds of application

One of the central motivations to study 1-cocycles β : G is their fundamental role in the construction of unitary representations on bosonic→ H Fock Hilbert spaces (e.g. [Is96]), continuous tensor product representations (e.g. [Del77, PS72b]) and Lévy processes/ convolution semigroups (e.g. [PS72, Par67]). What is common to these research areas is the use of infinitely positive definite kernels: In [Ne00, Prop. II.4.6], bosonic Fock Hilbert spaces are realized as reproducing kernel Hilbert spaces of some infinitely divisible positive definite kernel. As presented e.g. in [Del77, PS72b, AH78, St69], infinitely divisible positive definite functions lead to the construction of continuous tensor product representation. On the other hand, infinitely divisible positive definite functions on locally compact abelian groups G are precisely the Fourier transforms of infinitely divisible probability measures µ on the topological dual Gb (Bochner’s Theorem) and such measures generate convolution semigroups on Gb. For arbitrary topological groups G, the infinitely divisible positive definite functions ψ(g) ϕ : G C are precisely the functions ϕ(g) = e− , where ψ : G C is a negative definite→ function (Schönberg’s Theorem). Every such negative definite→ function ψ is of the following form (see [Gui72, Prop 4.5], [Ar69, Thm. 5.1] or [BHV08, Prop. 2.10.2] for the real-valued case):

1 2 ψ(g) = β(g) i γ(g) for all g G, 2 k k − ∈ where β : G is some 1-cocycle corresponding to a unitary representation (π, ) and γ : G →R is H a map satisfying the equation H → γ(gh) = γ(g) + γ(h) + Im β(g), π(g)β(h) for all g, h G. 〈 〉 ∈ This establishes the bridge between 1-cocycles β and inﬁnitely divisible positive definite functions ϕ. In particular, if (π, ) is an orthogonal representation, we may 1 2 H 2 β(g) choose γ 0. Then, the unitary representation deﬁned by ϕ(g) = e− k k via GNS 1 construction≡ is a cyclic Fock space representation . If the representation (π, ) = R ⊕ d x HX X (πx , x ) µ( ) decomposes as a direct integral for a suitable measure space ( , µ) H 1 We shall introduce the bosonic Fock representation (πt , t ) for a parameter t > 0 in Subsection 1 H 2.2. For t = 2 , one can show that the above GNS representation coincides with the subrepresentation of (πt , t ) that is generated by the vaccum vector Ωt := γt (0), because we have H 2 t β(g) πt (g)Ωt , Ωt = ϕt (g) = e− k k for all g G. 〈 〉 ∈ 3

R ⊕ d x and if the 1-cocycle β = X βx µ( ) decomposes into a direct integral of 1-cocycles 1 βx Z (G, πx , x ), then the GNS representation defined by ∈ H 1 2 1 R 2 2 β(g) 2 X βx (g) dµ(x) ϕ(g) = e− k k = e− k k is a cyclic continuous tensor product representation of G. Moreover, this representation extends to a representation of the group G(X ) of G-valued measurable step functions2. The analysis of the so-obtained unitary representations of G(X ) are considered in [Del77] as the principal motivation to study 1-cocycles β. They form a basic building block in the understanding of unitary representations of the group (X , G) of measurable functions X G (also called current group), of gauge groups M → of the form C ∞(X , G), where X is a smooth manifold (cf. [Is96]), and also for the groups L∞(X , G) which carry Lie-group structures (cf. [Gl03b]). In Quantum Field Theory, the structure theory of quantum fields leads to decomposition into so-called primary fields and infinitely divisible ones. The latter ones correspond to factorizable representations of the corresponding -algebras and their symmetry groups ([Ar69, BY75, He75, St68, St69b, St71b]). ∗

In the theory of negative definite functions, a fundamental problem is the classification of all nonzero negative definite functions ψ : G R that generate an extremal ray in the cone of (continuous) negative definite functions→ of a topological group G. These so-called extremal negative definite functions are precisely those of the form ψ(g) = 2 β(g) where the 1-cocycle β : G corresponds to an irreducible orthogonal representation.k k This is the Vershik–Karpushev→ H Theorem (cf. [VK84]) and was proven in [LSV04, Thm. 1]. We shall see below that ψ is a bounded function if and only if β is a 1-coboundary. This means that bounded extremal negative definite functions are (up to a positive factor) of the form

ψ(g) = Re(1 ϕ(g)), − where ϕ is a pure state of G, i.e. a normalized positive definite function generating an irreducible GNS representation. In view of this observation, the main problem in the classification of negative definite functions is to solve Problem 1.1 for irreducible orthogonal/unitary representations. The most famous classification result for negative definite functions ψ : G C is the Lévy-Khintchine formula. This is an integral representation formula for ψ→, which has been established first for locally compact abelian groups and later generalized to (discrete) abelian semigroups with involution ([BCR84]) and symmetric pairs (G, K) ([FH74, Thm. 3.3]). The Lévy-Khintchine formula for a locally compact abelian group G is also a classification result of convolution semigroups on G by virtue of the relation e tψ ( Par67 ). (µt )t>0 b µÒt = − [ ]

2This is the group of measurable functions f : X G that take only ﬁnitely many values in G. The group G embeds into G(X ) via the constante maps and→ the positive deﬁnite function ψ extends to G(X ) via the formula Z 1 2 ψ(f ) = exp βx (f (x)) dµ(x) . −2 X k k 4 1 Introduction

In the realm of locally compact groups, the study of the spaces H1 is of particular importance in geometric group theory and other areas, where groups with Kazhdan’s Property (T) play an important role (including e.g. ergodic theory and differential geometry). According to Kazhdan’s original definition, a locally compact group is said to have Property (T) if the trivial representation is an isolated point in the unitary dual Gb, endowed with the Fell topology. For compactly generated locally compact groups Kazhdan’s Property (T) is equivalent to the vanishing of the spaces H1 for every continuous unitary representation. This follows from the Delorme–Guichardet Theorem ([BHV08, Thm. 2.12.4]) that we describe below in some more detail. Moreover, Shalom’s Theorem ([BHV08, Thm. 3.2.1], see also [LSV04]) shows that the same characterization remains valid if one restricts to irreducible continuous unitary representations. Locally compact groups with Property (T) have the remarkable property that they are compactly generated (cf. [BHV08, Thm. 1.3.1]). In particular, discrete groups with Property (T) are finitely generated and one can turn each such a group into a metric space using the so-called word-metric which, roughly speaking, inter- pretes the distance of two given group elements g, h as the length of “shortest word” 1 for g− h with letters taken from a given finite generating subset. This can be visualized by the so-called Cayley graph which is a (colored) graph whose vertices are labeled by the group elements. The word distance is then the minimal number of edges connect- ing two given group elements. The idea is that the algebraic and geometric structures of the group are reflected in its Cayley graph.

Statement of the principal problems and main results

However, in this thesis, we will not pursue the issues of the preceding paragraphs but approach the following problem in the setting of inﬁnite-dimensional unitary groups:

Problem 1.2. 1. Which topological groups G do have the so-called Property (FH) that 1 H (G, π, ) = 0 for every continuous unitary representation (π, )? H { } H 2. Provided that G does not have Property (FH): Which unitary representations (π, ) 1 of G have H (G, π, ) = 0 ? H H 6 { } 3. Is there a systematic way to construct nontrivial 1-cocycles?

That only few modern research articles on first order group cohomology address Prob- lem 1.2 in the realm of infinite-dimensional Lie groups has to do with the observation that they are not locally compact. So far, quite a lot of techniques to analyze 1-cocycles have been created especially in the setting of locally compact groups (cf. e.g. [BHV08], [Ar69], [PS75],[PS72]). The proofs in [PS75], for example, rely heavily on structure theory of finite-dimensional semisimple Lie groups and on tools such as the Haar measure. Araki uses structure theory of finite-dimensional connected solvable Lie groups in order to derive in [Ar69] a general structure of their 1-cocycles. These methods do not apply to infinite-dimensional Lie groups so that the proofs in [PS75] and [Ar69] do not carry over to the infinite-dimensional setting. Thus, the main challenge for us 5 is to use or develop techniques which also apply to non-locally compact groups. It is instructive to illustrate this by the example of compact groups, which is both elementary and of fundamental importance for our thesis: Let G be a compact group. Then, any continuous 1-cocycle is a 1-coboundary. A direct way to prove this fact uses the existence of a normalized left-invariant Haar measure on G (cf. e.g. [PS72, Thm. 15.1] or [Ar69, Thm. 7.1]). If β : G is a continuous 1-cocycle asociated to the continuous unitary representation (→π, H), then we can define its “mean value ” v R g d g d g H := G β( ) , where denotes integration with respect to the Haar measure. For any g0, we apply the 1-cocycle equation (1) and compute (formally) Z Z Z π(g0)v = π(g0)β(g)d g = β(g0 g)d g β(g0)d g G G − G Z = β(g)d g β(g0) = v β(g0). G − −

This shows that β(g0) = v π(g0)v respectively that β = ∂v. A second way to prove the fact is of geometric nature− and does not make use of the Haar measure:

In order to introduce the second proof, we reformulate Problem 1.1 as a geometric problem. For any topological group G, a continuous 1-cocycle β associated to an orthogonal (or a unitary) representation (π, ) deﬁnes an isometric action A of G on by afﬁne maps H H A(g)v := π(g)v + β(g).

From this point of view, the 1-cocycle β = ∂v is a 1-coboundary if and only if v is a ﬁxed point for the afﬁne action A. Thus, Problem 1.1 can be reformulated as follows:

Problem 1.3. Which afﬁne isometric group actions

A : G Mot( ) = o O( ) [or o U( )] → H ∼ H H H H with continuous orbit maps have a fixed point, i.e. some v for which A(g)v = v holds for all g G? ∈ H ∈ Here, a first answer can be given: The affine action A has a fixed point if and only if the corresponding 1-cocycle β is bounded in the sense that supg G β(g) < (cf. Corollary 2.8). This is a consequence of the so called Bruhat–Tits∈ Fixedk Pointk Theorem∞ (cf. Theorem 2.7): The 1-cocycle β(g) = A(g)0 is the 0-orbit of A and, in case it is bounded, the closed convex hull is also a bounded and G-invariant subset. It has a (unique) “center of mass” which remains fixed under the group action. Now, if G is compact, then the expression β(g) is always bounded if β is continuous. The center k k v R g d g of mass coincides with the mean value = G β( ) . This is the second proof of the fact that the spaces H1 are trivial for compact groups. Note that the assertion of the Bruhat–Tits Fixed Point Theorem holds for arbitrary topological groups. In particular, it does not require the existence of a Haar measure. 6 1 Introduction

The previous discussion has shown that every continuous 1-cocycle β : G of a compact group G must be a 1-coboundary. In general, a topological group G→for H which 1 H (G, π, ) = 0 holds for any continuous unitary representation is called a group with PropertyH (FH){ }. From the geometric point of view, this means the following:

Property (FH): Every affine affine group action A : G Mot O ( ) ∼= o ( ) on a real Hilbert space with continuous orbit maps→ hasH a fixedH pointH (cf. Theorem 2.17). 3 H

We have just seen that compact groups have Property (FH). Compact groups even have the Rosendal’s Property (OB) (cf. [Ro09]).

Property (OB): Every continuous isometric group action on a metric space has bounded orbits. 4

For continuous afﬁne isometric group actions on a Hilbert space, this means that the corresponding 1-cocycle is bounded and thus a 1-coboundary. In particular, every group with Property (OB) is a group with Property (FH). Examples of groups with Property (OB) can already be found among the discrete groups, e.g. the group S N of all bijections N N (cf. [Berg06]). Interesting examples of non-locally compact 5 n Polish groups are homeomorphism→ groups of spheres (Homeo(S )) and of the Hilbert cube (Homeo([0, 1]N)) (cf. [Ro09]). An example relevant to our thesis is the unitary group U( ) of a complex Hilbert space and its subgroup U ( ) consisting of those unitaryH operators g for which the differenceH g 1 is a compact∞ H operator (cf. Proposition 3.29). Hence, there exist inﬁnite-dimensional− Lie groups for which every continuous 1-cocycle is automatically a 1-coboundary.

Among locally compact groups, there is another prominent class of groups with property (FH), namely the so-called groups with Kazhdan’s Property (T):

Property (T): There exists a compact subset K G and some positive constant " > 0 such that every continuous unitary representation⊆ (π, ), for which there exists H a unit vector v with supg K π(g)v v < ", has a non-trivial G-fixed G ∈ H 6 ∈ k − k vector, i.e. = 0 . H 6 { } 3This is the original definition of Property (FH). The acronym (FH) stands for “Fixed points for affine isometric actions on Hilbert spaces”. 4Recall that, if a topological group G acts on a metric space X by isometries, then the corresponding action G X X , (g, x) g.x is continuous if and only if every orbit map g g.x is continuous (for each x ×X ). → 7→ 7→ 5A Polish∈ group is a topological group for which the underlying topological space is a Polish space. In other words, the group topology is separable and completely metrizable 6 [BHV08, Thm. 1.2.5] shows that this definition is equivalent to Kazhdan’s original definition of the trivial representation of G being an isolated point in the unitary dual Gb. 7

Every compact group has Property (T) and every group with Property (T) has Property (FH). The implication (T) = (FH) was shown by Patrick Delorme in 1977 ([Del77]) and holds for arbitrary topological⇒ groups. The converse statement, which is due to Alain Guichardet, only holds for σ-compact locally compact groups. 7 The equivalence of Property (T) and Property (FH) for σ-compact locally compact groups is known as the Delorme–Guichardet Theorem. Finite-dimensional connected semisimple Lie groups are σ-compact locally compact groups and Theorem 3.5.4 in [BHV08] states that such a Lie group has Property (T) if and only if no simple factor of its Lie algebra is isomorpic to su(n, 1) or so(n, 1) (for n N). In particular, this happens for all simple connected Lie groups of real rank at∈ least two, which include e.g. the special linear groups SL(n, R) for n 3 or the special orthogonal groups SO(p, q) for p > q 2 and SO(p, p) for p 3.≥ In view of the Delorme–Guichardet Theorem, all these groups≥ have Property (FH).≥ This classification theorem supplements the results in [PS75] mentioned above. So far, our discussion has shown that there exist many groups with Property (FH), meaning that all their first order cohomology spaces H1 vanish. How do we find groups with the opposite property? The following insight provides a clue: Compact groups may be characterized as those topological groups, which are locally compact, amenable and have Property (T) (cf. [BHV08, Thm. 1.1.6]). Hence, any noncompact, amenable group which is σ-compact and locally compact does not have Prop- n n erty (FH). This applies e.g. to the abelian groups Z, R resp. Z , R for n N. In fact, the class of noncompact amenable groups without Property (FH) contains∈ even non-locally compact groups: The most accessible representatives can be found among (noncompact) abelian groups, direct limits of compact groups and those groups which contain such a direct limit as a dense subgroup. In the realm of irreducible representations, the study of abelian groups however is not much of interest: For abelian groups, every irreducible unitary representation is one-dimensional by Schur’s Lemma so that it may be identified with a character χ : G T. However, if χ = 1 is not the trivial character, then it is a straight forward computation→ to check that6 the corre- 1 1 sponding space H is trivial and, in case χ = 1 is trivial, the space H coincides with the space HomTopGrp(G, C) of continuous additive group homomorphisms (cf. [PS72, Thm. 16.1]). The same results extend to the class of nilpotent groups (cf. [PS72, Thm. 17.4]). This motivates us to investigate irreducible representations of direct limit (Lie) groups.

As already announced, our main focus lies on (Lie) subgroups of the unitary group U , where `2 , is an inﬁnite-dimensional complex separable Hilbert ( ) ∼= (N C) space.H The followingH idea is the basis of all our results: We have mentioned above that the full unitary group U( ) and its subgroup U ( ) both have Property (FH) H ∞ H 7The assumption of the σ-compactness is necessary: This follows from the fact that every locally compact group with Property (T) must be compactly generated (cf. [BHV08, Theorem 1.3.1]). In particular, every discrete group with Property (T) must be ﬁnitely generated, hence can be an at most countable group. But there exist uncountable discrete groups with Property (FH), such as S (also cf. N [BHV08, Remark 2.12.5]). 8 1 Introduction

(see also Theorem 3.30). Suppose that D G := U ( ) is a dense subgroup which carries a natural group topology that is ﬁner⊆ than∞ theH induced group topology. If (π, ) is a continuous unitary representation of G, then its restriction to D yields a continuousH unitary representation of D. Likewise, the restriction of a continuous 1- cocycle of G yields a continuous 1-cocycle of D and it is not hard to verify that the restriction map 1 1 Res : Z (G, π, ) Z (D, π, ), β β D 1H → 1 H 7→ | 1 is injective. Therefore, we have Z (G, π, ) Z (D, π, ). In view of Z (G, π, ) = 1 B (G, π, ), we expect that a suitableH choice⊆ of D yieldsH a topological groupH that admits unboundedH 1-cocycles. Indeed, the inﬁnitesimal setting provides a variety of such subgroups and the hypothesis turns out to be true for the following choices of D: For any p [1, ), we may put D := Up( ) which denotes the subgroup of those unitary operators∈ ∞g for which the differenceHg 1 is an operator of pth Schatten class. − For p0 < p, we have Up ( ) Up( ) so that we obtain a whole chain 0 H ⊂ H U1( ) ... Up( ) ... U ( ) U( ) H ⊂ ⊂ H ⊂ ⊂ ∞ H ⊂ H of dense subgroups of U ( ) and for smaller p, we expect larger 1-cohomology 1 ∞ H spaces H . This is illustrated by the following example: All groups Up( ), including H p = , act unitarily on the Hilbert space of Hilbert–Schmidt operators 2( ) via left multiplication∞ B H

Up( ) 2( ) 2( ), (g, X ) gX . H × B H → B H 7→ In Proposition 5.26, we will show that a continuous 1-cocycle β :Up( ) 2( ) is of the form β(g) = gX X for some bounded linear operator X H( →). If Bp >H2, then X is a qth Schatten class− operator, where q (2, ) is deﬁned∈ B byH the equation 1 1 1 ∈ ∞ p + q = 2 . Writing q( ) for the Banach space of qth Schatten class operators, we have B H

 , 1 p 2  ( ) 1 B H ≤ ≤ Z (Up( ), left multiplication on 2( )) = q( ), p > 2 H B H ∼ B H 2( ), p = . B H ∞ Hence, the quotient spaces H1 increase as the value of p decreases. The relation with direct limit Lie groups comes from the fact that all groups Up( ), including p = , contain the direct limit group U( ) := lim U(n) as a dense subgroup.H The U(n)∞de- ∞ note the compact Lie groups of unitary−→n n matrices embedded into U( ) with respect to a given complete orthonormal system× e of . Following ourH strategy, ( n)n N we obtain U( ) as a natural candidate for a group∈ withH nontrivial ﬁrst order cohomology spaces∞H1.

The main objective of our thesis is to turn this observation into a precise classiﬁcation result: Irreducible unitary representations of the compact groups U(n) have been clas- n siﬁed by the orbits of the weights λ Z under the Weyl group = Sn (cf. [GW98]). ∈ W ∼ 9

This means that every irreducible unitary representation of U(n) defines a unique high- n est weight λ Z and that two highest weight representations (πλ, λ) and (πλ , λ ) ∈ H 0 H 0 are (unitarily) equivalent if and only if the entries of their highest weights λ and λ0 coincide up to permutation. In [Ne98], this classification result has been extended to the direct limit group U( ) = lim U(n). For every infinite tuple λ ZN, one obtains ∞ ∈ a unitary highest weight representation−→ as a direct limit (πλ, λ) := lim(πλ(n) , λ(n) ). H H For each n N, (πλ(n) , λ(n) ) is the unitary highest weight representation−→ of U(n) with ∈ (n) H highest weight λ := (λ1, λ2,..., λn) consisting of the first n entries of λ. It follows from the classical Branching Laws (cf. [GW98, Thm. 8.1.1]) that (πλ(n) , λ(n) ) oc- H curs as a subrepresentation of (πλ(n+1) , λ(n+1) ) with multiplicity one. This justifies the H definition of (πλ, λ) as a direct limit of unitary representations. Two unitary highest H weight representations πλ, λ and πλ , λ are equivalent if and only if their high- ( ) ( 0 0 ) est weights belong to the sameH orbit of theH natural action of the Weyl group S ∼= (N) of finite permutations on ZN (cf. [Ne98], Theorem 6.15). In this thesis, we determineW for which highest weights λ ZN, the corresponding 1-cohomology space is trivial. ∈ 1 We will show in Theorem 7.23 that the space H (U( ), πλ, λ) vanishes only in the following three cases: ∞ H

Either λ = 0, or j λj 0 < , or j λj 0 < . { | ≤ } ∞ { | ≥ } ∞ In particular, for any nonzero λ with only finitely many nonzero entries, the corresponding 1-cohomology space H1 is nontrivial. The core ideas for the proof are the following: Any continuous 1-cocycle β :U( ) has the property that its restriction to the compact subgroup U(n) yields a∞ 1-coboundary.→ H We thus obtain a unique U(n) vector vn in the orthogonal complement of the subspace of U(n)-fixed vectors for which . As we shall see, the 1-cocycle is a 1-coboundaryH if and only if β = ∂vn β the sequence v converges in . From that one deduces a handy necessary and ( n)n N sufficient criterion∈ for the vanishingH of the space H1 for a given continuous unitary representation (π, ) of U( ): The representation (π, ) does not admit a nontrivial H ∞ H 1-cocycle if and only if there exists some n0 N such that every U(n0)-fixed vector is ∈ U(n0) U( ) automatically a U( )-fixed vector. In symbols, we write = ∞ . In case of ∞ H 8 H U( ) a unitary highest weight representation (πλ, λ) with λ = 0 , we have λ ∞ = 0 . H 6 U(n0) H { } In view of the direct limit structure λ = lim λ(n) , the case λ = 0 occurs if U n H H H { } and only if ( 0) 0 for every n n .−→ In other words, for no n n , the trivial λ(n) = 0 0 H { } ≥ ≥ representation of U(n0) occurs as a subrepresentation of the highest weight representation (πλ(n) , λ(n) ). This is a branching problem from U(n) to the subgroup U(n0) and can be solvedH by the classical Branching Laws in [GW98]. The reader should note that we do not construct unbounded 1-cocycles explicitly in the proof.

In a second step, we address the problem of constructing unbounded 1-cocycles. Prob- ably the easiest way to obtain examples is the following: If f : G C is a nonzero continuous additive group homomorphism and v is a nonzero→ G-fixed vector, then ∈ H β : G , g f (g) v 8 → H 7→ · The case λ = 0 corresponds to the trivial representation. 10 1 Introduction defines an unbounded 1-cocycle. This follows from the fact that any continuous additive group homomorphism f is either constant zero or unbounded. Hence, the group G HomTopGrp(G, ) is one source of nontrivial 1-cohomology spaces. However, if G H is a direct limit of compact groups, then we have HomTopGrp(G, C) = 0 and this construction does not apply. In the literature, on finds two particular strategies{ } to ob- G tain nontrivial 1-cocycles that apply particularly well to the case = 0 : First, if G = 0 , then the 1-coboundary map H { } H { } 1 ∂ : Z (G, π, ), v ∂v H → H 7→ 1 is injective with image B (G, π, ). The idea is to extend the representation π to some vector space E containing asH a G-invariant subspace such that the 1-coboundary map ∂ extends to a linear isomorphismH

1 ∂ : E Z (G, π, ), v ∂ v, → H 7→ where ∂ v is the 1-coboundary of the extended representation on E defined by v E. ∈ An instructive example is the left multiplication representation of Up( ) (for 1 p H ≤ ≤ 2) on the Hilbert space of Hilbert–Schmidt operators 2( ), as introduced above. This representation naturally extends to the Banach spaceB H ( ) of bounded linear operators via B H Up( ) ( ) ( ), (g, X ) gX H × B H → B H 7→ and every 1-cocycle β :Up( ) 2( ) has the form of a 1-coboundary of the representation on ( ). SuchH spaces→ BE Hhave also been constructed e.g. in [Ar69] for B H + solvable (finite-dimensional) Lie groups (denoted by D ) and also in [Gui71a] mainly for abelian groups (denoted by L0(G, E)). A second strategy is to obtain nontrivial 1- 1 cocycles by approximation of trivial ones. One way is to endow the space Z (G, π, ) with the topology of uniform convergence on compact subsets of G and to considerH the closure B1(G, π, ) of the subspace of 1-coboundaries. For locally compact groups, it 1 1 was shown in [Gui72bH ] that the space B (G, π, ) is closed in Z (G, π, ) if and only if the unitary representation (π, ) does not almostH have invariant vectors,H i.e. there H exists some " > 0 and some compact subset K G such that maxg K π(g)v v " v holds for all v . 9 Hence, one obtains⊆ unbounded 1-cocycles∈ k whenever− k ≥ thek representationk almost∈ H has invariant vectors. The simplest way of approximating 1-cocycles by 1-coboundaries is to add infinitely many 1-coboundaries β : P ∂ = n N vn ∈ 9That the subspace of 1-coboundaries is not always closed and that its closure is a linear subset of 1 Z (G, π, ) leads to the definition of the reduced 1-cohomology spaces as the quotient vector spaces H 1 H1(G, π, ) := Z (G, π, )/B1(G, π, ). H H H They play an important role because the vanishing of the spaces H1 for locally compact, compactly generated group G is equivalent to the vanishing of the spaces H1. Thus, the vanishing of the reduced 1- cohomology spaces characterizes Kazhdan’s Property (T) resp. Property (FH). This statement is known as Shalom’s Theorem (see e.g. [BHV08, LSV04]). At the same time, the reduced 1-cohomology spaces H1 “behave much better” than the 1-cohomology spaces H1. For instance, the vansihing of the spaces H1 remains stable under direct sum or integral decomposition (see e.g. [BHV08, Lem. 3.2.4]). 11

for suitably chosen vn . This idea is inspired by the classical Lévy–Khintchine Formula, which is a representation∈ H formula for continuous negative definite functions of locally compact abelian groups. As already pointed out, for any topological group G, a continuous 1-cocycle β : G defines a continuous negative definite function 2 ψ(g) := β(g) . The 1-cocycle→ equation H (1) reveals that ψ is normalized (ψ(e) = 0) k k 1 and symmetric (ψ(g− ) = ψ(g)) and one can show that any continuous, normalized, symmetric negative definite function ψ : G R 0 is of that form (cf. [BHV08, Prop. 2.10.2]). From this perspective, we obtain a→ second≥ reformulation of Problem 1.1 in terms of negative definite functions.

Problem 1.4. Which continuous, symmetric, normalized, negative definite functions ψ : G R 0 are unbounded? → ≥ If such a ψ is a bounded function, then it is of the form ψ(g) = ϕ(e) ϕ(g), where ϕ : G R is a continuous positive definite function. Any such function−ϕ is bounded with upper→ bound ϕ(e). A particular example is given by ϕ(g) := Re χ(g) for any continuous group character χ : G T. If G is a locally compact abelian group, then we have → Z ϕ(g) = q(g) + (1 Re χ(g))dν(χ), Gb 1 − \{ } where q : G R 0 is a quadratic form and ν is the Lévy measure on Gb 1 . This is the classical→ Lévy–Khintchine≥ Formula (cf. [FH74]). It shows that there\{ are} two sources of unbounded negative definite functions, namely quadratic forms and Lévy measures. Nonzero quadratic forms are always unbounded whereas the integral is not automatically unbounded: If a Lévy measure does not have “sufficiently mass” in a re- gion around the trivial character χ = 1, then the integral is a bounded function on G. Transferred to the context of 1-cocycles, we expect a continuous 1-cocycle β : G G → H to decompose into β = βf ix + βsum, where βf ix : G is an additive group homomorphism into the G-fixed vectors of and βsum→should H be any kind of sum of 1-coboundaries. The part βf ix is simply obtainedH by projecting β onto the subspace G of G-fixed vectors and yields an unbounded 1-cocycle provided that it is nonzero.

TheH interesting part is therefore βsum.

There are unitary representations of a topological group G whose 1-cocycles are defined by an infinite sum of 1-coboundaries. The most natural example is the following: 1 Suppose that (π, ) is a (continuous, unitary) representation with H (G, π, ) = 0 . We obtain a (continuous,H unitary) representation π on H { } ∞ · M 2 := d = ` (N, ) n N ∼ ∞ · H ∈ H H by taking (countably) infinitely many copies of π and buildung the direct sum. Pro- jecting any continuous 1-cocycle β : G onto the nth copy of yields an element Z 1 G, , and thus,→ by ∞ assumption, · H a 1-coboundary H βn ( π ) βn = ∂vn with v .∈ Hence, β decomposesH into the sum β P ∂ . We may view n = n N vn ∈ H ∈ 12 1 Introduction

`2 , `2 , as a tensor product of Hilbert spaces and the above repre- (N ) ∼= b (N C) sentationH ( H π⊗, ) can be considered as a tensor product of the representation 2 (π, ) with∞ the · trivial∞ · H representation of G on ` (N, C). More generally, finite tensor productH representations of G with two or more factors also admit 1-cocycles that are defined by infinite sums of 1-coboundaries and which are promising candidates for unbounded 1-cocycles (cf. Lemma 8.8). The problem here is to decide whether such an infinite sum of bounded 1-cocycles actually defines an unbounded 1-cocycle. In our thesis, we specialize to the following situation: We consider the unitary subgroup G U for 1 p < and where `2 , is some complex separable = p( ) ∼= (N C) Hilbert space.H Then≤G acts∞ on via the identicalH representation H G , (g, v) gv. × H → H 7→ We will show in Section 5.3 that the identical representation of G on does not admit nontrivial 1-cocycles (cf. Proposition 5.24). However, the unitaryH representation bn on the n-fold (n 2) Hilbert tensor product ⊗ , defined by π(g)v1 vn := gv1 gvn, does≥ admit unbounded 1-cocycles.H We construct some of⊗ them · · · ⊗ explicitly⊗ as · infinite· · ⊗ sums of 1-coboundaries. Yet, it is still an open problem to determine bn all unbounded 1-cocycles β :Up( ) ⊗ . In our second main result, we link this observation to unitary highest weightH → representations: H

A unitary highest weight representation (πλ, λ) of U( ) extends to the group U1( ) H ∞ H if and only if the weight λ is bounded, i.e. λ `∞(N, Z) (cf. Proposition III.7 and Theorem III.4 in [Ne98]). Moreover, the proof∈ of Proposition III.10 in [Ne98] shows that (πλ, λ) extends to the group Up( ) for p (1, ] if and only if the weight λ has finiteH support, which is to say it hasH only finitely∈ many∞ nonzero entries. Again, the problem is to determine for which weights λ, the corresponding first cohomology spaces are trivial. For p = , we already know that all 1-cohomology spaces vanish so that we assume that p∞ [1, ). Suppose that the highest weight λ has finite support and that all entries λn∈ 0∞ are nonnegative. Then, the corresponding highest weight space occurs as a U≥ -invariant subspace of bn for n : P λ . λ p( ) ⊗ = i N i N H H H bn ∈ ∈ If n 2, then we construct unbounded 1-cocycles β :Up( ) ⊗ which remain ≥ H → H unbounded after projecting them onto the subspace λ. Therefore, the correspond- 1 H ing 1-cohomology spaces H (λ) are nontrivial. The dual representation (πλ∗ , λ∗) is also a unitary highest weight representation. It corresponds to the highest weightH λ. 1 1 There is a natural antilinear isomorphism between the spaces H (λ) and H ( λ) −and 1 we conclude that also H ( λ) is nontrivial. If λ ZN has finite support,− then we + − ∈ can split λ = λ λ− into its positive and negative part. If both parts are nonzero, − then (πλ, λ) may be identified with the tensor product (πλ+ , λ+ )b(πλ∗ , λ∗ ). Also in this case,H we construct unbounded 1-cocycles. To sum up,H if λ has⊗ finite− H support,− 1 then the corresponding 1-cohomology space H (λ) is nontrivial except in the following three cases: The trivial representation of Up( ) (which corresponds to the case H λi = 0 for all i N), the identical representation of Up( ) on (which corresponds ∈ H H to λi = δi,j for all i and some fixed j N) and the dual representation of Up( ) on ∈ H ∗ (which corresponds to λi = δi,j for all i and some fixed j N). Moreover, in H − ∈ 13

1 case H (λ) = 0 , we construct a whole sequence of unbounded 1-cocycles and we 1 will show that6 { the} spaces H (λ) are inﬁnite-dimensional.

In a last step, we treat the case where λ is bounded with infinite support and where the corresponding highest weight representation (πλ, λ) of U( ) extends only to the H ∞ group U1( ). At first sight, the problem here is that the space λ occurs as a subspace of an “infiniteH tensor product” of and it is much harder to realizeH it concretely than it is in the finitely supported case.H The more surprising it is that one does not need any concrete realization for the construction of unbounded 1-cocycles: For bounded and infinitely supported λ, the corresponding highest weight representation (πλ, λ) 1 H of U1( ) does not admit nontrivial 1-cocycles and we have H (λ) = 0 . The idea H { } is to study the action of the “maximal torus” T1 of U1( ) which is defined as the subgroup of diagonal operators w.r.t. a fixed complete orthonormalH system of . The H abelian group T1 decomposes the highest weight space λ into (one-dimensional) H weight spaces. We thus obtain a sequence of characters χ : T1 T. The fact that λ is infinitely supported causes the characters χ to be at a “great distance”→ from the trivial character. Similar to the Lévy–Khintchine formula, this leads to the boundedness of the corresponding 1-cocycles of the abelian group T1 and also of the group U1( ). In 1 particular, we find H (λ) = 0 . H { }

Outline of our thesis

The structure of our thesis is as follows: Section2 contains various preliminaries. In a first step, the reader is introduced to the fundamental terminology used in our work. In particular, this terminology comprises the notion of a continuous 1-cocycle and that of the first order cohomology spaces. In a second step, we focus on the interplay between 1-cocycles β : G , affine isometric group actions A : G Mot( ) and negative definite functions→ψ H: G R. We shall recapitulate the well-known→ Bruhat–TitsH Fixed Point Theorem (Theorem→ 2.7), the representation of negative definite functions in terms of 1-cocycles (Theorem 2.11) and the construction of unitary representations

πt on the corresponding Fock Hilbert space t for t > 0 from the afﬁne isometric action A. In a ﬁnal step, we present a generalizedH version of the famous Delorme– Guichardet Theorem. We introduce the notion of Generalized Property (T) and prove that it is equivalent to Property (FH) (cf. Theorem 2.17). This result contains the classical Delorme–Guichardet Theorem as a special case (cf. Corollary 2.22).

Section3 is devoted to the study of the topological behaviour of continuous 1-cocycles β : G . Subsection 3.1 treats the so-called Automatic Continuity Problem. This is the question,→ H for which groups and unitary representations, a given 1-cocycle β : G is automatically continuous. Here, we assume that lim is the pointwise→ β = n ∂vn limitH of a sequence of continous 1-coboundaries. If G is a Baire→∞ group (e.g. a completely metrizable or a locally compact group), then β is automatically continuous (cf. Proposition 3.9). This result will be used in the later course of this thesis. In Appendix 14 1 Introduction

A, we treat the Automatic Continuity Problem in a more general way and see that, provided that G is a Baire group, then every weakly continuous or locally bounded 1-cocycle is automatically continuous (Proposition A.7). This continuity criterion is often quite handy. In Subsection 3.2, we study two particular boundedness concepts for topoplogical groups which both imply Property (FH). The first one is Rosendal’s Property (OB) in the sense of [Ro13] and the second one is that of a bounded topological group which goes back to the work of Hejcman ([Hei59]) and Atkins ([At91]). For connected topological groups, these two concepts coincide and a connected Banach– n Lie group G is bounded if and only if if G = exp(Br ) for some integer n N and some closed ball Br with radius r > 0 around the origin of the Lie algebra∈ g of G (Proposition 3.25). From that we derive that the unitary groups U( ) and U ( ) are bounded and thus have Property (FH) (Theorem 3.30). Some technicalH details∞ H of this subsection are carried out in AppendixB, where we study subsets of topological groups “with Rosendals Property (OB)” that we call coarsely bounded subsets. Finally, in Subsection 3.3, we turn to the problem that a bounded 1-cocycle β : G E with values in a Banach space E need not be automatically a 1-coboundary. That→ this is true for Hilbert spaces follows from the Bruhat– Tits Fixed Point Theorem which does not apply in arbitrary Banach spaces. Yet, it generalizes to reflexive Banach spaces (Theorem 3.34). If E is only a dual Banach space and if G is amenable, then we will see that Theorem 3.38 substitutes the Bruhat–Tits Fixed Point Theorem. In this case, every continuous group action of G on E by weak- -continuous affine isometries and with bounded orbits has a fixed point. ∗

In Section4, we treat our initial problem, i.e. Problem 1.1, from a Lie theoretic perspective: If the topological group G is a Lie group and if the corresponding representation (π, ) is “sufficiently smooth”, then it makes sense to ask whether the un- boundednessH of a “sufficiently smooth” 1-cocycle β : G can be seen on the Lie algebra level. To make things more precise, we assume that→ HG is a connected Banach– Lie group and that (π, ) is a norm-continuous unitary representation of G. By the Automatic SmoothnessH Theorem, the representation is then automatically smooth in the sense that π : G U( ) is a morphism of Banach–Lie groups. Likewise, every continuous 1-cocycle→ β H: G is automatically smooth (Lemma 4.30) and its d derivative dβ X : β exp→t HX , for X g : L G , is a Lie algebra 1-cocycle. ( ) = d t t=0 ( ( )) = ( ) Therefore, this observation links the· 1-cohomology∈ of Lie groups to the 1-cohomology of Lie algebras. If (ρ, ) is a representation of a Lie algebra g on a Hilbert space , then a map ω : g His called a Lie algebra 1-cocycle if H → H ω([X , Y ]) = ρ(X )ω(Y ) ρ(Y )ω(X ) for all X , Y g − ∈ and ω is a Lie algebra 1-coboundary if it is of the form ω(X ) = ρ(X )v for some v . 1 One denotes the vector space of Lie algebra 1-cocycles as Z (−g, ρ, ), its subspace∈ H of 1 Lie algebra 1-cocboundaries as B (g, ρ, ) and defines the Lie algebraH 1-cohomology space as the quotient space H

1 1 1 H (g, ρ, ) := Z (g, ρ, )/B (g, ρ, ). H H H 15

d In our setting ρ X : π exp t X is the derivative of π and the 1-cocycle β ( ) = d t t=0 ( ( )) is a 1-coboundary if and only if its derivative· dβ is a Lie algebra 1-coboundary. The key observation (Proposition 4.34) is that taking the derivative yields a linear map

1 1 d : Z (G, π, ) Z (g, dπ, ), β dβ, H → H 7→ which is an isomorphism whenever G is simply-connected or whenever the subspace G G of G-invariant vectors is trivial ( = 0 ). This means that, in these two cases, continuousH Lie algebra 1-cocycles alsoH integrate{ } to continuous group 1-cocycles, so that 1-cohomology on the Lie group level is the same as 1-cohomology on the Lie algebra level. This key observation is used in the following sections of our thesis. The rather strong assumption of norm-continuity of the representation π is justified by the fact that the unitary highest weight representations of the unitary groups Up( ) are norm-continuous. It is also worth noting that the corresponding general linearH 1 group GLp( ) := GL( ) ( + p( )) is the universal complexification of the H H ∩ 1B H unitary group Up( ) = U( ) ( + p( )) and that the unitary highest weight H H ∩ B H representations of the groups Up( ) extend to GLp( ). We will show that, in general, the 1-cohomology spaces H1Hremain stable underH universal complexification of Banach–Lie groups resp. Banach–Lie algebras (cf. Proposition 4.40 resp. Proposi- 1 tion 4.37). Therefore, our results on the 1-cohomology spaces H of Up( ) extend H to the group GLp( ). All these points are carried out in Subsection 4.2. Subsection 4.1 intends to informH the reader about recent trends and developments in Lie theory, based on [GlNe18, Ne06, Gloe05], which are relevant for our thesis. It provides some background material on the Lie group structure of the unitary groups occuring in this thesis. They all belong to the class of locally exponential Lie groups and arise as subgroups of the unit group of a Mackey -complete -algebra. This background chapter is continued in Subsection 5.2 of the next section,∗ after the introduction of the Schatten class operators in Subsection 5.1.

In Section5, we determine explicitly all continuous 1-cocycles corresponding to three particular unitary representations of the unitary groups Up( ) for 1 p < . These representations are H ≤ ∞

1.) π1 :Up( ) , (g, v) gv (the identical representation), H × H → H 7→ 2.) π2 :Up( ) 2( ) 2( ), (g, A) gA (the left multiplication representation), H × B H → B H 7→

1 3.) π3 :Up( ) 2( ) 2( ), (g, A) gAg− (the conjugation representation). H × B H → B H 7→

The identical and the conjugate representations are unitary highest weight representations of Up( ). The representation π1 corresponds to the highest weight λ = H (1, 0, 0, 0, . . .) and π3 corresponds to λ = (1, 1, 0, 0, 0, 0, . . .). The construction of their continuous 1-cocycles relies on ideal properties− of the pth Schatten class operators. For this reason, we introduce the Schatten class operators in Subsection 5.1 and 16 1 Introduction recapitulate their fundamental properties. We will also make use of the structure of the Lie group Up( ) and of its complexiﬁed Lie algebra glp( ) = p( ) and we provide the necessaryH details in Subsection 5.2. The main ideaH for theB determinationH of the 1-cocycles corresponding to the above representations is to exploit the results from the previous section and to determine the Lie algebra 1-cocycles of the respective derived representations:

1.) ρ1 : glp( ) , (X , v) X v, H × H → H 7→ 2.) ρ2 : glp( ) 2( ) 2( ), (X , A) XA , H × B H → B H 7→ 3.) ρ3 : glp( ) 2( ) 2( ), (X , A) XA AX = [X , A]. H × B H → B H 7→ −

The interesting point here is to see that glp( ) = p( ) carries the structure of a H B H Banach-algebra and that we obtain on 2( ) the structure of a p( )-bimodule via B H B H left and right multiplication, by virtue of the ideal structure of 2( ). This enables us to relate the problem to the ﬁrst order cohomology of operatorB algebras,H where one distinguishes between inner and outer derivations of Banach bimodules instead of trivial and nontrivial 1-cocycles: Suppose that ω3 : glp( ) 2( ) is a continuous Lie algebra 1-cocycle corresponding to ρ3. Then it satisﬁesH → the B relationH

ω3([A, B]) = [ω3(A), B] + [A, ω3(B)] for all A, B glp( ), ∈ H where [A, B] = AB BA is the Lie bracket of glp( ) = p( ). This means that ω3 − H B H is a Lie derivation of the Banach bimodule 2( ). We will show (cf. Lemma 5.17) B H that ω3 is already a derivation of 2( ), i.e. that it satisfies the product rule B H ω3(AB) = ω3(A)B + Aω3(B) for all A, B glp( ). ∈ H Therefore, the problem reduces to finding all derivations of glp( ) on 2( ). In Subsection 5.3, we generalize this observation in order to apply thisH techniqueB H to all three cases simultaneously. We find that the Lie algebra 1-cocycles have the following form:

1.) ω1(X ) = X v for some v . In particular, every ω1 is a Lie algebra 1-coboundary. ∈ H 2.) ω2(X ) = XB for some B ( ), if p 2, and B ∈ B H ≤ ∈ B 3.) ω3(X ) = [X , B] for some B ( ). ∈ B H 1 1 1 If p > 2 in the cases 2.) and 3.), we have B q( ) with q defined by p + q = 2 . ∈ B H Section6 opens the second part of our thesis that is devoted to the study of 1-cocycles of unitary highest weight representations (abbr.: HWR). Subsection 6.1 summarizes the basic theory of unitary HWR of involutive split Lie algebras and addresses to the reader who is unfamiliar with the concept of a highest weight representation. In Sub- section 6.2, we focus on the involutive split Lie algebra gl : S gl n , where ( ) = n N ( ) ∞ ∈ 17 gl(n) can be identified with the Lie algebra of complex n n-matrices. The main result here is Theorem 6.21. It determines whether a given unitary× HWR of gl( ) integrates to a representation of its Lie group GL( ). To each such unitary HWR∞ one associates a highest weight λ RN and the representation∞ integrates to GL( ) if and only if the corresponding weight∈ λ ZN is integer-valued. Such an integrated∞ representation is then called a unitary HWR∈ of GL( ) and its restriction to the subgroup U( ) is called a HWR of U( ). Moreover, Theorem∞ 6.21 states that a unitary HWR of U∞( ) ∞ 2 ∞ extends to the Banach completion U1(` ) if and only if λ `∞(N, Z) is bounded and it 2 ∈ extends to the group Up(` ) for p > 1 if and only if the highest weight λ is finitely supported. The details of the proof are rather technical and are carried out in Appendix C in the context of general weight representations. Finally, Subsection 6.3 deals with direct limit representations. We call (ρ, V ) a direct limit representation of gl( ), if V S V is an increasing union of gl n -invariant subspaces V . As shown∞ in = n N n ( ) n Proposition∈ 6.29, every unitary HWR of gl( ) is a direct limit representation. It then follows from Proposition 6.31 that also each∞ unitary HWR of U( ) is a direct limit representation (cf. Remark 7.14) and this will be used in Section7∞.

In Section7, we turn to study of 1-cocycles of direct limit groups. We call a topological group G a direct limit group if G S G is the union of a strictly increasing = n N n ∈ sequence of topological subgroups Gn such that the group topology on G coincides with the direct limit topology defined by the Gn. Here, interesting 1-cocycles β are those for which the restriction to each subgroup Gn yields a 1-coboundary. We call these 1-cocycles conditional and derive in Subsection 7.1 their general structure (Proposi- tion 7.5). In particular, each conditional 1-cocycle arises as the pointwise limit of some sequence of 1-coboundaries. In Subsection 7.2, we apply these results to the unitary highest weight representations π , of the unitary group U S U n , ( λ λ) ( ) = n N ( ) which is a direct limit of the compactH finite-dimensional unitary∞ groups U(∈n). We derive our first main theorem (Theorem 7.23) which determines, for which highest 1 weights λ, the corresponding 1-cohomology space H (U( ), πλ, λ) is trivial. ∞ H In Section8, we collect our results concerning the highest weight representations 2 (πλ, λ) of the Banach-completions Up(` ) with finitely supported highest weights. H In Subsection 8.1, we realize the highest weight Hilbert space λ as a subspace of a 2 n 2 n ∗ H tensor product of the form (` )⊗ (`− )⊗ for suitable positive integers n, n∗ N. This is based on the results in [GW98⊗ ] for highest weight representations of the finite-∈ dimensional unitary groups U(n) and on the results in [Ol90] for the construction of irreducible unitary representation of U( ). In Subsection 8.2, we describe a way to construct unbounded 1-cocycles in a finite∞ tensor product representation (π, ) of some Baire group G. These are 1-cocycles of the form H X β(g) = an[π(g)en en], n N − ∈ where e denotes a suitable orthonormal system in and a a sequence of ( n)n N ( n)n N complex numbers∈ that make the sum converge for everyH g G. In∈ Subsection 8.3, ∈ 18 1 Introduction this enables us to construct unbounded 1-cocycles for unitary highest weight representations π , of the Baire group U `2 for λ : P λ 2. The spaces ( λ λ) p( ) = n N n 1 2 H | | ∈ | | ≥ H (Up(` ), πλ, λ) vanish if λ < 2 and we obtain our second main theorem (Theo- rem 8.13). H | |

Finally, we go a step further and study the 1-cohomology spaces of general norm- 2 continous unitary representations of the group Up(` ) in Section9. Here, the case p > 1 is easy to handle because every norm-continuous unitary representation of 2 Up(` ) decomposes into a (discrete) direct sum of highest weight representations. This has been observed in [Ne98, Thm. III.14]. From our second main result from the previous section, we thus derive that the vanishing of the ﬁrst cohomology spaces 2 of norm-continuous unitary representations of Up(` ) is rather rare. The only such representations are listed up in Theorem 9.17. The case p = 1 is much more com- 2 plicated, since arbitrary norm-continuous unitary representations of U1(` ) in general do not discretely decompose into highest weight representations. For an example, 2 we refer the reader to the discussion of a type II1 representation of U1(` ) in [Ne98, Ex. IV.9] based on [Vo76]. Therefore, we restrict ourselves to the case of a norm- 2 continuous unitary weight representation (π, ) of U1(` ). This means that the sub- H 2 group T1 of diagonal operators (the “maximal torus”) in U1(` ) discretely decomposes 2 into one-dimensional eigenspaces. Thus, the weight representations of U1(` ) con- tainH all highest weight representations as a special case. Since the Lie derivative of a 2 norm-continuous weight representation of U1(` ) is a weight representation of the Lie algebra u `2 with respect to the maximal Lie algebra torus h : t `1 , , we 1( ) = 1 ∼= (N R) may apply our results of Section4 and transfer our studies on the Lie algebra level. In Subsection 9.1, we observe that for any weight representations ρ : g End(V ) of a Lie algebra g with maxial torus h g and with root decomposition, the→ restriction map ⊆ 1 1 Res : Z (g, ρ, V ) Z (h, ρ h, V ), ω ω h → | 7→ | is injective whenever no root of the Lie algebra g occurs as weight of the representation 1 1 (ρ, V ) (Lemma 9.1). If this is the case, then H (h, ρ h, V ) = 0 implies H (g, ρ, V ) = | { } 1 0 . This is our motivation to study the vanishing of the spaces H (h, ρ h, V ) in some { } | more detail in Subsection 9.2. For of certain weight representations of h = t1 = 1 ∼ ` (N, R), we obtain a necessary and sufﬁcient criterion for the vanishing of the spaces H1 (Proposition 9.9) and this result is applied in Subsection 9.3, where we observe 2 that, for any norm-continuous unitary representation (π, ) of U1(` ), the following implication holds (Proposition 9.13): H

H1 T , , 0 H1 U 2 , , 0 . (2) ( 1 π T1 ) = = ( 1(` ) π ) = | H { } ⇒ H { } 2 This insight should not be too much of a surprise since every element in U1(` ) is conjugate to an element of T1. In Subsection 9.4, we specialize to those unitary weight representations (π, ) that are generated by a weight vector corresponding to an inﬁnitely supported weight.H In this case, the above implication reveals that 1 2 H (U1(` ), π, ) = 0 (Theorem 9.19). In particular, this applies to every unitary H { } 2 highest weight representation (πλ, λ) of U1(` ) with an inﬁnitely supported bounded H 19 highest weight λ. This completes our list of unitary highest weight representations of 2 1 the unitary groups Up(` ) with vanishing 1-cohomolgy spaces H .

However, what is left open is the case of arbitrary norm-continuous unitary representa- 2 2 tions of U1(` ). In view of (2), one strategy to find more representations on U1(` ) with vanishing 1-cohomology spaces is to restrict them to the abelian subgroup T1. In the general case, a restriction to T1 does not necessarily lead to a discrete decomposition into one-dimensional weight spaces but very often to “continuous” decompositions. Here, the right tool to cope with these decompositions is a spectral measure. This leads to the study of the spaces H1 for unitary representations of abelian groups that are defined by some spectral measure. In case of a cyclic representation, such a repre- 2 sentation has a canonical representation on the space L (Gb, µ), where Gb is the unitary dual of the abelian topological group G (Bochner’s Theorem). For σ-compact locally compact abelian groups this problem has already been solved in [FV14], where a necessary and sufficient criterion for the vanishing of H1 is given in terms of a spectral condition on the respective measure µ. A second open problem is the case of arbitrary 2 continuous unitary representations of Up(` ) (for 1 p < ). These representations need not be smooth any more, so that one cannot≤ automatically∞ apply the Lie theoretic tools of our thesis. In a subsequent project, it might be interesting to study Problem 1.1 for the following representations: The factor representations of U( ) which are of type In or II1 have been classified in terms of their corresponding charac-∞ ters χ in [Vo76, Thm. 1 in §4] and each such representation extends to a continuous 2 unitary representation of the group U1(` ) ([Vo76, Thm. 3 in §4]). It would be interesting to know for which characters χ, the corresponding 1-cohomology space H1 vanishes. It seems that the methods developed in this thesis, in the realm of (highest) weight representations, are not entirely appropriate to address factor representations.

Some results of this PhD thesis have been summarized and published in the preprint [NeH16]: The content of Sections 6 and 7 in [NeH16] appears here in Section8. Our Section7 in this thesis is based on Sections 3 and 4 of [NeH16]. 20 2 Preliminary facts on 1-cocycles

2 Preliminary facts on 1-cocycles

In the first part of this section we introduce the basic concepts of our work and explain the underlying terminology. In the second part, we relate the notions of a 1-cocycle β : G to that of an affine isometric group action A : G Mot( ) and to that of a negative→ H definite function ψ : G R. Every 1-cocycle occurs→ asH the zero orbit of an affine isometric action β(g) = →A(g)0 and β it is a 1-coboundary if and only if the corresponding action A has a fixed point in . Moreover, the squared norm 2 ψ(g) := β(g) is a negative definite function withHψ(e) = 0 and, conversely, every such negativek definitek function is of that form. The 1-cocycle β is a 1-coboundary if and only if the associated negative definite function ψ is bounded. In the last part, we present the well-known Delorme–Guichardet Theorem in a generalized version. In its original version it states the equivalence of Property (T) and Property (FH) for σ-compact locally compact groups. Property (T) and Property (FH) are defined for arbitrary topological groups with Property (T) being stronger than Property (FH), but they are not always equivalent. Therefore, we generalize the notion of Property (T) in a natural fashion and it turns out that this Generalized Property (T) is equivalent to Property (FH) for all topological groups (Theorem 2.17). This observation is then used to obtain another characterization of Property (FH) (Proposition 2.21), but which only holds for a restricted class of topological groups including connected Banach– Lie groups, direct limits of compact groups and σ-compact locally compact groups. These groups have Property (FH) if and only if every 1-cocycle that arises as a certain infinite sum of 1-coboundaries must be itself a 1-coboundary. Conversely, whenever such a group does not have Property (FH), this motivates us to construct unbounded 1-cocycles as infinite sums of 1-coboundaries as done later in Section8.

2.1 Basic deﬁnitions

We denote by G a group, by g a Lie algebra over K = R, C with Lie bracket [ , ] and by V a vector space over K. If the Lie bracket is identically zero, then the Lie· algebra· g is called abelian, so that the abelian Lie algebras are precisely the vector spaces. We write End(V ) for the vector space of linear endomorphisms on V . The natural composite of two linear endomorphisms yields an algebra structure on End(V ) with unit 1 : V V, v v. Its group of invertible elements is denoted by GL(V ). If V is a normed→ vector7→ space with norm , then we endow V with the natural norm- topology and a linear endomorphism A :k·kV V is continuous if and only if its operator → norm A := sup v 1 Av < is ﬁnite. We write (V ) for the linear space of continuousk k or boundedk k≤ klineark endomorphisms∞ and noteB that the operator-norm turns (V ) into a normed vector space. If V is a Banach-space, then so is (V ). If V is Ba complex vector space, then we write V for the corresponding complexB conjugate vector space, endowed with the same additive group strcture but with the antilinear scalar multiplication.

Deﬁnition 2.1. i)A group topology on G is a Hausdorff–topology for which the 2.1 Basic deﬁnitions 21

group multiplication G G G and the group inversion G G are continuous maps. A vector topology×on V→is a Hausdorff–topology for which→ the vector addition V V V and scalar multiplication K V V are continuous maps. A Lie algebra× topology→ is a vector topology on g for× which→ the Lie bracket [ , ] : g g g is a continuous (bilinear) map. · · × →

ii) A group G is called a topological group if it is endowed with a group topology. Accordingly, a Lie algebra (a vector space) g (V ) is called a topological Lie algebra (a topological vector space) if it is endowed with a Lie algebra topology (a vector topology). iii)A representation of a topological group G on a topological vector space V is a group homomorphism π : G GL(V ). For every v V , the corresponding orbit map is given by G V, g π→(g)v. A linear subspace∈W V is called G-invariant or π-invariant or simply→ invariant7→ if π(G)W W. We usually⊆ denote a representation by π or by (π, V ).A representation of⊆ a topological Lie algebra g on V is a homomorphism of Lie algebras ρ : g End(V ). The deﬁnition of an orbit map and of an invariant subspace is analogous→ to the group case. We usually denote a Lie algebra representation by ρ or by (ρ, V ). iv) A representation π : G GL(V ) is continuous if all orbit maps are continuous. If V is a normed space→ with norm , then the representation is called norm- continuous if every operator π(g) k·k(V ) is a continuous linear endomorphism such that the map π : G GL(V∈) Bis continuous w.r.t. the norm-topology on (V ). Note that norm-continuous→ representations are continuous. The deﬁnition ofB a continuous resp. norm-continuous representation of a topological Lie algebra is analogous.

v) More generally, a topological group G is said to act continuously on a topological space X if the corresponing action map G X X , (g, x) g.x is continuous. In this case, we call the group action continuous× →. If (X , d) 7→is a metric space and if G acts on X by isometries, then the action is continuous if and only if all orbit maps G X , g g.x are continuous. → 7→ vi) A continuous representation π : G End(V ) resp. ρ : X End(V ) is called irreducible if 0 and V are the only→ closed (!) invariant subspaces.→ If F V is a ﬁnite subset{ } of V , then the invariant subspace generated by F is the smallest⊂ closed(!) invariant subspace containing F 10. vii) Let π : G End(V ) be a continuous representation. A 1-cocycle is a map β : G V satisfying→ the 1-cocycle equation →

β(g g0) = β(g) + π(g)β(g0) for all g, g0 G. ∈ A 1-cocycle β is called a 1-coboundary if it is of the form β(g) = v π(g)v for − some v V . We then write β = ∂v. The vector space of all continuous (!) 1- ∈ 10which is to say the intersection of all closed invariant subspaces. 22 2 Preliminary facts on 1-cocycles

1 1 11 cocycles is denoted by Z or, more precisely, by Z (G, π, V ). The subspace of 1-coboundaries is denoted by B1. Note that every 1-coboundary is a continuous 1-cocycle since the representation is continuous, so that we have B1 Z 1. The 1 1 1 quotient space H := Z /B is called the 1-cohomology space of the representation.⊆ viii) Let ρ : g End(V ) be a continuous Lie algebra representation. A Lie algebra 1-cocycle is→ a map ω : g V satisfying the equation → ω([X , X 0]) = ρ(X )ω(X 0) ρ(X 0)ω(X ) for all X , X 0 g. − ∈ A Lie algebra 1-cocycle β is called a Lie algebra 1-coboundary if it is of the form ω(X ) = ρ(X )v for some v V . As in the group case, we then write ω = ∂v and the vector− space of all continuous∈ (!) Lie algebra 1-cocycles is denoted by Z 1. The spaces B1 and H1 are deﬁned in the same way as in the group case.

We ﬁx the rule that, if no topology is speciﬁed on G, g or V , one endows all objects with the discrete topology respectively. This is what we call the algebraic context.

In our context, the term direct limit group is used in the following sense: Definition 2.2. i) We call a topological group G a direct limit group if there exists a strictly increasing sequence of topological subgroups [ G1 G2 ... Gn G = Gn ⊂ ⊂ ⊂ ⊂ n N ∈ whose union is G and such that the group topology of G coincides with the direct 12 limit topology on G. In this case we write G = lim Gn. ii) Likewise, we define a direct limit Lie algebra and−→ a direct limit vector space and write g = lim gn respectively V = lim Vn. −→ −→ iii) If G is a Lie group and all Gn are Lie subgroups such that the Lie group topology of G coincides with the direct limit topology, then we call G a direct limit Lie group 13 and write G = lim Gn. −→ From a categorical point of view, direct or inductive limits are, roughly speaking, objects that are defined by a direct or inductive system via a universal property. Since our focus lies on recognizing explicitly given groups of operators as direct limit groups, and not on the construction of such, we prefer working with the elementary definition above.

11 1 1 Occasionally, we will also use the notations Z (G, V ), Z (G, π) etc. In general, one 1 writesZ ([list of parameters]), where the parameter list specifies the corresponding representation and depends on the context. 12 The direct limit topology is the final topology on G for which all inclusion maps Gn , G are continuous. In particular, a map f : G X into a topological space X is continuous if and only→ if all restrictions f : G X are continuous.→ Gn n 13Beware not| to confuse→ our concept with the concept of a direct limit in the category of Lie groups, as used in the work of H. Glöckner and K–H. Neeb (see e.g. [Gloe05? , Ne06]). 2.1 Basic definitions 23

Unitary representations

Definition 2.3. i) We call a positive definite, nondegenerate Hermitian form , : V V K an inner product on V . We will stick to the convention that 〈·an·〉 inner× product→ is linear in the first argument and antilinear in the second. An p inner product defines a norm on V via v := v, v , thus turning V into a normed linear space. k k 〈 〉

ii) An involution on a (topological) Lie algebra is a (continuous) antilinear14 involutive antihomomorphism∗ : g g. In particular, this means that we have ∗ → [X1, X2]∗ = [X2∗, X1∗] and (X ∗)∗ for all X , X1, X2 g. ∈ iii) A Lie algebra representation ρ : g End(V ) is called unitary if there exists an involution on g and a scalar product→ , on V such that ∗ 〈· ·〉 ρ(X )v, v0 = v, ρ(X ∗)v0 holds for all v, v0 V and X g. 〈 〉 〈 〉 ∈ ∈ iv) A representation π : G End(V ) is called unitary if there exists a scalar product , on V such that → 〈· ·〉 1 π(g)v, v0 = v, π(g− )v0 holds for all v, v0 V and g G. 〈 〉 〈 〉 ∈ ∈ v) A vector space V with some inner product is called a Pre-Hilbert space or an inner product space. Any pre-Hilbert space occurs as a dense subspace of unique (up to isometric isomorphisms) Hilbert space such that the inner product of extends the inner product of V . We writeH = V and call the Hilbert spaceH completion of V . Any continuous linear endomorphismH A : V H V on V uniquely extends to a unique continuous endomorphism A : on→ with the same operator-norm value. In particular, representationsHπ : →G H (VH), ρ : g (V ) on a pre-Hilbert space V by bounded operators uniquely extend→ B to representations→ B π : G ( ), ρ : g ( ) on the Hilbert space completion by bounded operators.→ B H → B H H

We will basically work with continuous 1-cocycles in Hilbert spaces or Banach spaces. Our standard notation for Hilbert spaces is . We only consider real and complex Hilbert spaces. In the following, we assumeH that (π, ) is a continuous unitary representation of the topological group G. The followingH elementary observations will be used:

Assume that (π0, 0) is another continuous unitary representation of G and that H A : 0 is a continuous linear (antilinear) operator intertwining the represen- H → H tations π and π0. This means that π0(g)A = Aπ(g) holds for all g G. We obtain a canonical linear (antilinear) map ∈

1 1 A : Z (π, ) Z (π0, 0), β Aβ, H → H 7→ 14 If K = R, then antilinear is the same as linear. 24 2 Preliminary facts on 1-cocycles which maps 1-coboundaries to 1-coboundaries and thus induces a linear (antilinear) 1 map [A] on the quotient spaces H such that the following diagram commutes

1 A 1 Z (π, ) Z (π0, 0) H H

1 [A] 1 H (π, ) H (π0, 0). H H 1 1 If A is a (anti-) linear isomorphism, then also the spaces Z (π, ) and Z (π0, 0) are (anti-) linearly isomorphic. The same holds for the H1quotientH spaces. NoteH that the commutant of the representation (π, ) H

π(G)0 := A ( ) : Aπ(g) = π(g)A for all g G , { ∈ B H ∈ } 1 1 which is a von-Neumann algebra in ( ), acts naturally on Z (π, ) and H (π, ). B H H H 15 Lemma 2.4. i) If the representations (π, ) and (π0, 0) are unitarily equivalent , then the corresponding spaces Z 1 and HH1 are isomorphic.H

ii) If (π∗, ∗) is the dual representation of (π, ), then we have H H 1 1 1 1 Z (π∗, ∗) = Z (π, ) and H (π∗, ∗) = H (π, ). H ∼ H H ∼ H iii) If = 1 b 2 decomposes into two π-invariant closed subspaces, then we have H H ⊕H 1 1 1 1 1 1 Z (π, ) = Z (π, 1) Z (π, 2) and H (π, ) = H (π, 1) H (π, 2). H ∼ H ⊕ H H ∼ H ⊕ H

Proof. i) Put A = U, where U : 0 is the unitary intertwiner. Then also 1 H → H 1 1 1 U − : 0 is a unitary intertwiner and U − : Z (π0, 0) Z (π, ) is the H → H 1 1 H →1 1 H inverse function of U : Z (π, ) Z (π0, 0). Likewise, [U − ] : H (π0, 0) 1 H → H1 1 H → H (π, ) is the inverse function of [U] : H (π, ) H (π0, 0). H H → H ii) Consider the antilinear isomorphism

A : ∗, v A(v) := v∗, H → H 7→

where v∗ is deﬁned by v∗(w) := w, v for all w . Then, A intertwines π 〈 〉 ∈1 H 1 and π∗ and induces antilinear isomorphisms A : Z (π, ) Z (π∗, ∗) and 1 1 H → H1 [A] : H (π, ) H (π∗, ∗). The complex conjugate vector spaces Z (π, ) 1 H → H H and H (π, ) induce the above linear isomorphisms. H iii) The linear map

1 1 1 Z (π, 1) Z (π, 2) Z (π, ), (β1, β2) β1 + β2 H ⊕ H → H 7→ 15 This means that there exists a surjective linear isometry U : 0 intertwining the representa- H → H tions π and π0. 2.1 Basic deﬁnitions 25

is an isomorphism: The subspaces 1 and 2 are mutually orthogonal. Thus, H H β1 = β2 implies β1 = β2 = 0. If Pi ( i = 1, 2) denotes the orthogonal projections − 1 onto i, then Pi commutes with π and for every β Z (π, ) we obtain Piβ 1 H 1 ∈ H ∈ Z (π, i). In view of P1 + P2 = , we have β = P1β + P2β. Thus, the above linear map isH an isomorphism. It deﬁnes the linear map on the quotient spaces

1 1 1 H (π, 1) H (π, 2) H (π, ), ([β1], [β2]) [β1 + β2] = [β1] + [β2], H ⊕ H → H 7→ which is then also a linear isomorphism.

L Suppose that = c j J j decomposes into infinitely many π-invariant closed sub- H ∈ H 1 spaces and denote by Pj the orthogonal projections onto j. Then every β Z (π, ) P H1 ∈ H satisfies β = j J Pjβ and βj := Pjβ Z (π, j) so that we find 1 Q 1 ∈ ∈ H Z (π, ) j J Z (π, j). However, we do not have equality in general because, H ⊆ ∈ H 1 P 2 for an arbitrary choice of βj Z (π, j), the sum j J βj(g) need not exist or, if ∈ it exists, might not define a∈ continuousH map on G. To determine those βj for which P β = j J βj defines a continuous 1-cocycle is often a delicate problem. A closely re- ∈ 1 lated problem is the following: If H (π, j) = 0 for all j J, then we may not 1 H { } ∈ conclude that H (π, ) = 0 . However, the converse is true since j being π-invariant implies H { } H ⊆ H

1 1 1 1 Z (π, j) Z (π, ) and H (π, j) H (π, ) H ⊆ H H ⊆ H for all j J. ∈ Remark 2.5. Let G be a topological group and π : G U( ) a unitary representation on a Hilbert space . A vector v is called continuous→ Hif the corresponnding orbit H ∈ H map g π(g)v is continuous. We denote by cont the G-invariant linear subspace of continuous7→ vectors. We claim that cont is aH closed subspace. If v is the limit of a sequence v , then theH orbit map g π g v is the uniform limit of ( n)n N cont ( ) ∈ ⊂ H 7→ the continuous orbit maps g π(g)vn and therefore continuous. This shows that cont is a closed G-invariant linear7→ subspace. The projection of π onto cont yields a continuousH unitary representation by construction. H

Now, assume that G0 is another topological group and ϕ : G0 G is a continuous → group homomorphism. Then, π0 := π ϕ deﬁnes a continuous unitary representation ◦ of G0 on . Again, we obtain a canonical linear map H 1 1 1 Z (ϕ) : Z (G) Z (G0), β β ϕ, → 7→ ◦ 1 which maps 1-coboundaries to 1-coboundaries and thus induces a linear map H (ϕ) on the quotient spaces H1 such that the following diagram commutes

1 1 Z (ϕ) 1 Z (G) Z (G0)

1 1 H (ϕ) 1 H (G) H (G0). 26 2 Preliminary facts on 1-cocycles

A special case arises if G0 G is a subgroup of G and ϕ : G0 , G is the canonical ⊆ → inlusion. That ϕ is continuous means that G0 carries a topology which is ﬁner than 1 the subspace topology inherited from G. In this case, Z (ϕ) is simply a restriction map and we write 1 1 Res : Z G Z G0 , β β G . ( ) ( ) 0 1 → 7→ | Instead of H (ϕ), we write

1 1 [Res] : H (G) H (G0), [β] [β G ]. → 7→ | 0 1 Lemma 2.6. i) If ϕ : G0 G has a dense image in G, then the linear maps Z (ϕ) and 1 H (ϕ) are injective. →

1 1 ii) If ϕ : G0 G is an isomorphism of topological groups, then Z (ϕ) and H (ϕ) are isomorphisms→ of vector spaces.

1 1 iii) If D G is a dense subgoup of G, then Res : Z (G) Z (D) and 1 1 [Res] :⊆H (G) H (D) are linear isomorphisms. → →

Proof. i) If β ϕ = 0, then β(g) = 0 for all g ϕ(G0). Since ϕ(G0) G is a dense subgroup◦ in G and since β : G is continuous,∈ it follows that⊆ β = 0. 1 Thus,Z (ϕ) is injective. In particular, if→β Hϕ is a 1-coboundary, then so is β. This 1 1 means that H (ϕ)[β] = [0] implies [β] =◦ [0]. Hence, H (ϕ) is injective.

1 1 1 1 1 1 ii) If ϕ− : G G0 exists, then Z (ϕ− ) is the inverse function of Z (ϕ) and H (ϕ− ) 1 is the inverse→ function of H (ϕ). iii) That Res and [Res] are injective follows from part i). The relation β(g) β(h) = 1 k − k β(g− h) shows that every continuous 1-cocycle β : D is a uniformly continuous map into a Hilbert space. Thus, there exists a unique→ H uniformly continuous extension β : G . The continuous map → H G G , (g, h) β(gh) β(g) π(g)β(h) × → H 7→ − − vanishes on the dense subset D D, hence it is zero on G G. This reveals that also β is a 1-cocycle. Hence, Res×and [Res] are both surjective.×

2.2 1-cocycles, afﬁne isometric actions and negative deﬁnite functions

Let G be a topological group. For a real or complex Hilbert space , we write Mot( ) for the corresponding group of affine isometric actions, so thatH we have Mot H ( ) ∼= in the real case and Mot U in the complex case. WeH call a o ( ) ( ) ∼= o ( ) groupH H homomorphism A : G Mot(H ) anHaffine isometricH action of G on the Hilbert space . We say that A is continuous→ Hif all orbit maps H G , g A(g)v → H 7→ 2.2 1-cocycles, affine isometric actions and negative definite functions 27 for v are contiuous. Note that the zero-orbit β(g) := A(g)0 satisfies the 1-cocycle relation∈ H w.r.t. a group representation π on which is defined by the relation H A(g)v = π(g)v + β(g) for all g G and v . ∈ ∈ H Conversely, a given orthogonal or unitary representation π of G on and some corresponding 1-cocyle β define an affine isometric action A via the aboveH formula. An easy computation shows that the 1-cocycle β is a 1-coboundary of the form β(g) = v π(g)v if and only if v is a fixed point of the affine action A, i.e. we have A(g)v = v for− every g G. In the context of my PhD thesis, this naturally leads to the question, which (continuous)∈ affine group actions possess fixed points. A first answer is provided by the following theorem. Theorem 2.7 (Bruhat-Tits). For the continuous affine isometric action A, the following are equivalent:

i) A has a bounded orbit.

ii) A has a ﬁxed point. iii) Every orbit of A is bounded.

Proof. Cf. with [BHV08, Lemma 2.2.7 and Prop. 2.2.9]: i) = ii): Suppose that the orbit B := A(g)v : g G is bounded subset ⇒of . We claim that there exists x { and r ∈> 0} such ⊂ H that B is contained in H ∈ H the closed ball Br (x) := y : y x r and such that any y , for { ∈ H k − k ≤ } ∈ H which B is also contained in the ball Br (y), equals x. Then x is a ﬁxed point: For arbitrary h G, we put y = A(h)x and ﬁnd ∈ 1 A(g)v A(h)x = A(h− g)v x r, k − k − ≤ hence B Br (A(h)x) and therefore A(h)x = x. ⊆ Now, we construct the ball Br (x): Put r := inf s > 0 : ( y )B Bs(y) . { ∃ ∈ H ⊆ } Choose s, t r and δ > 0 arbitrary. If x, y are elements for which B Bs(x) and ≥ ⊆ B Bt (y), then we obtain the following estimate. First, choose some b B such ⊆ x+y ∈ that 2 b > r δ. Then, we use the parallelogram identity and calculate − − 2 2 x y = (x b) (y b) k − k k − −2 − k 2 2 = 2 x b + 2 y b (x b) + (y b) k − k2 k − k2 − k x−+y 2 − k = 2 x b + 2 y b 4 2 b 2 2 2 2 2 2 2 2 2sk +−2t k 4(rk δ−) k= −2(s r )− + 2(t r ) + (8rδ 4δ ). ≤ − − − − − Passing to the limit δ 0 yields

→ 2 2 2 2 2 x y 2(s r ) + 2(t r ) k − k ≤ − − For n , choose x with B B 1 x . Then, our estimate shows that N n r+ n ( n) the sequence∈ x is∈ a H Cauchy sequence.⊆ Let x be its limit in . By approx- ( n)n N ∈ H imation, we ﬁnd that B Br (x). If B Br (y) for some y , then we apply the above estimate with⊆s = t = r and⊆ obtain x = y. ∈ H 28 2 Preliminary facts on 1-cocycles ii) = iii): If x is a ﬁxed point of A, then we have for every v that ⇒ ∈ H A(g)v A(g)v A(g)x + x = v x + x . k k ≤ k − k k k k − k k k This means that every orbit of A is bounded. iii) = i): This is trivial. ⇒ As an immediate consequence, we obtain the next corollary which contains the statement that a continuous 1-cocycle β is a 1-coboundary if and only if it is bounded (w.r.t. the norm of the corresponding Hilbert space).

Corollary 2.8. For a continuous afﬁne isometric action A with corresponding continuous orthogonal or unitary representation π and continuous 1-cocycle β, the following are equivalent:

i) A has a ﬁxed point.

ii) The 1-cocycle β is a 1-coboundary.

2 iii) The continuous function ψ : G R, g β(g) is bounded. → 7→ k k

2 This corollary motivates us to take closer look at the function ψ(g) = β(g) : The function ψ : G R is a real-valued, negative definite function with ψ(e)k = 0. k → Definition 2.9. A function ψ : G R is called negative definite if the following conditions are satified: →

1 i) ψ is symmetric, i.e. we have ψ(g− ) = ψ(g) for all g G. ∈

ii) For arbitrary n N, group elements g1,..., gn G and coefﬁcients c1,..., cn R with Pn c 0,∈ we have ∈ ∈ i=1 i =

X 1 ci cjψ(gi− g j) 0 i,j ≤

We call a negative deﬁnite function ψ normalized if ψ(e) = 0.

Remark 2.10. Choosing n = 2, an arbitary g G and coefficients cg = 1 and ce = 1, we derive from condition ii) that ψ(g) ∈ ψ(e), which means that every negative− definite function is bounded from below.≥ In particular, every normalized negative definite function takes its values in the nonnegative real numbers. Note that ψ ψ(e) is also a negative definite function, which allows us to assume w.l.o.g. that ψ −is normalized. 2.2 1-cocycles, affine isometric actions and negative definite functions 29

For an affine isometric group action A : G Mot( ) on a real Hilbert space with → H 2 zero-orbit β(g) := A(g)0, we consider the function ψA(g) := β(g) . The relation k k 1 ψA(gh) ψA(g) ψA(h) = β(h), β(g− ) for all g, h G − − −〈 〉 ∈ reveals that the function ψA(g) satisfies condition ii) in Definition 2.9. In view of 1 1 β(e) = 0 and β(g− ) = π(g− )β(g), it is a normalized negative definite function. Moreover, the above equation− implies the inequality

Æ Æ Æ ψA(gh) ψA(g) + ψA(h) for all g, h G. ≤ ∈ The following theorem is the converse statement:

Theorem 2.11. Every negative deﬁnite, normalized function ψ : G R is of the form → ψ = ψA for some afﬁne isometric action A : G Mot( ) on a real Hilbert which is spanned by the elements β(g) := A(g)0 : g → G . TheH pair (A, ) is uniqueH up to G-equivariant linear isometry.{ ∈ } H

Proof. The proof is based on a GNS-construction similar to that related to positive deﬁnite functions. The details can be found in [BHV08, Prop. 2.10.2].

In a next step, we exploit GNS-constructions to see that the afﬁne isometric action A induces a unitary representation in a Fock-Hilbert space: Fix t > 0 and assume (for simplicity) that the Hilbert space is real. Then, there exists a complex Hilbert space

t , a continuous mapping H H 1 γt : S ( t ) := v t : v = 1 H → H { ∈ H k k } and a unitary representation

πt : G U( t ), → H which is continuous if A is continuous, such that

2 t v v0 a) γt (v), γt (v0) = e− − for all v, v0 , 〈 〉 k k ∈ H b) πt (g) γt = γt A(g) for all g G, ◦ ◦ ∈ c) the image γt ( ) is total in t . H H

The pair (πt , t ) is unique up to unique isomorphism. For details, we refer to [BHV08, H Prop. 2.11.1]. The vector Ωt := γt (0) t is called the vacuum vector. ∈ H Lemma 2.12. The affine isometric action A has a fixed point if and only if the unitary representation πt has a nonzero fixed point. 30 2 Preliminary facts on 1-cocycles

Proof. Assume first that v is a fixed point of A. We then have ∈ H πt (g)γt (v) = γt (A(g)v) = γt (v) for all g G. ∈ Since γt (v) = 1, the vector γt (v) is a nonzero fixed point of πt . Now, assumek k that A does not have any fixed points in . By Theorem 2.7, this means that every orbit A(G)v is unbounded. Fix an arbitraryH v . For each n N, we ∈ H ∈ thus find some gn G with A(gn)v n. Put wn := A(gn)v and observe that, for any w , we have ∈wn w k askn ≥ . Hence, ∈ H k − k → ∞ → ∞ 2 t wn w γt (wn), γt (w) = e− k − k 0 as n . 〈 〉 → → ∞

In view of the total image of γt in t , we conclude that the bounded sequence γ w weakly converges to 0. NowH let ξ G. We have ( t ( n))n N t ∈ ∈ H

0 lim γt wn , ξ lim γt A gn v , ξ = n ( ) = n ( ( ) ) →∞〈 〉 →∞〈 〉

lim πt gn γt v , ξ γt v , ξ . = n ( ) ( ) = ( ) →∞〈 〉 〈 〉 Since v was arbitrary and γt ( ) is total in t , we conclude that ξ = 0, hence that G H H t = 0 . H { } The following proposition is well-known (see e.g. [BHV08, Prop. 2.11.1 and Thm. 2.12.9]):

Proposition 2.13. Let A : G , (g, v) A(g)v be a continuous afﬁne isometric ×H → H 7→ action of G on a real Hilbert space . For t > 0, let (πt , t ) be the corresponding continuous unitary Fock representationH of G on the complex Fock–HilbertH space t . Then the following are equivalent: H

i) The affine action A has a fixed point in . H ii) The Fock representation πt has a nonzero fixed point in t . H iii) The Fock Hilbert space t has a nontrivial finite-dimensional G-invariant subspace. H Proof.i ) ii) : This is Lemma 2.12. ⇐⇒ i) iii) : Consider the affine isometric action ⇐⇒ A A : G , (g, v, w) (A(g)v, A(g)w). ⊕ × H ⊕ H → H ⊕ H 7→ Then A has a fixed point in if and only if A A has a fixed point in . By Lemma 2.12, the latter isH equivalent to the⊕ corresponding Fock representa-H ⊕ H tion having a nonzero fixed vector. We claim that this Fock representation is equivalent to the conjugation action

1 G 2( t ) 2( t ), (g, T) πt (g)Tπt (g)− × B H → B H 7→ 2.3 A generalized version of the Delorme–Guichardet Theorem 31

on the Hilbert space of Hilbert–Schmidt operators 2( t ) (cf. Subsection 5.1). To prove our claim, we introduce the map B H

Γt : 2( t ), (v, w) Γt (v, w) := γt (v) γt (w)∗ H ⊕ H → B H 7→ ⊗

and verify the conditions a) c): That Γt has a total image can be seen as follows. − If T 2( t ) is orthogonal to this image, then we have ∈ B H 0 T, Γ v, w Tr BΓ v, w Tr Tγ w γ v γ v , Tγ w = t ( ) 2( t ) = ( t ( )∗) = ( t ( ) t ( )∗) = t ( ) t ( ) t 〈 〉B H ⊗ 〈 〉H

for all v, w . That the image of γt is total in t implies T = 0. This proves condition c∈). H Moreover, we have H

Γ v, w , Γ v , w Tr γ v γ w γ w γ v t ( ) t ( 0 0) 2( t ) = ( t ( ) t ( )∗ t ( 0) ( 0)∗) 〈 〉B H ⊗ · ⊗ 2 2 t v v w w v , v w , w e 0 + 0 . = γt ( 0) γt ( ) t γt ( 0) γt ( ) t = − − − 〈 〉H 〈 〉H k k k k This proves a). To check condition b) we use πt (g)γt (v) = γt (A(g)v) and obtain

1 πt (g)Γt (v, w)πt (g)− = πt (g)γt (v) γt (w)∗πt (g)∗ = γt (A(g)v) γt (A(g)w) ⊗ ⊗

= Γt (A(g)v, A(g)w) for all v and g G. ∈ H ∈ Now, assume that T 2( t ) is a nonzero ﬁxed point of the conjugation action. In view of the relation∈ B H

1 πt (g)Tπt (g)− = T πt (g)T = Tπt (g) ⇐⇒

for all g G, this means that T belongs to the commutant πt (G)0. Hence, ∈ also the compact positive operator T ∗ T belongs to the commutant. Since T 1 is nonzero, we find an eigenvalue λ > 0 of T ∗ T. Then, ker(T ∗ T λ ) is a nontrivial, finite-dimensional and G-invariant subspace of t . Conversely,− · let P denote the orthogonal projection onto a nontrivial, finite-dimensionalH G-

invariant subspace of t . This means that H

0 = P πt (G)0 2( t ) 6 ∈ ∩ B H

and thus that P is a nonzero ﬁxed vector for the conjugation action on 2( t ). B H

2.3 A generalized version of the Delorme–Guichardet Theorem

As a last step of this subsection, we provide some details of the Delorme–Guichardet Theorem, following the ideas in [BHV08] and generalizing them at the same time.

Deﬁnition 2.14. Let S G be any subset and let " > 0. ⊆ 32 2 Preliminary facts on 1-cocycles i) If π : G U( ) is a unitary representation of G and v is a vector, then v is said to→ be (HS, ")-invariant if ∈ H

sup π(s)v v < " v . s S k − k k k ∈ ii) The pair (S, ") is called a Kazhdan pair if every continuous unitary representation π of G with some (S, ")-invariant vector automatically has a nonzero G-ﬁxed vector. iii) We write for the set of all subsets of G with the property that for every continuous negativeS deﬁnite function ψ : G R, the image ψ(S) is a bounded subset of R. In particular, every compact subset→ of G belongs to . S iv) The group G has Generalized Property (T) if there exists a Kazhdan pair (S, ") where S . If S in the Kazhdan pair (S, ") is compact, then we say that G has Property∈ S (T).∈ S

Remark 2.15. It is shown in [BHV08, Prop. 1.1.5] that (G, p2) is a Kazhdan pair for every topological group G. In particular, this observation shows that every compact group G has Property (T).

For a continuous afﬁne isometric action A : G Mot( ) on a real Hilbert space → H H and some t > 0, we denote by (πt , t ) the corresponding unitary representation on the Fock-Hilbert space. H

Lemma 2.16. Let t be a sequence of positive real numbers converging to 0. ( n)n N Then, the unitary representation∈ π, : L π , has a S, " -invariant vector ( ) = cn N( tn tn ) ( ) for all S and all " > 0. H ∈ H ∈ S Proof. Choose S and 0 and put v : 0 . In particular, we " > n = γtn ( ) tn ∈ S ∈ H ⊂ H have vn = 1. For g G, we calculate k k ∈ 2 2 g v v g 0 0 π( ) n n = πtn ( )γtn ( ) γtn ( ) k − k − 2 2 A g 0 0 2 1 e tn A(g)0 . = γtn ( ( ) ) γtn ( ) = − k k − − 2 Since ψ(g) := A(g)0 defines a real-valued continuous negative definite function k k 2 on G, we have sups S A(s)0 < by definition of . From tn 0, we thus derive that ∈ k k ∞ S →

lim π s vn vn 0 uniformly on S. n ( ) = →∞ k − k Hence, we ﬁnd some n N such that sups S π(s)vn vn < ". ∈ ∈ k − k Theorem 2.17. The following statements are equivalent:

i)G . ∈ S ii) G has Generalized Property (T). 2.3 A generalized version of the Delorme–Guichardet Theorem 33

iii) G has Property (FH), i.e. every continuous afﬁne isometric action on a real Hilbert space has a ﬁxed point.

1 iv)H (G, π, ) = 0 for every continuous orthogonal representation π : G ( ). H { } → H 1 v)H (G, π, ) = 0 for every continuous unitary representation π : G U( ). H { } → H Proof.i ) = ii) : If G , then the fact that (G, p2) is a Kazhdan pair implies that G has⇒ Generalized∈ Property S (T). ii) = iii) : Assume that G does not have Property (FH). This means that we find ⇒some continuous affine isometric action A : G Mot( ) without fixed points. L Consider the unitary direct sum representation→ π, H: π 1 , 1 . For ( 0) = cn N( n n ) H ∈ H G all n N, for the corresponding Fock space representation, we have 1 = n ∈ G H 0 by Lemma 2.12. Consequently, we have 0 = 0 . On the other hand, { } H { } Lemma 2.16 tells us that the unitary representation (π, 0) has an (S, ")-invariant vector for all " > 0 and S . This shows thatHG does not have Generalized Property (T) which contradicts∈ S ii). iii) = iv) : This follows immediately from the fact that an affine isometric action ⇒has a fixed point if and only if the corresponding 1-cocycle is a 1-coboundary (cf. Corollary 2.8). iv) = i) : This follows from the fact that every continuous negative definite function 2 ⇒ψ : G R is of the form ψ(g) = A(g)0 for some continuous affine isometric action→A : G Mot( ) of G on somek realk Hilbert space . → H H iv) v) : On the one hand, one can always view a complex Hilbert space as a real ⇐⇒one and, thus, a corresponding unitary representation as an orthogonal one. On the other hand, if π : G ( ) is an orthogonal representation on a real Hilbert space, then we canonically→ H obtain a unitary representation π : G U via C ( C) complexification on the Hilbert space . Viewing π→, H as an C ∼= C ( C C) othogonal representation, we have H ⊗ H H

π , π, π, ( C C) = ( ) ( ) H ∼ H ⊕ H and thus H1 π , H1 π, H1 π, . ( C C) = ( ) ( ) H H ⊕ H We conclude that

H1 π , 0 H1 π, 0 . ( C C) = ( ) = H { } ⇐⇒ H { } Since Property (T) is stronger than Generalized Property (T), we obtain the Delorme part of the Delorme–Guichardet Theorem as a corollary of Theorem 2.17.

Corollary 2.18. If a topological group has Property (T), then it has Property (FH). 34 2 Preliminary facts on 1-cocycles

Example 2.19. In the following case, the condition that G is trivially satisfied: Assume that the topological group G is exotic in the sense that∈ S every continuous unitary representation is trivial, i.e. π(g) = 1 for all g G. This implies that every pair (S, ") is a Kazhdan pair. In particular, exotic groups∈ have Propery (T) and therefore Property (FH). That every continuous unitary representation is trivial also means that every continuous positive definite function ϕ : G C is constant (GNS-construction). → tψ If ψ : G is a continuous negative definite function, then ϕ := e− with t > 0 R 1 → n ψ is positive definite, hence constant. Now ψ = limn n(1 e− ) shows that ψ = 0. Therefore, for exotic groups, we have ψ(G) = 0 →∞for ever− continuous negative definite function ψ. In [Ba91, Chap. 2], one finds{ examples} of exotic groups among the class of abelian Banachh–Lie groups. They are of the form G = E/Γ , where E is an infinite dimensional Banach space and Γ E is a discrete subgroup of E. Another Homeo⊆ well-known example is the group +([0, 1]) of orientation preserving homeo- morphisms equipped with the compact-open topology. As shown by Megrelishvili in [Me01], this Polish group even does not have non-trivial continuous isometric actions in reflexive Banach–spaces.

We now turn to the Guichardet part which is based on the following observation: In order to show that a topological group G does not have Generalized Property (T) which is to say Property (FH), it is enough to construct one unbounded 1-cocycle. A naive way to construct unbounded 1-cocycles from bounded ones is the direct sum: Let (π, ) be a continuous unitary representation of G without nontrivial G-fixed vectors.H We consider countably many copies of this representation and form the direct sum π, : π, . ( ) = b n N( ) ∞ · ∞ · H ⊕ ∈ H Let v be a sequence in such that P π g v v 2 < for all g G. ( n)n N n N ( ) n n Then, β∈ g : P π g Hv v defines∈ ak 1-cocycle− fork the∞ representation∈ ( ) = n N ( ) n n ( π, ). If we∈ additionally− assume that G is a Baire group, e.g. a completely∞ · metrizable∞ · H group or a locally compact group (cf. Subsection 3.1), then β is automatically continuous because it is the pointwise limit of a sequence of continuous 1-coboundaries (cf. Proposition 3.9). We claim that the 1-cocycle β is a 1-coboundary if and only if P v 2 < : If β is a 1-coboundary, then there exist v with n N n n0 2 ∈ P v < andk βk g ∞P π g v v . Projecting onto each component, n N n0 ( ) = n N ( ) n0 n0 ∈ ∞ ∈ − G we find π(g)vn vn = π(g)vn0 vn0 for each n N and g G. Since = 0 , we − − ∈ ∈ H { } find vn = vn0 for all n. The converse statement is clear. Hence, if we can construct a 1-cocycle β this way from a sequence v with P v 2 , then G does not ( n)n N n N n = have Property (FH). Our aim is to prove the∈ converse∈ statement,k k ∞ namely that if it is not possible to construct unbounded 1-cocycle that way, then one may conclude that G has Property (FH). This is realized in Proposition 2.21 if G satisfies the assumptions (A1) and (A2) below which hold in particular for every connected Banach–Lie group, any direct limit of compact group and every every σ-compact locally compact group.

There is also a more operator theoretic version of the preceding observation: Instead of the direct sum representation ( π, ), one considers the Hilbert space ∞ · ∞ · H 2.3 A generalized version of the Delorme–Guichardet Theorem 35

2( ) of Hilbert–Schmidt operators on . The continuous unitary representation B H H (π, ) naturally deﬁnes a continuous unitary representation of G on 2( ) via H B H G 2( ) 2( ), (g, T) π(g)T. × B H → B H 7→ 1 Every 1-coboundary is then of the form β(g) = (π(g) )T for some T 2( ). A nontrivial 1-cocycle may be of the form β(g) := (π(g)− 1)B for some bounded∈ B H lin- G ear operator B ( ). Similarly, = 0 implies that− β is a 1-coboundary if and only if B is a Hilbert–Schmidt∈ B H operator.H We{ shall} also see in Proposition 2.21 that any group G satisfying the assumptions (A1) and (A2) has Property (FH) if it is not possible to construct unbounded 1-cocycles of the form β(g) := (π(g) 1)B for some bounded − linear operator B. Note that, in view of 2( ) = ∗ b , the unitary representation B H ∼ H ⊗H 1 on 2( ) is canonically isomorphic to the tensor product representation π on B H ⊗ ∗ b which can be viewed as a direct sum of copies of (π, ). If is separable, H ⊗H H H then the representations on 2( ) and on are isomorphic. B H ∞ · H Let G be a topological group satisfying the the following two assumptions:

S S (A1) : There exists an increasing sequence of subsets n n+1 such that G S S . ⊆ ∈ S = n N n ∈ (A2) : Whenever the pointwise limit lim of a sequence in B1 G, , ex- β = n ∂vn ( π ) ists, where (π, ) is a continuous unitary→∞ representation of G, then β is alreadyH a continuous 1-cocycle.H

Example 2.20. We will brieﬂy discuss some important classes of topological groups satisfying the assumptions (A1) and (A2):

i) Our first example are locally compact σ-compact groups: We will see in the next section that every Baire group satisfies assumption (A2), which will be a consequence of Proposition 3.9. The Baire Category Theorem (cf. [Bour58, Théorème 1 de 5]) implies that every locally compact and every completely metrizable group is a Baire group (cf. Example 3.2). Moreover, it is clear that every compact subset of G belongs to the set . S ii) Next, we claim that connected Banach–Lie groups are prominent examples: To see this, observe first that, a Lie group is a Baire group if and only if its Lie algebra is a Baire space (Corollary 3.5). In particular, every Banach–Lie group is a Baire group and thus satisfies assumption (A2). Now we show that assumption (A1) is satisfied: If A, B G belong to the set , then also the subsets A B and A B.

In particular, if G⊆can be written as a countableS union of subsets ∪An , then· S ∈ S Sn := m n Am for all n N form an increasing sequence of subsets which cover the≤ whole∈ group S G. ∈ If G is a connected Banach–Lie group with Banach–Lie algebra (g, ) and exponential function exp : g G, then every subset of the form exp( k·kX : X n ) → { k k ≤ } 36 2 Preliminary facts on 1-cocycles

with n N belongs to (cf. Lemma 3.24 and Remark 3.23 in the next section). That∈ G is connectedS means that every group element is the ﬁnite product of elements of the form exp(X ) with X g. Thus, we have ∈ [ m G = exp( X : X n ) n,m N { k k ≤ } ∈ so that G is covered by countably many subsets from . S iii) A third example is given by the direct limits of compact groups, i.e. G can be written as a union of an increasing sequence of compact subgroups. Here, we use the fact that assumption (A2) is also satisﬁed if G is a direct limit of locally compact groups (cf. Corollary 3.11).

Proposition 2.21. Let G S S be a topological group satisfying the assumptions = n N n (A1) and (A2). Then, the following∈ are equivalent:

n i) None of the pairs (Sn, 2− ) with n N is a Kazhdan pair. ∈ ii) There exists a continuous unitary representation (π, ) without nontrivial G-invariant vectors but with some orthonormal systemH e such that ( n)n N P π g e e 2 < for all g G. ∈ n N ( ) n n ∈ k − k ∞ ∈ iii) There exists a continuous unitary representation (π, ) without nontrivial G-invariant vectors and a bounded linear operator B : H which is not Hilbert– 1 H → H Schmidt but for which β(g) := (π(g) )B deﬁnes a continuous map G 2( ) into the Hilbert–Schmidt operators on− . → B H H iv) The group G does not have Generalized Property (T) = Property (FH).

In particular, G has Generalized Property (T) if and only if one of the pairs S , 2 n ( n − )n N is a Kazhdan pair. ∈

n Proof.i ) = ii) Assume that no pair (Sn, 2− ) is a Kazhdan pair. This means that we ⇒ G find continuous unitary representations πn : G U( n) with n = 0 but n 16→ H H { } with an (Sn, 2− )-invariant unit vector en n. Then the continuous unitary direct sum representation π, : L ∈ Hπ , satisfies G 0 . Every ( ) = cn N( n n) = group element g G S HS is contained∈ inH all S withHn N{ for} some = n N n n sufficiently large N∈= N(g) ∈N so that we obtain the estimate ≥ ∈ X 2 X 2 X 2 π(g)en en = π(g)en en + π(g)en en n N k − k n

X n 4(N 1) + 4− < . ≤ − n N ∞ ≥ 16 Note that n does not stand for a Fock Hilbert space here. H 2.3 A generalized version of the Delorme–Guichardet Theorem 37 ii iii Let e be a complete orthonormal system which extends e , i.e. ) = ) ( j)j J ( n)n N ⇒ ∈ ∈ we assume N J. Denote by ej e∗j the unique bounded linear operator on which is defined⊆ by e e v ⊗v, e e . Then, B : P e e is a boundedH j ∗j ( ) = j j = n N n n∗ linear operator which is⊗ not Hilbert–Schmidt.〈 〉 For every g∈ G⊗we have ∈ 1 2 X 2 π g B π g en en < ( ( ) ) 2( ) = ( ) k − kB H n N k − kH ∞ ∈ and our assumption (A2) implies that this expression is continuous in g. iii) = iv) The map β(g) := (π(g) 1)B defines a continuous 1-cocycle for the con- ⇒ − tinuous unitary representation on the Hilbert space 2( ) which is given by B H G 2( ) 2( ), (g, T) π(g)T. In view of the fact that B is not Hilbert– G Schmidt×B H and→ B =H0 , the 1-cocycle7→ β is not a 1-coboundary. The existence of a nontrivial 1-cocycleH { shows} that G does not have Property (FH). iv) = i) This follows from the definition of Generalized Property (T). ⇒ The following corollary is known as the Delorme–Guichardet Theorem (see e.g. [BHV08, Thm. 2.12.4]): Corollary 2.22. Assume that G is locally compact and σ-compact (e.g. compactly generated). Then, Property (FH) and Property (T) are equivalent.

Proof. That Property (T) implies Property (FH) is the content of Corollary 2.18. Con- versely, assume that G has Property (FH). That G is σ-compact means that G is the S S G union of a sequence of increasing compact subsets n n+1 . By the previous n ⊆ ⊆ Proposition 2.21, there is some n for which (Sn, 2− ) is a Kazhdan pair, so that G has Property (T).

In the next section, we will show that the full unitary group U( ) of a complex Hilbert space has property (FH). However, if is infinite-dimensional,H there are many naturalH unitary subgroups of U( ) whichH do not have Property (FH). H Example 2.23. Let be an infinite dimensional complex Hilbert space. We denote H by U2( ) the group of all those unitary operators g U( ) for which the difference 1 H ∈ H 1 g is a Hilbert–Schmidt operator, i.e. we put G := U2( ) := U( ) ( + 2( )). The− metric d g, g : g g turns G U Hinto a (completelyH ∩ B metriz-H ( 0) = 0 2( ) = 2( ) k − kB H H able) topological group. The group G acts naturally on and on 2( ) via the identical representation (g, v) g.v respectively via theH left multiplicationB H representation (g, T) gT. Clearly,7→ every bounded linear operator B ( ) which is not Hilbert–Schmidt7→ defines an unbounded continuous 1-cocycle for∈ B theH representa- 1 tion on the Hilbert space of Hilbert–Schmidt operators 2( ) via β(g) := (g )B. In particular, β(g) := (g 1) is such an unbounded 1-cocycle.B H If is not separa-− ble, then β cannot be approximated− pointwise by a sequence of 1-coboundaries:H If 1 β(g) = limn (g )Tn pointwise held for a sequence of Hilbert–Schmidt operators T , then→∞ the fact− that each Hilbert–Schmidt operator has a kernel of countable ( n)n N ∈ 38 2 Preliminary facts on 1-cocycles co-dimension combined with our assumption of not being separable would imply H the existence of a unit vector v with Tn v = 0 for all n N. Passing to the limit, we would find (g 1)v = β(g)v∈= H0 for all g G. But this∈ contradicts the fact there − ∈ is no nontrivial common fixed vector of U2( ) in . H H In view of the fact that not every unbounded 1-cocycle is the pointwise limit of a sequence of 1-coboundaries (cf. Example 2.23), the following corollary of Proposi- tion 2.21 is remarkable.

Corollary 2.24. Let G S S be a topological group satisfying the assumptions A1 = n N n ( ) and (A2). ∈ Then G has Property (FH) if and only if for any continuous unitary representation (π, ) of G, the pointwise limit lim (if it exists) of a sequence in B1 G, , deﬁnesH β = n ∂vn ( π ) a 1-coboundary. →∞ H

Proof. If G has Property (FH), then the pointwise limit β is a 1-coboundary by definition. If G does not have Property (FH), then by Proposition 2.21, there exists a continuous unitary representation π, with some orthonormal system e such ( ) ( n)n N that P π g e e 2 < holdsH for all g G. Then, β g : P π g ∈ 1 e n N ( ) n n ( ) = n N( ( ) ) n deﬁnes a∈ 1-cocyclek − fork the continuous∞ representation∈ on `2∈ , ,− where ∼= (N ) each en now belongs to a different copy of . The sum∞β ·converges H pointwiseH in , hence it is a continuous 1-cocycle byH(A2). However, β is not a 1-coboundary since∞ · H the en form an orthonormal sequence in , which means that they cannot form a square integrable sequence. ∞ · H 39

3 Topological Aspects

In this section, we collect some facts on the topological behaviour of a 1-cocycle β : G . Later in subsection 8.2, we shall construct unbounded 1-cocycles which arise as→ a H pointwise convergent sum of 1-coboundaries P . The 1-coboundaries β = n ∂vn will be continuous and the question arises whether is then automatically contin- ∂vn β uous. In our first subsection, we will show that this is true whenever the topological group G is a Baire group (cf. Proposition 3.9). Prominent representatives of this class of topological groups are all locally compact groups and all completely metrizable groups. In AppendixA, we will study the automatic continuity behaviour of 1-cocycles of Baire groups in more detail. In our second subsection, we introduce the class of topological groups with Property (OB) and discuss some important examples. Every continuous 1-cocycle β : G on a group G with Property (OB) is bounded, hence a 1-coboundary. In particular,→ H groups with Property (OB) form a subclass of Property (FH) groups. A particular subclass of Property (OB) groups form the bounded topological groups. We will show that the unitary group U( ) of a complex Hilbert space is bounded, as well as its subgroup U ( ) consistingH of those unitary operators g for which the difference g 1 is a compact∞ H operator (cf. Proposition 3.29). In the last subsection, we replace− the unitary representation (π, ) by an isometric action on a Banach space E. If E is reflexive, then every continuousH 1-cocycle β : G E of a group G with Property (OB) is automatically a 1-coboundary as in the case→ of a Hilbert space. However, this need no longer be true if the condition of reflexivity is dropped. We will see that one may relax the condition of reflexivity if one requires in return G to be also an amenable group (cf. Theorem 3.40). On the other hand, if G is a precompact group, then one does not need any requirement on the Banach space E.

3.1 Baire groups and automatic continuity

We always assume that a topological space X is nonempty.

Deﬁnition 3.1. Let X be a topological (Hausdorff) space. A subset A X is called ⊆

nowhere dense, if A◦ = . • ; meager17, if it is a union of a countable family of nowhere dense subsets. In • particular, every nowhere dense subset is meager.

nonmeager18, if it is not meager. • comeager19 if its complement X A is meager. • \ 17Synonyms used in the literature are of ﬁrst (Baire) category and exhaustible. 18resp. of second Baire category, resp. inexhaustible 19also called residual 40 3 Topological Aspects

The space X is called a Baire space, if every nonempty open subset of X is nonmeager. In partiuclar, the whole space X is nonmeager.

Example 3.2. i) Every completely metrizable space X is a Baire space.

ii) Every locally compact Hausdorff space X is a Baire space. This is usually referred to as the Baire Category Theorem and a proof can be found in [Bour58, Théorème 1 de 5]. For the topological space X , we note that being nonmeager does not imply being a ` Baire space: Suppose that X = X1 X2 is the disjoint union of a meager space X1 and a nonmeager space X2. Thus, X is not a Baire space and it is nonmeager because, otherwise X2 were meager. However, in the realm of topological groups, this distinction never occurs.

Lemma 3.3. For a topological group G, the following statements are equivalent:

i) There exists a nonempty open subset which is nonmeager.

ii) G is nonmeager. iii) Every nonempty open subset of G is nonmeager, i.e. G is a Baire space.

Definition 3.4. If a topological group G satisfies one of the equivalent definitions from Lemma 3.3, then it is called a Baire group.

Proof. (cf. [Gao09, Prop. 2.3.1]): i) = ii): Subsets of nowhere dense subsets are nowhere dense. From this it follows ⇒that subsets of meager subsets are again meager. Hence if G was meager, then also every nonempty open subset. ii) = iii): Again, we argue by contradiction: Assume that U G is a nonempty ⇒open meager subset. Then, for every x G the set xU is also⊂ nonempty, open S ∈ and meager and we have G = x G xU. We write for the set of all sets whose elements are nonempty pairwise∈ disjoint openM subsets of G such that FV implies V xU for some x G. We order by inclusion and see that∈ F is not empty⊆ and that every chain∈ has an upperM bound. Applying Zorn’s M S Lemma, we ﬁnd a maximal set = (Vi)i I in . Then V := i I Vi is an open subset of G and, by virtue of theF maximality∈ ofM , also dense.∈ Hence, G V is nowhere dense and closed. F \

On the other hand, every Vi is meager as a subset of some meager subset xU. S We ﬁnd nowhere dense subsets Fi,n such that Vi = n N Fi,n for all i I. Note ∈ ∈ that Fj,n Fi,n = for j = i because the Vi are pairwise disjoint. If we put F : S ∩ F , then; we have6 V S F and therefore n = i I i,n = n N n ∈ ∈ [ G = V (G V ) = (G V ) Fn. ∪ \ \ ∪ n N ∈ 3.1 Baire groups and automatic continuity 41

We now claim that all Fn must be nowhere dense. This means that G is meager contradicting the assumption of G being nonmeager. To see that Fn is nowhere dense, we also argue by contradiction: Suppose that W G is an open nonempty ⊆ subset such that Fn W is dense in W. We have W V = since V is open and ∩ ∩ 6 ; dense. Then, there is some index i I with W Vi = . Therefore, we may assume w.l.o.g. that W Vi. We obtain∈ ∩ 6 ; ⊆ W open W = Fn W Fn W Fn Vi = Fn,i. ∩ ⊆ ∩ ⊆ ∩

Since Fn,i does not contain interior points, W = follows contradicting our assumption on W. ; iii) = i): This is trivial. ⇒ The next corollary is useful, if one looks for Baire groups among the various types of Lie groups. A particular class of Lie groups which is under active investigation is that of locally convex Lie groups (cf. [Ne06]). This means that the underlying manifold is modeled over a locally convex vector space. In particular, the Lie algebra, which is the tangent space at the group identity, is locally convex.

Corollary 3.5. A locally convex Lie group G is a Baire group if and only if its Lie algebra L(G) is a Baire space.

Proof. Since G is a manifold, there exist open identity neighborhoods in G and L(G) which are homeomorphic. Therefore, item i) in Lemma 3.3 is either satisﬁed for both G and L(G) or for none of them. (Note that Lemma 3.3 applies to L(G) which is a locally convex Lie group with respect to the addition operation.)

Example 3.6. i)A Fréchet space is a locally convex vector space which is completely metrizable such that the metric is translation invariant. A Fréchet space is a Baire space by the Baire Category Theorem (Example 3.2) and every Fréchet–Lie group, i.e. a locally convex Lie group whose Lie algebra is a Fréchet space, is a Baire group by Corollary 3.5.

ii) Since every Banach-space is a Fréchet space, every Banach–Lie group is a Baire group. In particular, every ﬁnite-dimensional Lie group is a Baire group.

Baire category arguments provide a remedy for variuous technical problems because pointwise limits preserve (under mild assumptions) properties like local boundedness or continuity of functions on Baire spaces. For instance, the Uniform Boundedness Principle (cf. [Bour58, Théorème 2 de 5]) states that, for a family of continuous functions fi : X R with i I, the (lower-semicontinuous) pointwise supremum → ∈ f (x) := supi I fi(x) (provided that it exists) is almost everywhere locally uniformly bounded. This∈ means that the subset of points in X admitting a neighborhood on which f is uniformly bounded is an open dense subset (and in particular nonempty). 42 3 Topological Aspects

A closely related analog is Baire’s Theorem on functions of ﬁrst class 20. Here the pointwise supremum f is reprelaced by a pointwise limit of a sequence of continuous functions fn : X R and the assertion is that the points of continuity of f form a dense comeager (and→ in particular nonempty) subset of the Baire space X . We give a proof of this theorem which goes back to [Hau62, Chap. IX, 42]:

Theorem 3.7. Let X be a Baire space and Y be a metric space. If a map f : X Y is → the pointwise limit of a sequence of continuous functions fn : X Y with n N, then → ∈ the continuity points of f (x) = limn fn(x) form a comeager, dense (and in particular nonempty) subset of X . →∞

Proof. We write for the continuity points of f and := X for the discontinuity points. We denoteC the interior of a subset A X eitherD by Aor\C by Int(A). The idea is to show that is the countable union of closed⊆ subsets with no interior. Thus, is meager by deﬁnitionD and = because, otherwise the open subset was meagerD in X contradicting the BaireD space; condition. Therefore, the complementD is a comeager, dense subset of X . C

Let d : Y Y R be a metric on Y . For given x0 X , " > 0 and m, n N, we introduce× the subsets→ ∈ ∈

d(fn, fm) " := x X : d(fn(x), fm(x)) " , { ≤ } { ∈ ≤ } d(f , fm) " := x X : d(f (x), fm(x)) " , { ≤ } { ∈ ≤ } d(fm, fm(x0)) " := x X : d(fm(x), fm(x0)) " , { ≤ } { ∈ ≤ } d(f , f (x0)) " := x X : d(f (x), f (x0)) " . { ≤ } { ∈ ≤ }

Suppose that x0 d(f , fm) " . Then, the triangle inequality of the metric shows the inclusions ∈ { ≤ }

d(f , f (x0)) " d(fm, fm(x0)) " d(f , fm) 3" , { ≤ } ∩ { ≤ } ⊆ { ≤ } d(f , fm) " d(fm, fm(x0)) " d(f , f (x0)) 3" { ≤ } ∩ { ≤ } ⊆ { ≤ } and we note that these inclusions also hold if all subsets are replaced by their interior. We also note that we have always x0 Int( d(fm, fm(x0)) " ) since all fm ∈ { ≤ } are continuous. Hence, if x0 Int( d(f , f (x0)) " ), we choose m N such that ∈ { ≤ } ∈ x0 d(f , fm) " (which is possible by the pointwise convergence) and conclude ∈ { ≤ } that x0 Int( d(f , fm) 3" ). Conversly, if x0 Int( d(f , fm) " ) for some m N, ∈ { ≤ } ∈ { ≤ } ∈ we find that x0 Int( d(f , f (x0)) 3" ). By definition, the point x0 is a continuity ∈ { ≤ } point of f if x0 Int( d(f , f (x0)) " ) for all " > 0. Our discussion then shows that ∈ { ≤ } \ [ = Int( d(f , fm) " ). C ">0 m N { ≤ } ∈ 20A function is said to be of first class (of Baire) if it can be represented as the pointwise limit of a sequence of continuous functions. 3.1 Baire groups and automatic continuity 43

For " > 0, we introduce the subsets

O " : S Int d f , f " , ( ) = m N ( ( m) ) ∈ { ≤ } Am(") := n>m d(fn, fm) " ∩ { ≤ } and note that O(") is open by constuction whereas Am(") is closed by virtue of the continuity of f , f . Since f is the pointwise limit of the f , we obtain X S A " n m n = m N m( ) ∈ and Am(") d(f , fm) " and thus Am(") O("). Since O(") is monotone in ", we have ⊆ { ≤ } ⊆ \ \ 1 O " O = ( ) = n C ">0 n N ∈ and therefore

[ 1 [ [ 1 1 X X O A O . = = n = m n n D \C n N \ n N m N \ ∈ ∈ ∈ 1 1 The subsets Am n O n are closed without interior points so that is the countable union of closed nowhere\ dense subsets as announced. Note, thatD our proof did not rely on the Baire space structure on X so that we can conclude, that the discontinuity points of a pointwise limit of continuous functions with values in a metric space form a meager subset.

Theorem 3.7 is an important tool for the construction of nontrivial 1-cocycles:

Lemma 3.8. Let π : G U( ) be a (not necessarily continuous) unitary representation of any topological group→ G. AH 1-cocycle β : G is continuous if and only if its set of continuity points is non-empty. → H

Proof. Let g0 G be a continuity point of G. This means that, for a given " > 0, we ∈ ﬁnd an open neighborhood U of g0 such that β(U) β(g0) < ". In particular, we have k − k

1 1 1 1 β(g0− U) = β(g0− ) + π(g0− )β(U) = π(g0)β(g0− ) + β(U) = β(U) β(g0) < ". k − k 1 For arbitrary g, we consider the open g-neighborhood g g0− U and obtain

1 1 1 β(g) β(g g0− U) = β(g) β(g) π(g)β(g0− U) = β(g0− U) < ". − − − Since " and g were arbitrary, we conclude that β is continuos in every point g G. ∈ Proposition 3.9. Let G be a Baire group and π : G U( ) be a (not necessarily continuous) unitary representation. → H Assume that a map β : G is the pointwise limit of a sequence of continuous 1- → H cocycles βn : G with n N. Then, β is also a continuous 1-cocycle. → H ∈ 44 3 Topological Aspects

Proof. The 1-cocycle equation follows from

β g g β g π g β g lim βn g g βn g π g βn g 0. ( 0) ( ) ( ) ( 0) = n ( ( 0) ( ) ( ) ( 0)) = − − →∞ − − Here, we have only used the continuity of the unitary operator π(g). From Theo- rem 3.7, we conclude that β has a continuity point g0 G, so that the continuity of β follows from Lemma 3.8. ∈

Remark 3.10. Proposition 3.9 can be reformulated as follows: We endow the space Z 1 with the topology of pointwise convergence and denote by B1 the sequential (!) closure of the subspace of 1-coboundaries B1. Then, we have B1 Z 1 and one can 1 deﬁne the reduced 1-cohomology space as H1 := Z /B1. ⊆

Another observation is that the assertion of Proposition 3.9 is also true for direct limits of locally compact groups: Assume that G is a topological group with an ascending sequence of locally compact subgroups G G such that G S G . Then, n n+1 = n N n [Yam98, Theorem 2] shows that the direct limit⊆ topology on G is a group∈ topology. We call G a direct limit group if its group topology coincides with this direct limit topology.

Corollary 3.11. Let G = lim Gn be a direct limit group with an increasing sequence of locally compact subgroups G G and G be a unitary representation. −→n n+1 π : U( ) Then every pointwise limit of⊆ a sequence of continuous→ H 1-cocycles is also a continuous 1-cocycle.

Proof. Suppose that β(g) := limn βn(g) is the pointwise limit of a sequence of →∞ continuous 1-cocycles βn : G with n N. The map β : G is a continuous 1-cocycle if and only if all the→ restrictions H β∈ : Gn are continuous→ H 1-cocycles on the locally compact subgroups Gn. Since every Gn→is Ha Baire group (cf. Example 3.2), the latter follows directly from Proposition 3.9 which completes the proof.

To conclude this subsection, we introduce a theorem revealing a certain automatic continuity behaviour of 1-cocycles on Baire groups. Its proof and some more details are given in AppendixA.

Theorem 3.12. Let G be a Baire group and π : G U( ) be a continuous unitary representation of G. Then, for any 1-cocycle β : G → , theH following are equivalent: → H i) β is totally weakly continuous, i.e. there exists a total subset E such that for all v E, the maps G C, g β(g), v are continuous. ⊆ H ∈ → 7→ 〈 〉 ii) β is locally bounded, i.e. there exists an open identity-neighborhood e U G such ∈ ⊆ that supg U β(g) < . ∈ k k ∞ iii) β is continuous. 3.2 Boundedness of topological groups and Rosendal’s Property (OB) 45

3.2 Boundedness of topological groups and Rosendal’s Property (OB)

In the preceding subsection, we have seen that a topological group G has Property (FH) if and only if every continuous negative definite function ψ : G R has a bounded image in R. In this way, we view Property (FH) as a boundedness→ property. For topological groups, one finds a rich diversity of boundedness concepts in the literature and in this subsection, we take a closer look at two such important and wide-spread concepts: The first one is that of a bounded topological group in the sense of Hejcman ([Hei59]) and Atkins ([At91]). Their concept originates in the theory of uniform spaces: A uniform space is bounded if every uniformly continuous map into the real numbers is bounded. Adapting this to the setting of a topological group with its natural (left) uniformity, one obtains the notion of a bounded topological group. Every such group has Property (FH) and there are many interesting unitary Banach– Lie groups which are bounded. However, there is a second, larger class of groups with Property (FH), containing all bounded groups. These are the groups with Rosendal’s Property (OB). They were first introduced by C. Rosendal (cf. [Ro09], [Ro13]) as those groups with the property that every continuous isometric action on a metric space has bounded orbits. It is immediate that these groups have Property (FH). The interesting point is that there exist numerous handy topological characterizations of Property (OB) which do not exist (yet?) in a similar fashion for Property (FH) groups.

Theorem 3.13. For a topological group G, the following are equivalent:

21 i) Whenever G acts continuously by isometries on a metric space , every orbit is bounded.

ii) Whenever G acts continuously by afﬁne isomteries on a Banach space, every orbit is bounded.

22 iii) Any continuous length function ` : G R 0 is bounded, which is to say that the → ≥ image `(G) is a bounded subset of R 0. ≥ iv) Any continuous, left-invariant pseudo-metric d on G is bounded, which is to say that G has a bounded diameter with respect to d.

v If W W ... G S W is an exhaustive sequence of open subsets, then ) 1 2 = k N k ⊆n ⊆ ⊆ ∈ G = Wk for some integers k, n N. ∈ vi) The following two conditions hold: 1.) G cannot be written as a union of an increasing sequence of open proper subgroups.

21We say that G acts continuously on a metric space X if the action on X has continuous orbit maps. 22 1 A function ` : G R is called a length function, if `(g− ) = `(g), `(e) = 0 and `(gh) `(g) + `(h) holds for all g, h G→. Note that this already implies that `(g) 0 for all g G. ≤ ∈ ≥ ∈ 46 3 Topological Aspects

n 2.) If V is an open symmetric generating subset of G, then G = V for some n N. ∈ Proof. The equivalence of i) v) follows from Theorem B.3 with S = G. It remains to verify the equivalence v) −vi): That condition v) implies 1.) and 2.) is readliy ver- ified. Conversely, suppose⇐⇒ that W is an increasing exhaustive sequence of open ( k)k N ∈ 1 subsets of G. We assume w.l.o.g. that e W1. Put WÝk := Wk Wk− and observe that S ∈ ∩ S n WÝk Wßk 1 and that G = k WÝk. Hence, the open subgroups Gk := n WÝk form + N N n an increasing⊆ exhaustive sequence∈ in G, so that 1. implies that G G ∈ S W ) = k = n N Ýk n n ∈ for some k N and 2.) implies that G = WÝk = Wk for some n N. ∈ ∈ Definition 3.14. A topological group G is said to have Property (OB) if it satisfies one of the equivalent conditions of Theorem 3.13. Theorem 3.15. For a topological group G, the following are equivalent:

23 i) Every (left-) uniformly continuous function f : G R is bounded. → ii) For every open symmetric identity neighborhood U G, one finds a finite subset n F G and some n N such that G = FU . ⊆ ⊂ ∈ iii) G has Property (OB) and every open subgroup of G has finite index.

Proof. Apply Theorem B.9 with S = G. Definition 3.16. A topological group G is called bounded if it satisfies one of the equivalent conditions of Theorem 3.15. Example 3.17. i) Any compact group G is bounded: If U is an open symmetric iden- S tity neighborhood, then G = g G gU is an open covering of G which has a finite subcover G = FU. Hence, condition∈ ii) of Theorem 3.15 is satisfied for n = 1. A slight modification of this argument shows that also every pre-compact group G satisfies condition ii) with n = 1. Recall that a group is called pre-compact, if it is a (dense) subgroup of a compact group. A topological group G is called totally bounded if, for every open symmetric identity neighborhood U, there exists a finite subset F G such that G = FU. In particular, every totally bounded group G is bounded.⊆ We have just seen that every compact group is totally bounded and, as mentioned, also every pre-compact group. Weil shows the converse ([Weil37]): The totally bounded groups are precisely the pre-compact groups. One finds quite a number of equivalent characterizations of pre-compactness in the current literature. One of these characterization points out that a group G is pre-compact if and only if every continuous affine isometric action of G on a Banach-space has a fixed point (cf. [NguPe10]). That every pre-compact group G is bounded can also be shown using condition i) of Theorem 3.15: Every uniformly continuous function uniquely extends to a uniformly continuous function on the compact group which contains G as a dense subgroup. Therefore, it must be bounded.

23 A real-valued function f : G R is called (left-) uniformly continuous if, for every " > 0, there → 1 exists an open identity neighborhood U G such that f (g) f (h) < " holds whenever g− h U. Note that in particular, that every uniformly⊆ continuous| function− is continuous.| ∈ 3.2 Boundedness of topological groups and Rosendal’s Property (OB) 47

ii) In view of condition i) in Theorem 3.15, the pseudo-compact groups constitute another class of bounded groups. Recall that a topological group G is called pseudo- compact if every continuous real-valued function on G is bounded. It turns out that this is a subclass of the pre-compact groups: In [CoRo66, Thm. 1.1, Thm. 1.5], the pseudo-compact groups are characterized as those pre-compact groups for which every continous real-valued function is uniformly continuous. Remark 3.18. Assume that G is a connected group with Property (OB). Then, G does not have proper open subgroups, so that Theorem 3.15 implies that G is bounded. Hence, every connected group has Property (OB) if and only if it is bounded and condition vi) of Theorem 3.13 shows that this occurs if and only if, whenever U is an open symmetric identity neighbourhood, the whole group G is a ﬁnite power of U. Some authors, e.g. Neeb in [Ne13], even call a topological group bounded if, for every n open, symmetric identity neighborhood U, there exists an integer n N with G = U . This stronger version of boundedness goes back to the work of Olshanski∈ and Pickrell. Remark 3.19. Observe that, if G is a topological group with Property (OB) and

Φ : G G0 is a surjective continuous group homomorphism. Then, G0 also has Prop- → erty (OB): If `0 is a continuous length function on G0, then ` := `0 Φ is a continuous length function on G and hence bounded. The surjectivity of Φ shows◦ the boundedness of `0. Analogously, if G is bounded, then so is G0. In particular, if G is a topological group with Property (OB) and with identity component G0 (which is a closed normal subgroup in G), then also the quotient group G/G0 has Property (OB). The group G/G0 is discrete if and only if the identity component G0 is open. If this is the case, then G/G0 is either finite or uncountable. A countable discrete group never has Property (OB) because it can be exhausted by an increasing sequence of finite subsets (Theorem 3.13 v)). Thus, a topological group with countably infinitely many open connected components never has Property (OB). This is a curiosity about Property (OB) groups. Analogously, we see that a bounded topological group with open connected components can have at most finitely many connected components. Here, we use that the bounded discrete groups are precisely the finite groups (cf. Proposition 3.21).

The following remark compares the three boundedness concepts Property (FH), Prop- erty (OB) and Bounded Group. Remark 3.20. It is immediate that

BoundedGroup = Proper t y(OB) = Proper t y(FH). ⇒ ⇒ However, the corresponding inclusions are proper:

i) There are topological groups with Property (FH) but without Property (OB): In order to ﬁnd examples, we use the following fact. If G has Property (OB) and π : G GL(E) is a norm-continuous representation on a Banach space E, then π → is bounded in the sense that supg G π(g) < . This follows from the observa- ∈ k 1k ∞ tion that `(g) := log max π(g) , π(g− ) deﬁnes a continuous length function. For n 3, the special{k lineark groups SL(}n, R) all have Property (T) (see e.g. ≥ 48 3 Topological Aspects

[BHV08, Thm. 1.4.15]) and, thus, they have Property (FH) (Corollary 2.18). The n identical representation of SL(n, R) on E = R is an unbounded norm-continuous representation, so that SL(n, R) does not have Property (OB). ii) There are topological groups with Property (OB) but which are not bounded: We find examples within a certain class of groups with Property (OB) which goes back to Bergman ([Berg06]): A group G has strong uncountable cofinality (also called the Bergman property) if, for any increasing chain W W ... G S W 1 2 = k N k ⊆ ⊆ ⊆ ∈ n of subsets covering the whole group G, there exist n, k N such that G = Wk . In other words, the groups with strong uncountable cofinality∈ are precisely the 24 discrete groups with Property (OB) (see also [Ro09, Thm. 2.1]). Observe that no countable group can have strong uncountable cofinality whereas a discrete group is bounded if and only if it is finite. A basic example of a group having strong uncountable cofinality is given by the infinite permutation group S : S = N ∞ of all bijections on N (cf. [Berg06]). This group cannot be bounded as a discrete group because it is not finite.

Among the locally compact groups, the concept of boundedness coincides with the notion of compactness.

Proposition 3.21. For a locally compact group G, the following are equivalent:

i) G is compact.

ii) G has Property (OB) and G is σ-compact. iii) G is bounded.

Proof.i ) ii): If G is compact, then G is trivially σ-compact and G has Property (OB)⇐⇒ since any continuous length function on G is bounded. Suppose, conversely, that G is σ-compact and has Property (OB). Choose an open symmetric identity neighborhood V such that V is compact in G. Note that any compact subset of G can be covered by ﬁnitely many left translates of V . Therefore, since G is exhausted by an increasing sequence of compact subsets, we obtain an increasing sequence of ﬁnite subsets F such that G S F V . Condition ( k)k N = k N k ∈ n n ∈ v) of Theorem 3.13, then shows that G = (FkV ) = (FkV ) . The right-hand side is compact, hence so is G.

24Therefore, Rosendal’s Property (OB) is sometimes called the Topological Bergman property. The expression strong uncountable cofinality can be explained as follows: Every group is the union of a directed system of (finitely generated) subgroups. If it cannot be written as a union of a countable directed system of proper subgroups, then it is said to have uncountable cofinality. In this sense, a group G has strong uncountable cofinality if and only if G has uncountable cofinality and G is Cayley- bounded in the sense that, for every symmetric and generateing subset V , there exists an intger n N n with G = V . From this point of view, groups with strong uncountable cofinality form a counterpart∈ to the countable direct limit groups G = lim Gn. Corollary 7.8 in chapter7 and [BHV08, Prop. 2.4.1] show that countable direct limits of open subgroups−→ are examples of groups which do not have Property (FH). 3.2 Boundedness of topological groups and Rosendal’s Property (OB) 49 i) iii): If G is compact, then G is bounded because every uniformly continuous ⇐⇒real-valued function on G is bounded. Conversely, if G is bounded and V an open n n symmetric identity neighborhood with compact closure, then G = F V = F V for some finite subset F and some integer n N. Again, the right-hand· side· is compact and thus, G is compact. ∈

Analogously to the notion of a bounded group, respectively, a group with Property (OB), we now introduce the notion of a bounded subset S G of a topological group resp. that of a subset S “ with Property (OB)” which we will⊆ call coarsely bounded.

Definition 3.22. Let G be a topological group and S G be any subset of G. ⊆ i) We say that S is a bounded subset if, for any open symmetric identity neighborhood U, one finds a finite subset F G and some integer n N such that S FU n. ⊂ ∈ ⊆ ii) We call S coarsely bounded if, for any increasing exhaustive sequence W of ( k)k N n ∈ open subsets in G, there exist integers k, n N such that S Wk . ∈ ⊆ Remark 3.23. The reader should note that every bounded subset is coarsely bounded and every coarsely bounded subset belongs to the set of those subsets S G for which every continuous real-valued negative definite functionS on G has a bounded⊆ image (cf. Remark B.11). We will also use the fact that finite products of (coarsely) bounded subsets are again (coarsely) bounded subsets. A direct argument can be given as follows: Assume that S1, S2 are bounded subsets of G. Let U be an identity n2 neighborhood. There exists a finite subset F2 and an integer n2 with S2 F2U . By virtue of the continuity of the group conjugation, we find an open identity⊆ neighbor- 1 hood U 0 U with f2− U 0 f2 U for all f2 F2. Now, one finds a finite subset F1 and ⊆ ⊆ n1 ∈ n1 n2 n1+n2 an integer n1 such that S1 F1(U 0) . Hence, S1 S2 F1U 0 F2U F1 F2U . If W is an exhaustive⊆ increasing sequence of· open⊆ subsets· and S⊆, S are both ( k)k N 1 2 ∈ n coarsely bounded, then we find some integers k, n large enough such that S1 Wk n 2n ⊆ and S2 Wk . Hence S1S2 Wk . This shows that the product S1S2 is a (coarsely) bounded⊆ subset whenever S⊆1 and S2 are (coarsely) bounded.

In the following, let G be a Banach–Lie group with Lie algebra g and exponential function exp : g G. For any r > 0 we deﬁne the ball →

Br := x g : x r g. { ∈ k k ≤ } ⊆

We write G0 for the identity component of G. Recall that G0 is an open connected normal subgroup in G.

Lemma 3.24. i) The subset S := exp(Br ) G is bounded for any r > 0. ⊆

ii) If the identity component G0 is bounded, then, for any r > 0, there exists an integer n n = n(r) N such that G0 = exp(Br ) . ∈ 50 3 Topological Aspects

Proof. i) Let U be an identity neighborhood. Since the exponential map is a local diffeomorphism, there exists some " > 0 with exp(B") U. Then, we ﬁnd some r ⊆ n n n N with n ". Hence, Br Bn" = nB" and thus exp(Br ) exp(B") U . ∈ ≤ ⊆ ⊆ ⊆ ii) The subset exp(Br ) G0 contains an open symmetric identity neighborhood U G0. In view of Remark⊆ 3.18, the boundedness of the connected group G0 means⊆ n n that there exists an integer n N with G = U = exp(Br ) . ∈

Proposition 3.25. For a Banach–Lie group G with identity component G0, the following are equivalent: i) G is bounded. ii) G has Property (OB) and G has only finitely many connected components. iii) G0 is a bounded group of finite index. iv) There exists some r > 0, a finite subset F G and an integer n N such that n ⊆ ∈ G = F exp(Br ) . ·

Proof. For a Banach–Lie group G, the identity component G0 is an open, connected, normal subgroup. i) ii): Since G0 is open, the condition that G has only finitely many connected ⇐⇒components is equivalent to the condition that every open subgroup of G has finite index. Therefore, the equivalence of i) and ii) is immediate from Theo- rem 3.15. i) = iii): Let U G0 be any open symmetric identity neighborhood in G0. Since ⇒G0 is open, U⊆is also an open symmetric identity neighborhood in G. Since G is bounded, there exists a finite subset F G and some integer n N with n n ⊆ ∈ G = F U , whence G0 = (F G0) U . This shows that G0 is a bounded group. · ∩ · The particular choice U = G0 yields that G0 has finite index. iii) = iv): Since G0 has finite index, there exists a finite subset F G such that ⇒ ⊆ n G = F G0. Pick any r > 0. By Lemma 3.24 ii), we have G0 = exp(Br ) for some n N and· iv) follows. ∈ iv) = i): Since the subset exp(Br ) is a bounded subset in G (Lemma 3.24 i)) and ⇒since finite product of bounded subsets are bounded subsets (Remark 3.23), it follows that G is a bounded subset in itself and this means that G is a bounded group. 3.2 Boundedness of topological groups and Rosendal’s Property (OB) 51

Problem 3.26. Proposition 3.25 states that every bounded Banach–Lie group has a bounded identity component. Is there a Banach–Lie group with Property (OB) but with an unbounded identity component? Hint: This never happens if G has countably many connected components or if G decomposes as a semi-direct product of its identity component G0 with the group G/G0 of connected components.

Remark 3.27. In general, boundedness (or Property (OB)) cannot be inherited to arbitrary subgroups. If G is bounded and H is a subgroup, then H is bounded as a subset of G but not necessarily as a group. For a counterexample, consider a countably infinite, discrete group G (e.g. G = Z). The group G topologically embeds into 2 the unitary group U(` (G)) via the left regular representation. We shall see later in 2 Proposition 3.29 that U(` (G)) is a bounded group. On the other hand, the discrete group G cannot have Property (OB) because G can be exhausted by an increasing sequence of finite subsets (Theorem 3.13 v)). However, there are a few exceptions: If H is a subgroup whose closure H is a finite index subgroup, then H is bounded (has Property (OB)) if and only if G is bounded (has Property (OB)) (cf. Proposition B.16). In particular, this implies to all dense subgroups and to all open finite index subgroups of G. Note that, if G is a bounded group, then every open subgroup H has finite index (Theorem 3.15 iii)). For Banach–Lie groups with finitely many connected components, Proposition 3.25 reads as follows:

Corollary 3.28. Let G be a Banach–Lie group with ﬁnitely many connected components.

We write G0 for the identity component of G. Then the following are equivalent:

i) G is bounded.

ii) G has Property (OB). iii)G 0 is bounded.

n iv) There exists some r > 0 and some integer n N such that G0 = exp(Br ) . ∈ Using spectral calculus, one easily finds various infinite dimensional unitary groups which are bounded such as the unitary group U( ) of a complex Hilbert space or the orthogonal group O( ) of a real Hilbert space.H The following example is basically [Ne13, Prop 1.3], butH the author uses the stronger concept of boundedness (cf. Remark 3.18). Using the weaker concept of boundedness, the list of bounded groups given in [Ne13, Prop 1.3] can be extended by the orthogonal group O( ) of an infinite dimensional real separable Hilbert space and by its subgroup O H( ) consisting of those orthogonal operators g for whichH the difference g 1 is a compact∞ H operator. − Proposition 3.29. The following groups are bounded:

i) The unitary group U( ) of a von Neumann algebra . A A 52 3 Topological Aspects

ii) The unitary groups U( ) and U ( ) := g U( ) : g 1 ( ) for a complex Hilbert space H. ∞ H { ∈ H − ∈ K H } H iii) The unitary groups U and U for a quaternionic Hilbert space . H( ) H, ( ) H ∞ H H iv) The orthogonal groups O( ) and O ( ) for a real Hilbert space . H ∞ H H Proof. i) Let be a von Neumann algebra. Its unitary group G := U( ) is a Banach–LieA group with Banach–Lie algebra g = u( ), the skew-hermitianA elements, and with exponential function exp : g G givenA by the traditional power series. Below, we use spectral calculus in order→ to show that every unitary element g G is of the form g = exp(X ) for some skew hermitian element X g ∈ ∈ with X π. This shows that G = exp(Bπ) is a bounded group. Recallk thatk ≤ for any normal element X , there exists a unique spectral mea- ∈ A sure resp. a unique spectral integral PX : ∞(spec(X )) with PX (id) = X . For any bounded, measurable function f :L spec(X ) C →C A, we put ⊆ → Z f (X ) := zdPX (Z) := PX (f ). spec(X )

Recall that PX is a -algebra homomorphism so that we have, for bounded mea- ∗ surable functions f , f 0

PX (f + z0 f 0) = PX (f ) + z0 PX (f 0) for z0 C, ∈

PX (f f 0) = PX (f )PX (f 0) and PX (f )∗ = PX (f ).

For the norm of PX (f ), we have the estimate PX (f ) f . For a unitary element g, we have spec(g) T. Let L : T ] kπ, π] ik the ≤k boundedk∞ measurable L ⊆ → − function for which E = id. Then, in view of L = L, the operator X := Pg (L) = R − L(z)dP(z) is skew-hermitian with X π. The spectral integral of X is given T k X k ≤ by PX (f ) = Pg (f L) and we obtain E = PX (E) = Pg (E L) = Pg (id) = g. ◦ ◦ ii) As a special case, we obtain that the unitary group G = U( ) of a complex Hilbert space is bounded. We denote by ( ) := ( H) the closed ideal of compact operatorsH on and claim thatB the∞ H unitaryK Banach–LieH subgroup U ( ) := U( ) (1 + H ( )) is also bounded: To this end, recall that a normal∞ H operatorH X∩ ( B∞) isH compact if and only if its spectrum spec(X ) is ∈ B H discrete in C× := C 0 and PX ( z ) projects onto a ﬁnite-dimensional subspace of for all z spec\{ (}X ) 0 .{ One} uses this fact to conclude that, if g 1 is H ∈ \{ } − compact, then X := Pg (L) is compact. The Banach–Lie algebra of U ( ) is given by the skew-hermitian compact operators and its exponential function∞ H is again given by the traditional power series. This shows that G = U ( ) also ∞ H satisﬁes G = exp(Bπ) so that it is bounded. iii) We want to show that the analogous result for quaternionic unitary groups U ,U corresponding to a Hilbert space over the quaternions . H( ) H, ( ) H H ∞ H 3.2 Boundedness of topological groups and Rosendal’s Property (OB) 53

This is reduced to the complex case by the following observation: Let be a complex Hilbert space and J : a surjective antilinear isometry.H Assume H → H 1 that a unitary operator g U( ) satisfies J gJ − = g. This means that the corre- ∈ H 1 sponding spectral integral Pg satisfies JPg (f )J − = Pg (f ∗) where f ∗(z) := f (z). The bounded measurable map L : T ] π, π] i satisfies L∗ = L on T 1 . → − \{− } Hence, if Pg ( 1 ) = 0, then {− } 1 1 JXJ − = JPg (L)J − = Pg (L∗) = Pg (L) = X .

Now, let be a Hilbert space over the quaternions H with canonical basis 1, i, j, i j Hwhere the elements i, j satisfy the relations { } 2 2 i = j = 1 and i j = ji. − − Let g U and put : ker g 1 , : . We accordingly ob- H( ) 0 = ( + ) 1 = 0⊥ tain a∈ decompositionH g H 1 g . We considerH H as a complex Hilbert = 0 1 1 − H ⊕ H space and note that a complex-linear operator X1 on 1 is H-linear if and only if J X J 1 X for the complex-antilinear surjective isometryH J : j1 on . 1 1 1− = 1 1 = 1 1 Since P 1 0 by construction, we obtain g exp X for someH -linearH g1 ( ) = 1 = ( 1) H skew-hermitian{− } operator X with X and which is compact when g 1 1 1 π 1 1 H is compact. Further, we obtain g =k expk ≤(X ) for the H-linear, skew-hermitian− operator X : i 1 X of norm X . Note that X is compact if g 1 is = π 0 1 π H compact because then,⊕ the 1-eigenspacek k ≤ 0 is ﬁnite-dimensional. This shows− that both groups G U − ,U Hsatisfy G exp B and, thus, are = H( ) H, ( ) = ( π) bounded. H ∞ H

iv) If the real Hilbert space is finite-dimensional, then the group O( ) is a compact group and in particularH a bounded group. For infinite-dimensionalH real Hilbert spaces , the situation is more complicated: For an element g O( ) we likewise considerH the decomposition g 1 g for : ker∈ g H1 = 0 1 0 = ( + ) and : . Consider the complexification− H ⊕of andH g U 1 = 0⊥ 1,C 1 1,C ( 1,C) of g H. An operatorH X on leaves the realH subspaceH invariant∈ ifH and 1 1,C 1,C 1 only if J X J 1 X forH the antilinear isometry J onH which satis- 1,C 1,C 1,−C = 1,C 1,C 1,C fies J 2 1 and whose fixed point set is given by the real subspaceH . Since 1,C = 1 g leaves invariant, we have J g J 1 g . By construction,H we have 1,C 1 1,C 1,C 1,− = 1,C H C Pg 1 0, so that we find a skew-hermitian operator X1, with X1, π, 1,C ( ) = C C J X{−J }1 X and g exp X . Projecting onto the real subspace≤ , 1,C 1,C 1,− = 1,C 1,C = ( 1,C) 1 C H we obtain g1 = exp(X1) for a skew-symmetric operator X1 with X1 π and which is compact whenever g 1 is compact. Now, we focus onk thek ≤operator 1 1 1 − H g0 = on 0 which is the more tricky part. If 0 is infinite-dimensional or finite-dimensional− H with even dimension, then weH find an orthogonal complex 2 structure, i.e. a skew-symmetric bounded linear operator J0 satisfying J0 = 1. − Then X := πJ0 X1 is skew-symmetric with X π and satisfies exp(X ) = g. If 0 is odd-dimensional,⊕ then 1 is infinite-dimensionalk k ≤ because is assumed toH be infinite-dimensional andH we can write H

g 1 g 1 g 1 1 1 . = ( 0 ) 1 = ( 0 1) ( 0 1 ) ( ) − H ⊕ H ⊕ · H ⊕ − H · − H 54 3 Topological Aspects

Then we can choose complex structures J on 1 and with exp(πJ) = 1. 3 H 3 H 1− This shows that g exp(Bπ) and O( ) exp(Bπ) is bounded. If g is ∈ H 1 ⊆ − compact, then the space 0 = ker(g + ) is finite-dimensional, hence either even- or odd-dimensional.H In the odd-dimensional case, the previous argument does not work because the operator 1 is not in O ( ). The deeper reason is that O ( ) has two connected− components (see∞ H[Ne02, Cor. II.15]): ∞ H The identity component O ,0( ) consists of those operators g for which 0 ∞ is even-dimensional and, for elementsH from the other component, 0 is odd-H H dimensional. For elements g O ,0( ), we have g = exp(X ) where the skew- ∈ ∞ H symmetric operator X = πJ0 X1 with X π is compact since J0 lives on the finite-dimensional subspace ⊕ and X kis compact.k ≤ Choose o : 1 1 (with 0 1 = 1 H 1 − ⊕ H 0 being one-dimensional) and put F := , o . Then, O ( ) = F O ,0( ) = H { } ∞ H · ∞ H F exp(Bπ) shows that O ( ) is bounded. If is finite-dimensional, then O(· ) = O ( ) is a compact∞ H group and thereforeH bounded. H ∞ H

The following conclusion is relevant for our discussion of the ﬁrst cohomology spaces of inﬁnite dimensional unitary groups.

Theorem 3.30. Let G U ( ),U( ) for a complex Hilbert space and let (π, ) be a continuous unitary∈ representation { ∞ H H of G.} Then, H

1 H (G, π, ) = 0 . H { }

Proof. The assertion follows from the fact that G is a bounded group (Proposition 3.29 ii)) and from the fact that every bounded group has Property (FH) (Remark 3.20 and Theorem 2.17).

As we shall see in the following chapters, the unitary groups U( ) and Up( ) for 1 p < are not bounded because they have non-trivial 1-cohomology∞ spaces.H ≤ ∞ Remark 3.31. Theorem 3.30 states that the groups U( ) and U ( ) have Property (FH) which is to say Generalized Property (T). It is naturalH to ask∞ H whether they also have Property (T) which is stronger than Property (FH). If is separable, then Bekka shows in [Be03] that the group U( ) indeed has PropertyH (T), even strong Property (T), meaning that U( ) has a ﬁniteH Kazhdan set. In contrast to that, Pestov shows in a recent preprint [Pe17H] that the group U ( ) cannot not have Property (T): This is due to the fact that any extremely amenable∞groupH 25 which admits a faithful continuous unitary representation cannot have Property (T) (cf. [Pe17, Cor. 6.5]). This applies in particular to the group U ( ) which is extremely amenable as shown in [GrMi83] (see also [Pe06, Cor. 4.1.17∞ ]H).

25A topological group G is extremely amenable if every continuous action of G on a compact space admits a ﬁxed point. For details, we refer the reader to [Pe06]. 3.3 A short remark on bounded orbits and ﬁxed points in Banach spaces 55

3.3 A short remark on bounded orbits and ﬁxed points in Banach spaces

One of the central properties of 1-cocycles of unitary representations on a Hilbert space is that every bounded 1-cocycle automatically is a 1-coboundary. This is a consequenceH of the fact that evey continuous affine isometric action on a Hilbert space with bounded orbits has a fixed point (Theorem 2.7). The idea behind this result is that every bounded orbit is contained in a ball B(x, r) := v : x v r which is uniquely characterized by the property that every y { ,∈ for H whichk − thek ball≤ }B(y, r) also contains the orbit, coincides with x. It is easily seen∈ H that, then x must be a fixed point of the affine isometric action. This result is also known as the “Lemma of the centre”. It is natural to convey these concepts to Banach spaces: If A E is a bounded subset of a Banach space E, put r := r(A) := inf s > 0 : ( v E)A ⊂B(v, s) . A point x E is called a Chebyshev center of A, if A {B(x, r).∃ The∈ question⊆ now} is which Banach-spaces∈ have unique Chebyshev centers.⊆ This is the case for every uniformly convex Banach-space (cf. [Am78, Thm. 5]), so that every affine isometric group action on a uniformly convex Banach space with bounded orbits has a fixed point. This result can be extended to all reflexive Banach spaces, but with different means: Recall that every uniformly convex Banach space is reflexive, which is precisely the content of the Milman–Pettis Theorem. As mentioned in [Am78], every reflexive Banach space also admits Chebyshev centres but they are not always unique. Hence, the method above does not work in general. The right tool for reflexive Banach spaces is the Ryll–Nardzewski Fixed Point Theorem (cf. [Ry67]): Theorem 3.32. Let E be a locally convex vector space and S be a semigroup of continuous affine linear endomorphisms on E. 26 Assume that K E is a nonempty, weakly compact, convex and S-invariant subset. Then, K contains an S-fixed⊆ vector if one of the following two conditions holds:

i) S is commutative.

ii) S is noncontracting on K. 27

Proof. First, we observe that we may assume w.l.o.g. that S is a semigroup of linear (!) endomorphisms: Otherwise, consider the locally convex vector space E R and put s.(x, λ) := (s.x s.0 + λs.0, λ) for x E and λ R. Note that s.x s.0 is⊕ linear in − ∈ ∈ − x and that s0.(s.x s.0 + λs.0) = s0s.x s0s.0 + λs0s.0 + (1 λ)s0.0. Hence, we obtain a semigroup of continuous− linear operators− on E R which− is again denoted by S. It is clear that S is commutative on E R if S is commutative⊕ on E. On K 1 , we have s.(x, 1) = (s.x, 1), so that S is noncontracting⊕ on K 1 if S is noncontracting⊕ { } on K. ⊕ { } 26 Pn Pn This means that each s S is a continuous map s : E E satisfying s. i 1 λi xi = i 1 λis.xi for n = = x E and coefﬁcients λ ∈ with P λ 1. Note that→ s is a continuous linear endomorphism if i i R i=1 i = and∈ only if s.0 = 0. ∈ 27The semigroup S is called noncontracting on K if 0 does not belong to the closure of the set s.x s.y : s S whenever x = y and x, y K. Equivalently stated, there exists a continuous seminorm{ p−on E ∈ } 6 ∈ (depending on x and y) for which infs S p(s.x s.y) > 0. ∈ − 56 3 Topological Aspects

If K is a nonempty, weakly-compact, convex and S-invariant subset, then so is K 1 . Finally, S has a ﬁxed point in K if and only if it has a ﬁxed point in K R. ⊕{ } The remainder of the proof is devided into three steps: ⊕

Step 1: Suppose that s : E E is a continuous linear endomorphism leaving a

nonempty, weakly compact,→ convex subset K invariant. Choose x0 K and put x : 1 Pn sk.x for each n . Thus, we obtain a sequence ∈x in n = n+1 k=0 0 N ( n)n N K. The weak compactness of K implies∈ that there exists a point x K such∈ that every weakly open neighborhood of x contains a subsequence of∈ x . Let ( n)n N ∈ f : E R be a continuous linear functional. Since f is bounded on the weakly → compact subset K, we obtain a constant C := supx K f (x) < . For every " > 0, the weakly open x-neighborhood ∈ | | ∞ " \ " Uf ,"(x) := y E : f (y) f (x) < y E : f (s.y) f (s.x) < ∈ | − | 3 ∈ | − | 3 contains a subsequence of x . In particular, we find some index n with ( n)n N N 2C " ∈ ∈ xn Uf ,"(x) and n 1 < 3 . We derive the estimate ∈ + 2" f (s.x) f (x) 3 + f (s.xn) f (xn) | − | ≤ 2" | 1 f sn+−1.x |f x = 3 + n+1 ( 0) ( 0) 2" 2C | − | 3 + n 1 < ". ≤ + The estimate holds for any choices of ", thus showing that f (s.x) = f (x) holds for all continuous linear functionals f . Since the topological dual of a locally convex vector space separates the points, we conclude that s.x = x. Step 2: The first step shows that, for each s S, the subset Ks of s-fixed points in K is nonempty. Note that Ks is again convex∈ and weakly compact (as a weakly

closed subset of K). Next, we show that for ﬁnitely many s1, s2,..., sn S, the n intersection T Ksi is nonempty: If S is commutative, then we have s∈ Ks1 i=1 2( ) Ks1 , so that Step 1 shows that Ks1 Ks2 is nonempty. Therefore, the claim follows⊆

by iteration. If the si are all noncontracting∩ on K, then one considers the mean 1 Pn operator m := n i 1 si which is also a continuous linear operator leaving K = m invariant. It is shown in [Ry67] that every ﬁxed point x K of m is a common ﬁxed point for all si. ∈

S T s Step 3: Consider the subset K := s S K K. In view of Step 2, a standard compactness argument shows that KS∈is nonempty⊆ which is to say that S has a common ﬁxed point in K.

Remark 3.33. If E is a Banach space (or more generally a normed vector space) and K E is a non-empty, weakly compact, convex subset, then any group of afﬁne isometries⊂ on E leaving K invariant is automatically noncontracting on K and thus, has a common ﬁxed point in K.

Theorem 3.34. Let E be a reflexive Banach space. Then, every continous affine isometric action of a topological group G on E with bounded orbits has a fixed point. 3.3 A short remark on bounded orbits and fixed points in Banach spaces 57

Proof. Let A : G E E, (g, v) A(g)v denote such an affine isometric group action of G on E. Denote× by→K the weak7→ closure of the convex hull of some bounded orbit. By construction, K is invariant under the action A and is contained in some closed ball B(0, r) with r > 0. The weak compactness of B(0, r) is equivalent to the reflexivity of the Banach space E. Hence, we conclude that K is a weakly compact, convex and A-invariant subset. Hence, there exists a fixed point in K by Theorem 3.32 and Remark 3.33.

Nonetheless, there are continuous affine isometric actions in (non-reflexive) Banach- spaces with bounded orbits but without fixed points: A simple example is given by the Banach-space E = ( ) of compact operators on a complex, separable Hilbert space . The group G =KU H( ) = U( ) (1+ ( )) acts on E via G E E, (g, T) HgT + 1 g. This defines∞ H a continuousH ∩ affineK isometricH action with× bounded→ orbits.7→ But the only− candidate for a fixed point would be the identity operator which is not in ( ). Nevertheless, what survives is that every continuous affine isometric action ofK aH compact group has a fixed point: Let G be a compact group with normalized, left invariant Haar measure µ and let A : G E E, (g, v) A(g)v be a continuous affine ×E → v 7→R A g d g isometric action on some Banach-space . Then, := G ( )0 µ( ) is a fixed point in E, because we have Z Z Z A(g)v = A(g)A(h)0dµ(h) = A(gh)0dµ = A(h)0dµ = v. G G G This fixed point property remains true for precompact groups, i.e. dense subgroups of compact groups, because the continuous group action extends to the compact closure (cf. [NguPe10, Prop. 2.4]). One can even show that every non-precompact topological group admits a fixed-point free continuous action by affine isometries on a Banach- space (cf. [NguPe10, Thm. 5.8]). Thus, the topological concept of precompactness can be characterized geometrically via a fixed-point property. The next step is to replace the precompact groups by the larger class of amenable groups. Here, we will obtain a similar fixed point theorem but with some constraints on the Banach-space.

In our context, it is natural to deﬁne amenability as a ﬁxed point property.

Definition 3.35. We call a topological group G amenable if it has the following fixed point property: Let G act on a locally convex topological Hausdorff vector space E by continuous affine linear endomorphisms. If K E is a nonempty, compact, convex, G- invariant subset with right-uniformly continuous⊆ orbit maps, then K contains a G-fixed point.

Remark 3.36. We roughly check that our definition of amenability is equivalent to the existence of a left-invariant mean, which is the more convenient way to define amenability: For a topological group G, we denote by RUCB(G) the linear space of right-uniformly continuous, bounded functions ξ : G R. We consider the linear space E of all linear functionals I : RUCB(G) R, endowed→ with the weak- -topology. We call a linear functional I E a mean of G →if it is monotone (i.e. I(f ) 0∗ for f 0) ∈ ≥ ≥ 58 3 Topological Aspects and normed in the sense that I(1) = 1. The Banach–Alaoglu Theorem shows that the subset K of all means is a weak- -compact subset. It is nonempty since every functional I(f ) := f (g) with g G is∗ a mean. The group G acts naturally by continuous ∈ linear endomorphisms on E via g.I(ξ) := I(ξg ), where ξg denotes the right-uniformly continuous function ξg (h) := ξ(gh) (for g, h G and ξ RUCB(G)). If I is a mean, then the orbit map g g.I is right-uniformly∈ continuous∈ w.r.t. the weak- -topology on E. Hence, if G is amenable,7→ it admits a fixed point in K, i.e. a mean ∗I with the property that I(ξg ) = I(ξ) for all g G and ξ RUCB(G). Any such mean is called left-invariant. Conversely, assume that∈ G admits∈ a left-invariant mean I and that K is a nonempty, convex, compact, G-invariant subset of a locally convex vector space E on which G acts be continuous affine linear endomorphisms. Pick some x0 K. If the orbit map g g.x0 is right-uniformly continuous, then, for any continuous∈ lin- 7→ ear functional f : E R, the map g f (g.x0) is an element of RUCB(G) and we → 7→ obtain a linear functional f I(g f (g.x0)). Applying the Hahn–Banach Separa- 7→ 7→ tion Theorem, one finds (a unique) x K for which I(g f (g.x0)) = f (x) holds for all f . Then x is a G-fixed vector: Since∈ G acts by continuous7→ affine linear endomorphisms, the map x g.x g.0 is a continuous linear endomorphism. Hence, for every continuous linear7→ functional− f and every g G, we have ∈ f (g.x) f (g.0) = I(h f (gh.x0) f (g.0)) = I(h f (h.x0)) f (g.0)I(1) − 7→ − 7→ − = f (x) f (g.0). − Since the continuous linear functionals on a locally convex vector space separate the points, we conclude g.x = x for every g G. For more details on the proof, the reader is referred to [Ri67, Thm. 4.2]. ∈ Example 3.37. 1.) Every compact group is amenable: If the group G is compact and K E is a nonempty, compact, convex, G-invariant subset with continuous orbit maps,⊆ then, for x, y K with x = y, the subset g.x g.y : g G is compact in E and does not∈ contain 0.6 Hence, the action{ of− the compact∈ group} G is noncontracting on K and thus has a fixed point in K by Theorem 3.32. An important consequence of the amenability of compact groups is the existence of a left-invariant Haar-measure. For compact groups, the space RUCB(G) and the space C(G) of continuous functions on G coincide because continuous functions on compact spaces are uniformly continuous. If I is a left-invariant mean, then I f R f d we have ( ) = G µ by the Riesz-Markov Representation Theorem, where µ is the Haar measure on G.

2.) Every abelian group is amenable by Theorem 3.32.

3.) Every open subgroup of an amenable group is amenable: This is the content of [Ri67, Thm. 3.2]. If H G is an open subgroup, then every right-uniformly continuous bounded function⊆ f : H R extends to a right-uniformly continuous bounded function on G via f (g) :=→0 for all g / H. In this sense we may view RUCB(H) as a subspace of RUCB(G). If there∈ exists a left-invariant mean I of G on RUCB(G), then its restriction to the subspace RUCB(H) is a left-invariant mean of H. 3.3 A short remark on bounded orbits and ﬁxed points in Banach spaces 59

4.) If Φ : G G0 is a continuous sujective group homomorphism and G is amenable, → then so is G0. In particular, amenability is inherited to quotient groups.

5.) If D G is a dense subgroup, then D is amenable if and only if G is amenable: This⊆ follows from the fact that every uniformly continuous and bounded function on D has a unique uniformly continuous extension to G. Thus, we may identify the spaces RUCB(D) with RUCB(G).

6.) If there exists an exhaustive increasing sequence G G ... G S G of 1 2 = n N n ∈ amenable subgroups Gn, then G is amenable: Let K⊆and ⊆E as in⊆ Definition 3.35 and denote by Kn the nonempty compact subset of Gn-fixed vectors in K. The Kn form a decreasing sequence of compact subsets so that, by compactness of K, the subset K : T K is nonempty. Every x K is a G-fixed point. = n N n ∞ ∈ ∈ ∞ 7.) For a separable, complex Hilbert space , the unitary groups U( ) := lim U(n) 1 H ∞ and Up( ) := U( ) ( + p( )) for 1 p are amenable: This−→ is a consequenceH of 1.)H, 5.)∩and 6.)B. H ≤ ≤ ∞

Theorem 3.38. Let E be a dual Banach-space in the sense that it is the (topological) dual of some Banach space F and suppose that G is an amenable group. Then, every continous group action of G on E by weak- -continuous afﬁne isometries and with bounded orbits has a ﬁxed point. ∗

Proof. Let A : G E E be a continuous affine isometric action of the amenable group G on the Banach× space→ E. Let v E and denote by K the weak- -closure of the convex hull of the bounded orbit A(G, v∈). By the Banach–Alaoglu Theorem,∗ the convex subset K is compact in the weak- -topology of E. Since G acts continuously by isometries, every orbit in K is automatically∗ right-uniformly continuous w.r.t. the norm of E and, in particular, right-uniformly continuous in the weak- -topology on E. By assumption, every affine isometry A(g) is weak- -continuous so that∗ the amenability of the group G implies the existence of a fixed point∗ in K.

Remark 3.39. The requirement of E being a dual Banach space cannot be dropped as our Example 3.37 7.) of the amenable group G = U ( ) acting on the Banach space ( ) = ( ) by bounded orbits but without fixed∞ H vectors shows. B∞ H K H Theorem 3.40. Let π : G Iso(E) be a continuous isometric action of a topological group G with Property (OB)→ on a Banach space E. In the following three cases, we have 1 H (G, π, E) = 0 : { } i) The group G is amenable, E is a dual Banach space and every affine map π(g) : E E is weak- -continuous. → ∗ ii) The Banach space E is reflexive. iii) The group G is precompact. 60 3 Topological Aspects

1 Proof. For any β Z (G, π, E), we consider the continuous affine isometric action A(g)v = π(g)v +β∈(g). Then the vector v E is a fixed point of A if and only if β(g) = v π(g)v holds for all g G. Since G has∈ Property (OB), the action A has bounded orbits.− Then Theorem 3.38∈ , Theorem 3.34 and [NguPe10, Prop. 2.4] respectively imply that the action A has a fixed point in E and thus, that β is a 1-coboundary.

Remark 3.41. The requirement of π being isometric may be replaced by the condition that π : G GL(E) is a continuous representation for which supg G π(g) < . If this is the→ case, then we may replace the norm on E by the∈ equivalentk k norm∞ k·k v supg G π(g)v with respect to which π is isometric. Note that the boundedness condition7→ ∈ isk automaticallyk satisﬁed if G is compact and π : G GL(E) is a continuous linear representation (Proposition B.6). →

Corollary 3.42. Let π : G GL(E) be a continuous linear representation of a compact 1 group G on a Banach space→ E. Then, H (G, π, E) = 0 . { } Proof. In view of the preceding remark, this follows directly from Theorem 3.40. An alternative argument goes as follows: Let β : G E be a continuous 1-cocycle. Then its image β(G) is a compact subset in E. Hence,→ the closed convex hull K of β(G) is also compact. The afﬁne action A(g)v := π(g)v + β(g) leaves K invariant. That A has a ﬁxed point in K is now a direct consequence of G being amenable. 61

4 Lie Theoretic Aspects

The aim of Lie theory is to establish a bridge between global, analytic properties of a Lie group G and local, algebraic properties of its Lie algebra g, which is the tangent space Te(G) at the group identity e of G. This works particularly well for finite-dimensional Lie groups, where we have a so-called exponential map exp : g G. If a Lie group is modelled on an infinite-dimensional locally convex space, so that→ its Lie algebra is a locally convex Lie algebra, then one cannot expect to have a smooth exponential map which is a local diffeomorphism, as in the finite dimensional case, and which describes all smooth one-parameter homomorphisms R G in terms of a generator sitting in the Lie algebra. The problem is that vector fields→ on locally convex manifolds need not posses integral curves. If one imposes the existence of such an exponential function, then one calls the corresponding Lie group a locally exponential Lie group. It is remarkable that the cornerstones of finite dimensional Lie Theory (such as the Automatic Smoothness Theorem or the Integrability Theorem) carry over to locally exponential Lie groups. We shed light on this in Subsection 4.1 and we use these results in Subsection 4.2 to transfer the problem of finding nontrivial 1-cocycles on a Lie group G to its Lie algebra g: Let π : G U( ) be a norm-continuous unitary representation of a connected locally exponential→ LieH group G. Then, every continuous 1-cocycle β : G is smooth and its derivative → H

d dβ(X ) := β(exp(t X )) for X g d t t=0 · ∈ deﬁnes a continuous Lie algebra 1-cocycle corresponding to the derived representation 1 1 dπ of π. The derivative d maps the space Z (G, π) into Z (g, dπ) and we will see that G it is a linear isomorphism either if G is 1-connected or if = 0 (Proposition 4.34). This means that, in these two cases, the converse passage,H namely{ } from the Lie algebra level to the group level, also works.

4.1 The class of locally exponential Lie groups

The class of locally exponential Lie groups contains all unitary and linear Lie groups that we are dealing with in this thesis. This is our motivation to brieﬂy recall the most important Lie theoretic results concerning locally exponential Lie groups. As we shall see, the requirement of a locally diffeomorphic exponential map ensures that many important results from the theory of Banach–Lie groups carry over to locally exponential Lie groups, such as the Automatic Smoothness Theorem, the Lie Subgroup Theorem or the Integrability Theorem. Most of this material is taken from [Ne06]. We require the reader to be familiar with the standard analytical concepts of locally convex topological vector spaces and locally convex manifolds (see e.g. [KM97]). Deﬁnition 4.1. i)A locally convex Lie group G (which we will simply call a Lie group) is both a group and a smooth (real) manifold modeled over a (real) locally convex vector space such that the group multiplication and the group inversion 62 4 Lie Theoretic Aspects

are smooth maps.

A map ϕ : G G0 of (locally convex) Lie groups is called morphism of (locally convex) Lie groups→if it is a smooth group homomorphism. It is called an isomorphism 1 of (locally convex) Lie groups if it is bijective and both ϕ and ϕ− are morphisms of (locally convex) Lie groups.

ii)A locally convex Lie algebra g over the ﬁeld K R, C is both a Lie algebra and a locally convex topological K-vector space such∈ that { the} Lie bracket of g is continuous.

A map ψ : g g0 of locally convex Lie algebras is called a morphism of locally convex Lie algebras→ if it is a continuous Lie algebra homomorphism. It is called an 1 isomorphism of locally convex Lie algebras if it is bijective and both ψ and ψ− are morphisms of locally convex Lie algebras.

For every (locally convex) Lie group G with group identity e G, the tangent space ∈ Te(G) can be endowed with a natural locally convex Lie algebra structure. In fact, every element X Te(G) uniquely extends to a left-invariant vector field X l : G ∈ → T(G) such that X l (e) = X . Hence, Te(G) can be identified with the Lie (sub-) algebra of left-invariant vector fields of G. Moreover, Te(G) can be identified with the underlying locally convex vector space E of the manifold structure of G by the tangent map Te(ϕ) : Te(G) E of a local chart ϕ : U E in an open e-neighborhood. Note that Te(ϕ) is → → a linear isomorphism. One shows that the Lie bracket on Te(G) is continuous for the locally convex vector topology inherited from E (cf. [Ne06, Rem. II.1.8]). Definition 4.2. Let G be a (locally convex) Lie group.

i) The locally convex Lie algebra L(G) := Te(G) is called the Lie algebra of G. If ϕ : G G0 is a morphism of (locally convex) Lie groups, then we write L(ϕ) := → Te(ϕ) : L(G) L(G0) for the corresponding tangent map, which is a morphism of locally convex→ Lie algebras28.

ii) We call a smooth map expG : L(G) G an exponential map of G if, for every → X L(G), the smooth curve γX (t) := expG(tX ) is a one-parameter group with ∈ γ0X (0) = X . iii) If the Lie group G has an exponential map expG which, in addition, is a local diffeomorphism in 0 L(G), then we say that G is locally exponential resp. a locally exponential Lie∈ group.

28 This follows from the usual argument with related vector ﬁelds: For g G and X , Y Te(G), we ∈ ∈ have X l (g) = Te(λg )X and Yl (g) = Te(λg )Y . Since ϕ : G G0 is a group homomorphism, we ﬁnd → ϕ λg = λϕ(g) ϕ which entails ◦ ◦ T(ϕ) X l = (L(ϕ)X )l ϕ and T(ϕ) Yl = (L(ϕ)Y )l ϕ ◦ ◦ ◦ ◦ and thus T(ϕ) [X l , Yl ] = [(L(ϕ)X )l , (L(ϕ)Y )l ] ϕ. ◦ ◦ Evaluation at g = e yields L(ϕ)[X , Y ] = [L(ϕ)X , L(ϕ)Y ]. 4.1 The class of locally exponential Lie groups 63

Remark 4.3. i) If G is a Lie group, then any smooth one parameter group γ : R G → is uniquely determined by its value X := γ0(0). This follows from the so called Uniqueness Lemma 29 and reveals that there exists at most one exponential func-

tion expG. In fact, one deﬁnes the exponential function via expG(X ) := γX (1), provided that γX exists for all X g. ∈ ii) Suppose that G is a connected locally exponential Lie group with exponential map G G G X X G G expG : L( ) . Then 0 := expG( ) : L( ) Grp is an open subgroup in and thus coincides→ with G. This shows that∈ connected locally exponential Lie groups are generated (as a group) by the image of the exponential map.

Theorem 4.4. [Automatic Smoothness Theorem] Let G, G0 be locally exponential Lie groups and ϕ : G G0 be a group homomorphism. The following statements are equivalent: →

i) ϕ is continuous.

ii) ϕ is smooth. iii) There exists a continuous homomorphism ψ : L(G) L(G0) of Lie algebras such → that ϕ expG = expG ψ ◦ 0 ◦ If this is the case, then we have ψ = L(ϕ).

Proof. cf. [Ne06, Thm IV.1.18] and [Gloe02, Prop. 2.4]

Remark 4.5. Note that the Automatic Smoothness Theorem generalizes the fact that every continuous one parameter group γ : R G is smooth and of the form γ(t) = → expG(tX ), where X := γ0(0) L(G), which lies at the heart of its proof. ∈ Note that each Lie group is a topological group. We adopt the convention that a group topology is Hausdorff.

Corollary 4.6. Every topological group G carries at most one structure of a locally exponential Lie group.

Lie subgroups

Deﬁnition 4.7. We call a closed subgroup H G of a locally exponential Lie group G a locally exponential Lie subgroup (or simply a Lie⊆ subgroup) if H carries a (unique) locally exponential Lie group structure for which the underlying group toplogy coincides with topology induced by G.

29 The Uniqueness Lemma can be found e.g. in [Ne06] as Lemma II.3.5. One applies this Lemma with N = R, viewing G as the subgroup of all left translations λg in Diff(G), and ﬁnds that there exists at most one smooth one parameter group γX : R G with γ0X (0) = X . → 64 4 Lie Theoretic Aspects

Theorem 4.8. [Lie Subgroup Theorem] Let G be a locally exponential Lie group and H G be a closed subgroup. Then, ⊆ e L (H) := X L(G) : expG(RX ) H { ∈ ⊆ } is a closed Lie subalgebra of L(G) and H is a locally exponential Lie subgroup if and only if there exists an open 0-neighborhood V L(G) such that expG V is a diffeomorphism ⊆ e | onto an open e-neighborhood in G and expG(V L (H)) = expG(V ) H. We then have e H H and e . ∩ ∩ L( ) = L ( ) expH = expG L (H) | Proof. See e.g. [Ne06, Lemma IV.3.1, Thm. IV.3.3] and [Gloe02, Prop. 2.13]. Alter- natively, see [GlNe18, Prop. 8.3.3, Lemma 8.3.10, Prop. 8.3.11].

The following are special cases of the Lie Subgroup Theorem:

Corollary 4.9. Let G, G0 be a locally exponential Lie groups.

i) Let ϕ : G G0 be a smooth group homorphism. If H0 G0 is a Lie subgroup of G0, → 1 ⊆ 1 then H := ϕ− (H0) is a Lie subgroup of G with Lie algebra L(H) = L(ϕ)− (L(H0)). In particular, H := ker(ϕ) is a Lie subgroup with Lie algebra L(H) = ker(L(ϕ)). ii) Let θ : G G be a smooth automorphism of G. Then, the ﬁxed point set → θ H := G := g G : θ(g) = g { ∈ } is a Lie subgroup with Lie algebra

L θ L(H) := X L(G) : L(θ)(X ) = X = LG ( ). { ∈ } Proof. i) This is a straight forward application of Theorem 4.8 (see [GlNe18, Prop. 8.3.12] or [Ne06, Prop. IV.3.4]). ii) Consider the smooth group homomorphism

ϕ : G G G, g (θ(g), g). → × 7→ Using Theorem 4.8, one easily shows that the diagonal group ∆G := g, g G G : g G G is a Lie subgroup of the product Lie group G G with ( ) ∼= Lie{ algebra∈ ×∆g g∈. The} ﬁxed point group can be expressed as Gθ ϕ 1×∆G so ∼= = − ( ) that the assertion follows from i).

Universal covering groups

Deﬁnition 4.10. Let G, Ge be topological groups and q : Ge G be a continuous group homomorphism. Then, q is said to be a covering morphism→ of topological groups if it is an open, surjective map with discrete kernel. 4.1 The class of locally exponential Lie groups 65

Deﬁnition 4.11. We call a topological group G 1-connected if it is path-connected and simply-connected30. A topological group G is said to be semilocally 1-connected if there exists an open path-connected e-neighborhood U such that each loop f : [0, 1] U with f (0) = f (1) = e is homotopic in G to the constant loop at e. → Theorem 4.12. If G is a connected and semilocally 1-connected topological group, then there exists a 1-connected topological group Ge and a covering morphism q : Ge G of → topological groups. The pair (Ge, q) is unique in the sense that, if (Ge0, q0) is another such pair, then there exists an isomorphism ϕ : Ge Ge0 of topological groups with q = q0 ϕ. → ◦ Therefore, we call Ge the universal covering group, q the universal covering morphism and the pair (Ge, q) the universal covering of G.

Proof. See e.g. Sätze 78, 79 and 80 in [Pon58]. See also Theorems III.8.1, III.8.4 and Exercise III.8.5 in [Bre93].

Note that each (locally convex) Lie group is semilocally 1-connected since we can ﬁnd an open e-neighborhood which is homeomorphic to some open convex subset of a locally convex vector space. Proposition 4.13. Let q : Ge G be a covering morphism of topological groups. If Ge or G is a Lie group, then the other→ group has a unique Lie group structure such that q is a morphism of Lie groups which is a local diffeomorphism.

Proof. This is [Ne06, Cor. II.2.4].

We conclude that every connected (locally convex) Lie group G has a universal covering (Ge, q) consisting of a 1-connected (locally convex) Lie group Ge and a smooth covering morphism q : Ge G which is a local diffeomorphism. This implies in particular that the tangent map→L(ϕ) : L(Ge) L(G) is an isomorphism of locally convex Lie algebras. This allows us from now on→ to identify these two Lie algebras L(Ge) = L(G) q G and assume that L( ) = idL(G). The universal covering group e is uniquely determined up to isomorphism. The existence of the universal cover also reveals that every connected Lie group is the quotient of a 1-connected Lie group by some discrete normal subgroup. In order to show that the universal covering group Ge is locally exponential whenever G is, we need the Lifting Theorem. Theorem 4.14. [Lifting Theorem] Let W be a nonempty path-connected, locally path- connected and simply-connected topological space and suppose that q : Ge G is a cover- → ing morphism of topological groups. If f : W G is a continuous map with f (w0) = e → for some w0 W, then there exists a continuous map fe : W Ge for which the following diagram commutes∈ → f W G.

q fe Ge

30This means that each loop is homotopic to the constant path. 66 4 Lie Theoretic Aspects

The map fe is uniquely deﬁned by the conditions

f = q fe and fe(w0) = e. ◦ Proof. This follows directly from [Bre93, Theorem III.4.1].

If G is a connected locally exponential Lie group with exponential function exp : L(G) G, then the Lifting Theorem (with W = L(G) = L(Ge)) shows that → exp factors through the covering morphism q by a continuous function expg : L(G) = L(Ge) Ge. Since q is smooth and a local diffeomorphism, we conclude that expg is smooth→ and a local diffeomorphism at 0, both properties being inherited from exp. 2 Applying the uniqueness assertion of the Lifting Theorem to W = R and 1 fe(s, t) := expg(sX )expg(tX )[expg((s + t)X )]− for some fixed X L(Ge), we find that q fe e, so that fe is constant, revealing ∈ ◦ ≡ that expg(tX ) defines a homomorphism. Thus, the universal covering group carries a unique locally exponential Lie group structure and we have

exp = q exp.g ◦ The Lifting Theorem is also used to show that ker(q) is isomorphic to the fundamental group π1(G) of the Lie group G: Suppose that q : Ge G is a covering morphism of → topological groups and suppose that Ge is 1-connected. For every eg ker(q), one can ∈ choose a continuous path γeg : [0, 1] Ge from e to eg because Ge is path-connected e → by assumption. Then, the path γg := q γeg is a continuous loop in G and we deﬁne e ◦ e ϕ(eg) := [γg ] π1(G). By virtue of Ge being simply connected, this deﬁnes a map e ∈ ϕ : ker(q) π1(G), eg [q γeg ]. → 7→ ◦ e 1 Indeed, if ηg is another path from e to g, then the loop γ ηg is homotopic to the ee e e e−eg ee constant curve so that ∗ 1 1 1 . = γe−g ηeg = [γeg ]− [ηeg ] e ∗ e e e Hence, we have γ η and therefore γ η . [eeg ] = [ eeg ] [ eg ] = [ eg ] Proposition 4.15. The map ϕ : ker q π G , g q γ is an isomorphism of ( ) 1( ) e [ eeg ] groups. → 7→ ◦

Proof. If g , g ker q , then we can choose γ : γ γ . This shows that ϕ is a e1 e2 ( ) eeg1 eg2 = eeg1 eeg2 ∈ · · group homomorphism. If [γ] π1(G), then there exists a unique path γe : [0, 1] Ge ∈ → with γ = q γe and γe(0) = e by the Lifting Theorem. We have γe(1) ker(q) since ◦ ∈ γ(1) = e. Therefore, ϕ is surjective. If [γ] = [η] π1(G), then there exists a homo- 2 topy H : [0, 1] G with ﬁxed endpoints and such∈ that γ = H( , 0) and η = H( , 1). 2 By the Lifting Theorem,→ there exists a unique homotpoy He : [0, 1]· Ge with H = q· He → ◦ and He(0, 0) = e. The continuous curves t He(0, t) and t He(1, t) have their image in the discrete subset ker(q), hence they are7→ both constant7→ maps. This shows that the curves γe := He( , 0) and ηe := He( , 1) satisfy γe(0) = ηe(0) = e and γe(1) = ηe(1). In particular, this argument· shows that· ϕ is injective. 4.1 The class of locally exponential Lie groups 67

Theorem 4.16. [Integrability Theorem] Let (G, g, exp) be a connected locally exponential Lie group with universal covering (Ge, q) and let G0 be a Lie group with exponential function exp0 : L(G0) G0. If ψ : L(G) L(G0) is a continuous homomorphism of Lie algebras, then there→ exists a unique smooth→ group homomorphism : G G ϕe e 0 with L . There exists a unique smooth group homomorphism : G G→with (ϕe) = ψ ϕ 0 → L(ϕ) = ψ if and only if ker(q) ker(ϕe). ⊆

Proof. This is [Ne06, Thm. IV.1.19] together with the observation that our convention q L( ) = idL(G) implies the equivalence

ϕe = ϕ q L(ϕe) = L(ϕ) ◦ ⇐⇒ for any two smooth group homomorphisms : G G and : G G . ϕe e 0 ϕ 0 → → Lemma 4.17. Let G be a connected locally exponential Lie group with Lie algebra g := L(G) and exponential map exp : g G. Let Ge be the 1-connected universal cover- → ing group of G with exponential map expg : g Ge and with universal covering map q : Ge G. Suppose that H G is a connected Lie→ subgroup with Lie algebra h := L(H) and assume→ that the following⊆ two conditions are satisﬁed:

h i) ker(q) expg( ) Grp. ⊆ 〈 〉 ii) The inclusion map ι : H , G induces a group isomorphism →

π1(ι) : π1(H) π1(G), [γ] [ι γ] → 7→ ◦ of the corresponding fundamental groups.

Then, 1 h He := q− (H) = expg(X ) : X Grp 〈 ∈ 〉 is a 1-connected Lie subgroup of G and q : H H is a universal covering map of H. e He e | → Note that, for ﬁnite-dimensional Lie groups, Lemma 4.17 follows from the Homotopy 31 Group Theorem in [HiNe12, Thm. 11.1.15, Cor. 11.1.4, Prop. 11.1.10]

1 Proof. That He = q− (H) is a Lie subgroup follow from Corollary 4.9. In view of exp = h q exp,g the group He contains the subgroup expg( ) Grp. By assumption, ker(q) is contained◦ in this generated subgroup and we conclude〈 〉 that q 1 H exp h . In − ( ) = g( ) Grp 〈 〉 31 In the ﬁnite-dimensional case, our condition i) may be dropped: As stated in Remark 11.1.12 in [HiNe12], the map Ge G G/H factors to a covering qG/H : Ge/H1 G/H, where H1 denotes the identity component of→q 1 H→. The condition ker q exp h H→means that the covering q − ( ) ( ) g( ) Grp 1 G/H is an isomorphism, which is to say that G/H is simply⊆ 〈 connected.〉 ⊆ The diagram in Theorem 11.1.15 reveals that the vanishing of the fundamental group π1(G/H) is equivalent to the surjectivity of π1(ι) : π1(H) π1(G). Thus, our condition i) means that π1(ι) is surjective. → 68 4 Lie Theoretic Aspects particular, He is (path-) connected. To see that He is also simply-connected, consider the following commuting diagrams

π ι eι 1(e) He Ge π1(He) π1(Ge) = 0 { } q q π q π q He 1( He ) 1( ) | | ι π1(ι) H G π1(H) π1(G). ∼

Let γe be a continuous loop in He starting at the identity e. The commutativity of the right-hand diagram and the assumption that π1(ι) is an isomorphism shows that π q γ q γ 1 in π H . This means that there exists a homotopy in H 1( He )[e] = [ e] = 1( ) | ◦ of continuous loops from γ := q γe to the constant curve η e. Keeping in mind that continuous maps into the discrete◦ subgroup ker(q) Ge are≡ constant, we obtain ⊂ a homotopy of loops in Ge from γe to the constant curve ηe e by the Lifting Theorem (Theorem 4.14). The image of this lifted homotpy lies in H≡ q 1 H , showing that e = − ( ) γe is homomotpic to the constant curve in He. We conclude that He is 1-connected. In a last step, we show that He is isomorphic to the universal covering group of H which we denote by He0. Let qH : He0 H be the universal covering morphism. In view of the Integrability Theorem (Theorem→ 4.16), there exists an ismomorphism of Lie groups

ϕ : He He0 such that the following diagram commutes: → q He |

ϕ qH He He0 H. ∼

h id h id h

In particular, we conclude that q q is a covering morphism. This completes He = H ϕ the proof. | ◦

Since Banach–Lie algebras, ﬁnite-dimensional Lie algebras and Fréchet Lie algebras are special cases of locally convex Lie algebras, we introduce the following subclasses of (locally convex) Lie groups.

Definition 4.18. We call a Lie group G a Banach–Lie group (a finite-dimensional, Fréchet, ... Lie group) if its Lie algebra L(G) is a Banach–Lie algebra (a finite-dimensional, Fréchet, ... Lie algebra).

Remark 4.19. Every Banach–Lie group is locally exponential: If G is a Banach–Lie G G T group with exponential function expG : L( ) , then e(expG) = idL(G) follows from → the requirement that every smooth one-parameter curve γX (t) = expG(tX ) satisﬁes γ0X (0) = X . That expG is a local diffeomorphism in 0 now follows from the Inverse Function Theorem.

The unit group of a Mackey-complete -algebra ∗ A 4.1 The class of locally exponential Lie groups 69

All unitary groups appearing in our thesis can be viewed as a Lie subgroup of the unit group of certain unital -algebra: Let be a complex locally convex algebra. This means that is a complex∗ associative algebraA and carries a locally convex (Hausdorff) vector topologyA for which the bilinear multiplication , (a, b) ab is continuous 32. The algebra may or may not haveA a multiplicative × A → A unit7→ element which we will denote by 1. IfA does not have such a unit element, then we deﬁne unitization A its to be the direct sum + := C with algebra multiplication given A ⊕ A by (λ, a)(µ, b) := (λµ, λb + µa + ab). Clearly, + is a unital complex locally convex algebra. A

Deﬁnition 4.20. Let be a complex locally convex algebra. A i) If is unital, we call a continuous inverse algebra (resp. a CIA for short) if the A A 1 1 1 group × := a : ( a− )aa− = a− a = 1 of invertible elements in A { ∈ A ∃ ∈ A } 1 A is an open subset of and if the inversion ι : × , a a− is a continuous A 1 A → A 7→ map. Note that the multiplicative inverse a− is unique whenever it exists. If continuous inverse algebra33 A is not unital, then we call a if its unitization + is a CIA. A A

ii) We call Mackey complete if for every smooth curve ξ : [a, b] R there A ⊆ → A exists a smooth curve η : [a, b] such that η0 = ξ. → A Remark 4.21. i) There are numerous equivalent characterizations of Mackey completeness (cf. [KM97, Thm 2.14]). One of the characterizations states that is Mackey complete if and only if every Mackey–Cauchy sequence 34 in con-A verges. Since every Mackey–Cauchy sequence is a Cauchy sequence, we concludeA that every (sequentially) complete algebra is Mackey complete. A ii) Let be Mackey complete without multiplicative unit. Since every smooth curve Aa b ξ+ : [ , ] + = C is of the form ξ+ = ξ1 ξ, we conclude that the Mackey → A ⊕A ⊕ completeness of carries over to the unitization +. A A In the following, we write GL( ) := × for the group of invertible elements in a unital Mackey complete CIA. ForA a non-unitalA Mackey complete CQIA , we put A GL( ) := ( +)× ( 1 ). Our aim is to construct a natural locally exponential Lie groupA structureA ∩ on{ } the × A unit group GL( ). Here, the Mackey completeness plays a fundamental role because it provides aA Holomorphic Calculus which works as for Banach algebras and enables one to construct exponential and logarithm functions.

a If is unital, then we put + := . For we denote by A A A ∈ A Sp a a ( ) := λ C : λ1 / ( +)× { ∈ − ∈ A } 32 Some authors, e.g. H. Biller in [Bil10], only require separate continuity instead of joint continuity. 33Some authors use the expression continuous quasi-inverse algebra to distinguish the unital case from the non-unital one. We will not need this distinction in our thesis. 34A sequence a is called a Mackey–Cauchy sequence if there exists a bounded (absolutely convex) ( n)n N subset B and a net∈ µ converging to 0 such that a a µ B. ( nm)n,m N R n m nm ⊆ A ∈ ⊂ − ∈ 70 4 Lie Theoretic Aspects

spectrum of a the . The symbol ( +)× stands for the group of invertible elements in +. A A If Ω C is an open subset, then we write Ω for the algebra of holomorphic functions f : Ω⊆ C, equipped with the locally convexO topology of uniform convergence on compact→ subsets of Ω.

Theorem 4.22. [Holomorphic calculus] Let be a unital Mackey complete CIA and A Ω C be an open subset. Then Ω := a : Sp(a) Ω is an open subset of and ⊆ A { ∈ A ⊆ } A for all a Ω, the spectrum Sp(a) is a nonempty compact subset of C. For each a Ω, the following∈ A statements hold: ∈ A

i) There exists a unique continuous homomorphism of unital algebras Ω , O → A f f (a) which sends idΩ Ω to a Ω. Moreover, the map Ω Ω , 7→ ∈ O ∈ A O × A → A (f , a) f (a) is smooth. In particular, for each f Ω, the map Ω , a f7→(a) is smooth. ∈ O A → A 7→ ii) We have Sp(f (a)) = f (Sp(a)). P n iii If f has a power series expansion f z ∞ c z z on Ω, then the ) Ω ( ) = n=0 n( 0) ∈P O n − series ∞n 0 cn(a z01) converges to f (a). = − iv) If Ω, Ω0 are open subsets in C and f Ω, g Ω are holomorphic functions such ∈ O ∈ O 0 that f (Ω) Ω0, then we have (g f )(a) = g(f (a)). ⊆ ◦ The relation Sp(f (a)) = f (Sp(a)) is usually called the Spectral Mapping Theorem.

Proof. This is [Bil10, Thm 4.2 + Rem 4.3 + Cor. 4.4] and [Gloe02b, Lemma 4.3, Prop. 4.9, Thm. 4.10]. Proposition 4.23. If is a Mackey complete CIA, then the group GL( ) carries the structure of a locally exponentialA Lie group with Lie algebra and exponentialA map P an A exp : GL( ), a exp(a) := ∞n 0 n! . A → A 7→ = Proof. Suppose first that is unital: We briefly sketch the main points of the proof. For a more detailed version,A the reader is referred to [Ne06, Thm. IV.1.11] and to [Gloe02b, Thm. 5.6]. Since × is an open subset of , the manifold structure on × is given by the A A A identity chart × , . The smoothness of the multiplication on × is inherited from the bilinearityA → A of the multiplication on . The smoothnessA of the inversion is first established locally around 1 via theA von Neumann power series: Put 1 P n Ω : z : z 1 < 1 . Then, for each a , we have a ∞ 1 a which = C Ω − = n=0( ) defines{ a∈ smooth| − map| by} Theorem 4.22. To∈ A show that the inversion is− smooth ev- 1 1 1 erywhere on ×, we use the identity (a0a)− = a− a0− with a0 × and a Ω. Using the holomorphicA calculus (with Ω = C), one obtains a smooth∈ A exponential∈ map A P an exp a : ∞ . That t exp t a is a one-parameter homomorphism is a direct ( ) = n=0 n! ( ) 7→ · exp(t a) 1 consequence of the relation exp((t+t0) z) = exp(t z) exp(t0 z) and limt 0 t· − = a exp(t z) 1 · · · → follows from limt 0 t· − = z. To see that the exponential map is a local diffeomor- → phism, one constructs a logarithmic function on Ω with Ω := C ] , 0] and uses A \ − ∞ 4.1 The class of locally exponential Lie groups 71 part iv) of Theorem 4.22. We are left with the non-unital case: Consider the continuous algebra homomorphism

a ε : + = C C, (λ, ) λ. A ⊕ A → 7→ The restriction ε : of ε to the Lie group GL is a smooth +× +× C× ( +) = +× |A A → 1 A A homomorphim and we have GL( ) = ε − ( 1 ). Thus, Corollary 4.9i) proves the +× assertion. A |A { }

Remark 4.24. If ab = ba, then an approximation argument shows that also

exp(a) exp(b) = exp(b) exp(a).

We obtain a smooth group homomorphism ϕ : R ×, t ϕ(t) := exp(ta) exp(t b) → A 7→ with ϕ0(0) = a + b. Part iii) of Theorem 4.4 shows that ϕ(t) = exp(t(a + b)) so that t = 1 yields

exp(a + b) = exp(a) exp(b) for all a, b with ab = ba. ∈ A In order to speak about unitary elements in a complex locally convex algebra , we need an involution on . A ∗ A Deﬁnition 4.25. We call a complex locally convex algebra a continuous inverse - algebra (resp. a -CIA for short) if it is a CIA endowed withA a continuous, conjugate∗ ∗ 35 linear, anti-multiplicative involution : , a a∗. An element a is ∗ A → A 7→ ∈ A called normal if aa∗ = a∗a, self-adjoint if a∗ = a and skew-adjoint if a∗ = a. − Remark . 4.26 If is a -CIA without multiplicative unit, then its unitization + = A ∗ A C becomes a -CIA with respect to the involution (λ, a)∗ := (λ, a∗). ⊕ A ∗ 1 We will write U( ) := a GL( ) : a∗ = a− for the group of unitary elements in a -CIA and callA it the{unitary∈ groupA of . } ∗ A A Corollary 4.27. If is a Mackey complete -CIA, then its unitary group U( ) GL( ) carries a natural structureA of a locally exponential∗ Lie group with Lie algebraA ⊆ A

u( ) := a : a∗ = a , A { ∈ A − } the skew-adjoint elements of , and with exponential map A X an exp : u U , a exp a : ∞ . ( ) ( ) ( ) = n! A → A 7→ n=0

Proof. Consider the smooth homomorphism

θ : GL( ) GL( ), g g−∗. A → A 7→ θ We then have U( ) = GL( ) and the assertion follows from Corollary 4.9 ii). A A 35In the literature, not every author requires the involution to be continuous. ∗ 72 4 Lie Theoretic Aspects

The following two examples of Mackey complete -CIAs are of particular importance for our purposes. ∗ Example 4.28. i) Every Banach- -algebra is a Mackey complete CIA: Its completeness in particular implies∗ that it is MackeyA complete (cf. Remark 4.21). If A is unital, then its group of invertible elements × is open by the usual argument with von Neumann power series: On the open subsetA U := u : 1 u < 1 , 1 P 1 n { ∈ A k − k } the inversion is given by u− = ∞n 0( u) which is continuous in u. For every = − a ×, the left multiplication map λa : is a homeomorphism so that ∈ A A → A1 1 1 aU = λa(U) is an open subset. In view of (au)− = u− a− , we have aU × ⊆ A proving that × is an open subset on which the inversion is continuous. Hence A is a Mackey complete -CIA. If is not unital, then its unitization + = C Ais a unital Banach- -algebra∗ and weA conclude that is a Mackey completeA -CIA.⊕A In particular, every∗ finite-dimensional -algebra isA a Mackey complete CIA.∗ ∗ ii) Let := lim n be the strict direct limit of a strictly increasing sequence of A A S finite-dimensional−→ -subalgebras n such that = n n. Then is a Mackey complete CIA if it∗ is endowed withA the directA limit topology:A ThatA is a CIA follows from [Gloe02b, Prop. 9.5]. We summarize the main points ofA the proof. Assume that 1 is a multiplicative unit such that 1 n for all n N. First, one observes that∈ A the product is a direct limit∈ of A the n ∈n, i.e. we A × A A × A have = lim n n in the category of topological spaces (see e.g. [Hi01, A ×A A ×A Thm. 4.1]). Hence,−→ the continuity of the multiplication maps µn : n n n implies the continuity of the multiplication map µ : A ×Abecause→ A we A × A → A have µ = lim µn. The relation × n = n× reveals that × is open in the A ∩ A A A direct limit−→ topology. The continuity of the inversion maps ιn : n× n implies A → A the continuity of the inversion ι : × because we have ι = lim ιn. If is not unital, then we have for the unitizationA → A that lim Aand + = C = −→C n the argument applies to showing that is aA CIA. The⊕ Mackey A completeness⊕ A + −→ A A of = lim n follows directly from [KM97, Thm. 2.15]. The main point here is A A that the−→ image of a smooth curve ξ : [a, b] is a compact subset in and → A A hence contained in some finite-dimensional n (see e.g. [Ne04b, Lemma A.7]). Thus, the Mackey completeness of the finite-dimensionalA subalgebras n implies the Mackey completeness of . A A Example 4.29. i) Let be a complex Hilbert space. Applying Proposition 4.23 and H Corollary 4.27 to := ( ), the C ∗-algebra of bounded linear operators on , we obtain theA classicalB LieH group structures on GL( ) and U( ). The Lie H H H algebras are given by gl( ) := ( ) and u( ) := X ( ) : X ∗ = X , the skew hermitian operators.H TheB exponentialH functionH { is given∈ B byH the traditional− } P 1 n power series exp X : ∞ X . ( ) = n=0 n! ii) More generally, if E is a complex Banach space, then the general linear group

GL(E) := (E)× is a Banach–Lie group with Lie algebra (E) and an exponential map exp :B (E) GL(E) deﬁned by the usual power seriesB (Proposition 4.23). Let F E Bbe closed→ subspace. Then H := g GL(E) : (g 1).E F is a Lie ⊆ { ∈ − ⊆ } 4.2 Integration and differentiation of 1-cocycles 73

subgroup of G := GL(E) with Lie algebra L(H) = X (E) : X .E F . The argument is similar to the one in the proof of Lemma{ IV.12∈ B in [Ne04b⊆]: First,} H is a closed subgroup of G and we have

e L (H) := X (E) : exp(R X ) H = X (E) : X .E F , { ∈ B · ⊆ } { ∈ B ⊆ } which can be seen by differentiation and by the power series expansion of exp. Let V (E) be an open 0-neighborhood such that ⊆ B exp(V ) U := g GL(E) : g 1 < 1 ⊆ { ∈ k − k } and exp V is a diffeomorphism. Then, spectral theory (Theorem 4.22) shows that | 1 the inverse function log := (exp V )− is given by the converging power series | X 1 n+1 log g ∞ ( ) g 1 n. ( ) = − n ( ) n=0 − For g = exp(X ) exp(V ) H, we thus find X .E = log(g).E F by virtue of the power series expansion∈ of∩ log. Now, the assertion follows from⊆ Theorem 4.8. iii) If E is a complex Banach space, then also the semidirect product E o GL(E) is a locally exponential Lie group with (Banach–) Lie algebra E o (E). This is a special case of the more general fact that semi-direct productsB of Banach–Lie groups are Banach–Lie groups (cf. [GlNe18, Prop. 3.2.8]). Alternatively, one can E E v g argue as follows: Put + := C. Then, every element ( 0, 0) in the semidirect ⊕ g E product defines an invertible linear operator on + via g v g v v v E .( , λ) := ( 0 + λ 0, λ), for ( , λ) + · ∈ and g GL E satisfies g 1 .E E, 0 E. Conversely, every g GL E ( +) ( ) + ( ) ∼= ( +) g∈ 1 E E − ⊆ ∈ with ( ). + is of that form with − ⊆ 1 (g0.v, 0) := g.(v, 0) and (v0, 0) := (g ).(0, 1). − Therefore, 1 E o GL(E) = H := g GL(E+) : (g ).E+ E ∼ { ∈ − ⊆ } ii F E E and the assertion follows from part ) with := +. ⊆

4.2 Integration and differentiation of 1-cocycles

Let G be a locally exponential Lie group with Lie algebra g := L(G) and exponential function exp : g G. For a complex Banach space E, we consider a norm-continuous unitary representation→ π : G GL(E) of G on E. This means that π is a continuous group homomorphism w.r.t. the→ norm topology on GL(E). The norm topology turns GL(E) into a (Banach–) Lie group (cf. Example 4.29 ii)) and the Automatic Smooth- ness Theorem (Theorem 4.4) shows that π is a smooth group homomorphism whose d tX derivative dπ(X ) := d t t=0π(exp( )) deﬁnes a continuous representation of Lie algebras dπ : g gl(E) := | (E). There is an analogous version for continuous 1-cocycles 1 β Z (G, π→, E). B ∈ 74 4 Lie Theoretic Aspects

Lemma 4.30. Every continuous 1-cocycle β : G E is automatically smooth. We obtain a continuous Lie algebra 1-cocycle →

d dβ : g E, X β(exp(tX )) d t → 7→ t=0 and a linear map 1 1 d : Z (G, π) Z (g, dπ), β dβ. → 7→ Proof. This is due to the fact that every continuous 1-cocycle β deﬁnes a continuous group homomorphism of locally exponential Lie groups

A : G E o GL(E) = Aff(E), g A(g) := (β(g), π(g)). → ∼ 7→ The smoothness of A (Theorem 4.4) implies the smoothness of β, since the projection of the semi-direct product E oGL(E) onto the E-component is smooth. The Lie algebra of E o GL(E) is given by E o (E) (cf. Example 4.29 ii)) so that the derivative of A is a continuous homomorphismB of the Lie algebras

d dA : g E (E), X A(exp(tX )). o d t → B 7→ t=0

The derivative dA is of the form dA(X ) = (dβ(X ), dπ(X )), where dπ : g u( ) and dβ : g are continuous linear maps. For X , Y g, the relation → H → H ∈ dA X , Y dA X , dA Y ([ ]g) = [ ( ) ( )]u( )o H H translates into

dπ([X , Y ]g) = dπ(X )dπ(Y ) dπ(Y )dπ(X ) − dβ([X , Y ]g) = dπ(X )dβ(Y ) dπ(Y )dβ(X ), − which means that dπ is a continuous homomorphism of Lie algebras (namely the Lie–derivative of π) and dβ is a continuous Lie-algebra 1-cocycle.

The question now is under which conditions d is an isomorphism of vector spaces. Note that d maps Lie group 1-coboundaries to Lie algebra 1-coboundaries and that 1 1 the restriction d : B (G, π) B (g, dπ) is always surjective. If G is connected, then d is always injective. Indeed,→ if dβ = 0, then β : G E is a 1-cocycle β : G E → → which is constant on the path-components of G, hence constant on G = G0 with value β(e) = 0. By Theorem 4.16, d is surjective if G is simply-connected, so that every continuous Lie algebra 1-cocycle integrates to the universal covering group Ge of a connected Lie group G. It then just remains to discuss under which conditions this integrated 1-cocycle on Ge factors through the universal covering map q : Ge G. This is what we do in our next step. → 4.2 Integration and differentiation of 1-cocycles 75

Some algebraic preliminaries

Let G be a connected topological group and suppose that N Å G is a discrete normal subgroup. We assume that E is any complex vector space and that π : G GL(E) 1 → is any representation of G on E. We write Zal g (G, π, E) for the vector space of all 1-cocycles β : G E. → Lemma 4.31. i) We have N Z(G), where Z(G) denotes the center of G. ⊆ ii) If N ker(π), then the restriction ⊆ 1 G r : Zal g (G, π, E) HomGrp(N, E ), β β N → 7→ | G is a linear map, where E := v E : ( g G)π(g)v = v stands for the subspace of G-ﬁxed vectors. { ∈ ∀ ∈ }

1 Proof. i) For n N, consider the continuous map cn : G N, g gng− . Then, ∈ 1 → 17→ in view of N being discrete, Gn := g G : gng− = n = cn− ( n ) is an open { ∈ } { } subgroup of G. Since G is connected, we have G = Gn.

ii) For n, n0 N, we have β(nn0) = β(n) + π(n)β(n0) = β(n) + β(n0), so that ∈ β N : N E is a group homomorphism. For g G and n N Z(G), we have | → ∈ ∈ ⊆ β(gn) = β(ng) = β(g) + π(g)β(n) = β(n) + β(g) =⇒ π(g)β(n) = β(n). ⇒ G This shows that β(N) E . ⊆

Now, let G, G0 be two topological groups and suppose that G0 is connected. Assume that ϕ : G0 G is a continuous surjective group homomorphism with discrete kernel → ker(ϕ). If π : G GL(E) is a representation of G on E, then the pullback π0 := π ϕ → ◦ deﬁnes a representation of G0 on E and we obtain a natural linear map

1 1 1 Z (ϕ) : Zal g (G, π) Zal g (G0, π0), β β ϕ. → 7→ ◦ Lemma 4.32. We have an exact sequence of vector spaces

1 1 Z (ϕ) 1 r G 0 Zal g (G, π) Zal g (G0, π0) HomGrp(ker(ϕ), E ).

G 1 In particular, if E = 0 , then Z (ϕ) is a linear isomorphism of vector spaces. { } 1 Proof. The injectivity of Z (ϕ) follows from the surjectivity of ϕ because β ϕ = β 0 ϕ 1 ◦ ◦ implies β = β 0. Hence, it remains to to show that im(Z (ϕ)) = ker(r): In view of r Z 1 Z 1 r β ϕ ker(ϕ) = 0, we have (ϕ) = 0 and therefore im( (ϕ)) ker( ). Conversely, ◦ | 1 ◦ ⊆ let β 0 Zal g (G0, π0) such that ker(ϕ) ker(β 0). Then, the afﬁne maps A0(g)v := ∈ ⊆ π0(g)v + β 0(g) deﬁne a group homomorphism A0 : G0 E o GL(E) with ker(ϕ) → ⊆ 76 4 Lie Theoretic Aspects

ker(A0). Thus, A0 factors by ϕ through a unique group homomorphism A : G Aff E E GL E as in the diagram → ( ) ∼= o ( )

A0 G0 Aff(E).

ϕ A G

1 We have A(g)v = π(g)v +β(g) for some β Zal g (G, π) and putting v = 0 reveals that ∈ β 0 = β ϕ. ◦ Remark 4.33. Suppose that E is a complex topological vector space. Then, it makes 1 1 sense to consider the subspace Zcont (G, π, E) Zal g (G, π, E) of continuous 1-cocycles β : G E. As in the previous lemma, we obtain⊆ an exact sequence → 1 1 Z (ϕ) 1 r G 0 Zcont (G, π) Zcont (G0, π0) HomGrp(ker(ϕ), E ), provided that ϕ is, in addition, an open map. In other words, we require that

ϕ : G0 G is a covering morphism of topological groups. In this case, the group → G is isomorphic (as a topological group) to the quotient G0/ ker(ϕ) endowed with the quotient topology, as shown in the diagram

ϕ G0 G.

!ϕe ∃ G/ ker(ϕ)

The map ϕe is a continuous, bijective group homomorphism. Since ϕ is open, so is ϕe which means that it is a homeomorphism. Thus, the universal property of the quotient topology implies that a map f : G X into a topological space X is continuous if and → only if the composite f ϕ is continuous. Hence, if β 0 = β ϕ, then β is continuous ◦ ◦ if and only if β 0 is continuous. This shows that the proof of Lemma 4.32 carries over 1 1 to continuous 1-cocycles and we may replace Zal g by Zcont .

Integration of Lie algebra 1-cocycles

Now, we return to our initial setting and assume that G is a connected locally exponential Lie group with Lie algebra g and exponential function exp : g G. We apply the previous results to the universal covering morphism q : Ge G. The→ universal covering group Ge is a 1-connected locally exponential Lie group with→ the same Lie algebra L(Ge) = g and we have L(q) = idg. Hence, if π : G GL(E) is a norm-continuous rep- → resentation of G on a complex Banach space E, the pullback representation πe := π q ◦ of Ge on E, is also norm-continuous with the same derivative dπe = dπ : g (E). 1 → B We write βe for a continuous 1-cocycle in Z (Ge, πe). 4.2 Integration and differentiation of 1-cocycles 77

Proposition 4.34. i) Every continuous Lie-algebra 1-cocycle ω : g E integrates to a unique continuous 1-cocycle βe : Ge E with dβe = ω. In particular,→ the map → 1 1 de : Z (Ge, πe) Z (g, dπ), βe dβe → 7→ is an isomorphism of vector spaces. Moreover, there exists a unique continuous 1- cocycle β : G E with dβ = ω if and only if ker(q) ker(βe). → ⊆ ii) We have the following two exact sequences of vector spaces:

1 d 1 G G 0 Z (G, π) Z (g, dπ) HomGrp(ker(q), E ) = HomGrp(π1(G), E ), → → → ∼ 1 1 G G 0 H (G, π) H (g, dπ) HomGrp(ker(q), E ) = HomGrp(π1(G), E ), → → → ∼ 1 1 where the map d is the derivative d : Z (G, π) Z (g, dπ), β dβ. → 7→ Proof. i) Suppose that ω : g E is a continuous Lie algebra 1-cocycle. Then, ψ(X )v := dπ(X )v + β(X ) deﬁnes→ a continuous homomorphism of Lie algebras ψ : g aff E E E and the Integrability Theorem (Theorem 4.16) guar- ( ) ∼= o ( ) antees→ the uniqueB existence of a smooth Lie group homomorphism Ae: Ge Aff E E GL E with dAe ψ. Moreover, there exists a unique smooth ( ) ∼= o ( ) = homomorphism→ A : G Aff(E) with dA = ψ if and only if ker(q) ker(Ae). Since → ⊆ Ae and A are of the form Ae(g)v = πe(g)v + βe(g) and A(g)v = π(g)v + β(g) for some smooth 1-cocycles βe : Ge E and β : G E, the existence of βe and β → → is equivalent to the existence of Ae and A respectively. In view of πe = π q, the condition ker(q) ker(Ae) is satisﬁed if and only if ker(q) ker(βe). This◦ shows i). ⊆ ⊆

ii) This follows from the ﬁrst part together with the exact sequence from Lemma 4.32: We already know that the sequence

1 1 Z (q) 1 r G 0 Z (G, π) Z (Ge, πe) HomGrp(ker(ϕ), E )

r i is exact, where denotes the restriction βe βe ker(q). Part ) tells us that the 1 1 7→ | 1 derivative de : Z (Ge, πe) Z (g, dπ), βe dβe is an isomorphism, so that Z (q) → 1 7→ 1 1 factors through d: We have de− d(β) = β q = Z (q)β for all β Z (G, π) and 1 1 ◦ 1 ◦ ∈ thus Z (q) = de− d. Deﬁning e := r de− , we obtain a sequence ◦ ◦ 1 d 1 e G 0 Z (G, π) Z (g, dπ) HomGrp(ker(ϕ), E )

which is exact because the injectivity of d is inherited from the injectivity of 1 1 Z (q) = de− d and by virtue of ◦ 1 1 1 ker(e) = (r de− )− (0) = de(ker(r)) = de(im(Z (q))) = im(d). ◦ 78 4 Lie Theoretic Aspects

Applying the universal property of quotient vector spaces, we obtain the following commuting diagram

1 d 1 e G 0 Z (G, π) Z (g, dπ) Hom(ker(q), E )

pG pg id

1 D 1 E G 0 H (G, π) H (g, dπ) Hom(ker(q), E ),

1 where pG and pg denote the canonical surjections onto the quotients H (G) and 1 H (g). To see the exactness of the second row of the diagram, we calculate

1 1 1 1 1 1 pG− (ker(D)) = d− (ker(pg)) = d− (B (g, π)) = B (G, π) = pG− ( 0 ), { } which leads to ker(D) = 0 by virtue of the surjectivity of pG, and { } im(D) = im(D pG) = im(pg d) = pg(im(d)) = pg(ker(e)) = ker(E). ◦ ◦

Here again, we have used the surjectivity of pG and pg. This completes the proof.

1 1 Corollary 4.35. The derivative d : Z (G, π) Z (g, dπ) is an isomorphism if one of the following two conditions is satisﬁed: → a) The locally exponential Lie group G is connected and simply connected.

G b) The locally exponential Lie group G is connected and E = 0 . { } In particular, if one of these conditions holds, we have H1 G, π H1 g, dπ . ( ) ∼= ( ) Remark 4.36. If (π, E) is a norm-continuous unitary representation of a connected locally exponential Lie group G, then we have seen in Proposition 4.34 that every continuous Lie-algebra 1-cocycle integrates to a unique continuous 1-cocycle on the G Lie group, provided that E = 0 . In general, this condition cannot be dropped as already the following simple{ example} shows: Take G = T := z C : z = 1 with Lie algebra g = R and (π, E) as the trivial representation on{ ∈E = C|.| Then} 1 Z (T, id) = Hom(T, C) = 0 , since the group T is compact. But the continuous Lie algebra 1-cocycles are precisely{ } the linear maps R C and none of these integrates to a continuous 1-cocycle on T. → G In case that E = 0 , one can ask whether it is possible to split off the G-fixed vectors as in the Hilbert6 space{ } case. This can be done if E is reflexive and the representation is uniformly bounded in the sense that supg G π(g) < . This result is another consequence of the Ryll–Nardzewski Fixed Point∈ k Theoremk ∞ (Theorem 3.32) and a proof 36 G is given in [Shu17, Thm.4] . Hence, if E decomposes as E = E EG, where EG ⊕ 36 It is also shown in [Shu17] that every discrete non-amenable group G admits an isometric representation on a Banach space E such that the subspace of G-fixed vectors EG is not complemented in E. 4.2 Integration and differentiation of 1-cocycles 79 denotes the complement of EG, and if a continuous Lie algebra 1-cocycle ω : g E takes its values in EG, then it uniquely integrates to a continuous 1-cocycle β : G → E with dβ = ω. Now, suppose that Hom(G, R) = 0 , that E = is a complex Hilbert→ space and that (π, ) is a norm-continuous unitary{ } representationH of G. Then, every H G G continuous 1-cocycle β : G takes its values in ⊥, i.e. β(G) ⊥. In particular, a continuous Lie→ algebra H 1-cocycle ω : g H integrates to⊆ aH unique G → H 1-cocycle on the Lie group if and only if ω(g) ⊥ and we obtain the following commuting diagram where the horizontal maps⊆ areH isomorhisms of vector spaces:

1 d 1 G Z (G, π, ) Z g, dπ, ⊥ H H

1 1 G H (G, π, ) H g, dπ, ⊥ . H H

Universal complexiﬁcation

The Banach–Lie algebra g of a Banach–Lie group G is a Lie algebra over the field R. It might be useful to work with its Lie-algebra complexification g which is obtained C as follows: Let g be the complexification of g as a Banach-space. The Lie-bracket C of g is a R-bilinear map [ , ] : g g g so that we obtain a C-bilinear extension , : g g g . The two-· · and× trilinear→ maps on g g resp. on g g g [ ]C C C C C C C C C · · × → × × × Z , Z Z , Z Z , Z ( 1 2) [ 1 2]C + [ 2 1]C 7→ and Z , Z , Z Z , Z , Z Z , Z , Z Z , Z , Z ( 1 2 3) [ 1 [ 2 3]C]C + [ 2 [ 3 1]C]C + [ 3 [ 1 2]C]C 7→ vanish on the subspaces g g resp. on g g g, hence are both identically zero by the universal property of the× Banach–space× complexification.× Thus, , turns g [ ]C C · · into a complex Banach–Lie algebra. If g0 is another complex Banach–Lie algebra and ψ : g g0 a homomorphism of real Banach–Lie algebras, then there exists a unique complex→ linear map ψ : g g such that the following diagram commutes: C C 0 → ψ g g0.

!ψ C g ∃ C

The bilinear maps on g g C C × Z , Z ψ Z , Z and Z , Z ψ Z , ψ Z ( 1 2) C([ 1 2]C) ( 1 2) [ C( 1) C( 2)]g 7→ 7→ 0 coincide on g g, hence they are identical by the universal property of the Banach– space complexiﬁcation× and ψ thus becomes the unique homomorphism of complex C Banach–Lie algebras extending ψ. This is the universal property of the Banach–Lie 80 4 Lie Theoretic Aspects algebra complexiﬁcation. In particular, if ρ : g (E) is a (norm-continuous) Lie algebra representation of g on a complex Banach→ space B E, then there exists a unique (norm-continuous) complex linear representation ρ : g E such that ρ ρ. C C ( ) C g = Every (continuous) Lie-algebra 1-cocycle ω : g E is a R→-linear B map and extends| to a unique (continuous) -linear map ω : g → E such that the following diagram C C C commutes → g ω E.

!ω C g ∃ C The bilinear maps on g g C C × Z , Z ω Z , Z and Z , Z dπ Z ω Z dπ Z ω Z ( 1 2) C([ 1 2]C) ( 1 2) C( 1) C( 2) C( 2) C( 1) 7→ 7→ − coincide on g g, hence they are identical by the universal property of the Banach– space complexiﬁcation.× This turns ω into a (continuous) Lie algebra 1-cocycle for C the representation dπ , and we obtain the following proposition: ( C ) H Proposition 4.37. Let ρ : g (E) be a (norm-continuous) representation of the real Banach–Lie algebra g on a complex→ B Banach space E. Then every (continuous) Lie algebra 1-cocycle ω : g E uniquely extends to a (continuous) Lie algebra 1-cocycle ω : g E C C corresponding to→ the complex linear extension ρ : g E . In particular, we obtain→ C C ( ) linear isomorphisms → B

Z 1 g, ρ, E Z 1 g , ρ , E and H1 g, ρ, E H1 g , ρ , E . ( ) ∼= ( C C ) ( ) ∼= ( C C )

There is an analogous result on the Lie group side: We assume tacitly that all Lie groups are locally exponential.

Deﬁnition 4.38. i) Following [HiNe12], a complex Lie group is a real Lie group G whose Lie algebra g := L(G) is a complex Lie algebra and for which every adjoint automorphism Adg : g g, g G, is a complex linear automorphism of g. If → ∈ ad g G is connected, then the second condition is automatic since Ad(G) e ( ) Aut g . A smooth group homomoprhism ϕ : G G of complex⊆ Lie groups⊆ C( ) 0 is called holomorphic if L(ϕ) is complex linear. Accordingly,→ a smooth 1-cocycle β : G of a complex Lie group G is called holomorphic if the derivative dβ → H 1 is complex linear. We denote by Zhol the vector space of holomorphic 1-cocycles, 1 1 1 1 by Bhol its subspace of holomorphic 1-coboundaries and by Hhol := Zhol /Bhol the corresponding quotient vector space.

ii) Let G be a real Lie group. A complex Lie group G is said to be a (universal) C complexiﬁcation of G if there exists a smooth homomorphism η : G G with C the following universal property: For every smooth homomorphism ϕ→: G H into a complex Lie group H, there exists a unique holomorphic homomorphism→ 4.2 Integration and differentiation of 1-cocycles 81

ϕ : G H such that the diagram C C → ϕ G H

η !ϕ C G ∃ C commutes. Remark 4.39. The pair η, G is unique in the following sense: If η , G is another ( C) ( 0 C0 ) pair satisfying the universal property, then there exists a unique holomorphic isomorphism ϕ : G G such that ϕ η η . In particular, the universal complexiﬁcation C C C0 C = 0 of G is uniquely→ determined up to◦ isomorphism of complex Lie algebras.

Proposition 4.40. Let π : G GL(E) be a norm-continuous representation of the real Lie group G on a complex Banach→ space E. Assume that a universal complexiﬁcation η, G of G exists and let π : G GL E be the unique holomorphic representation ( C) C C ( ) with π η π. Then, every continuous→ 1-cocycle β : G E uniquely extends to a C = holomorphic◦ 1-cocycle β : G E in the sense that β η →β. In particular, we obtain C C C = linear isomorphisms → ◦

Z 1 G, π, E Z 1 G , π , E and H1 G, π, E H1 G , π , E . ( ) ∼= hol ( C C ) ( ) ∼= hol ( C C )

Proof. Consider the continuous afﬁne action A(g)v := π(g)v +β(g) which is a continuous (and hence smooth) group homomorphism A : G Aff E E GL E into the ( ) ∼= o ( ) complex Lie group Aff(E). By the universal property, we→ obtain a unique holomorphic homomorphism A : G Aff E with A η A. This holomorphic homomorphism C C ( ) C = is of the form A g v π→ g v β g for◦ a uniquely determined β Z 1 G , π , E C( ) = C( ) + C( ) C ( C C ) satisfying β η β. ∈ C = ◦ Consequently, this result leads to the question of which Lie groups have a universal complexiﬁcation. For Banach–Lie groups, this question has been answered in [GlNe03] and later generalized to the class of BCH–Lie groups (see [GlNe18, Thm. 8.7.9]). We only need the much weaker version below, stated as Lemma 4.41.

Let G be a connected real Banach–Lie group with Lie algebra g. Let G be connected C Banach–Lie group with Lie algebra g and suppose that there exists a smooth homo- C morphism η : G G with derivative L η ι, where ι : g , g is the natural C ( ) = C embedding of g into→ g . Then, in view of Theorem 4.12, there exist→ universal cover- C ing groups and universal covering maps q : G G , q : G G and there exists a C eC C e unique smooth homomorphism η : G G with derivative→ L η→ ι (Theorem 4.16). e e eC ( e) = The following diagram commutes: →

η G e G e eC q q C !η G ∃ G . C 82 4 Lie Theoretic Aspects

Consequently, we have η ker q ker q which implies that Γ : η ker q is a e( ( )) ( C) = e( ( )) discrete central subgroup of G . ⊆ eC Lemma 4.41. The group G is a universal complexiﬁcation of G with corresponding C smooth homomorphism η : G G if and only if η ker q ker q . C e( ( )) = ( C) → Proof. Let H be a complex Lie group and ϕ : G H be a smooth group homo- → morphism. Put ϕe := ϕ q : Ge H. On the Lie algebra side, the Lie derivative L ϕ : g h : L H has◦ a unique→ complex linear extension L ϕ : g h with ( e) = ( ) ( e)C C → → L ϕ ι L ϕ . ( e)C = ( e) ◦ In view of the Integrability Theorem (Theorem 4.16), we obtain a unique smooth homomorphism ϕ : G H with eC eC → ϕ η ϕ eC e = e ◦ by integration. In particular, ϕ is holomorphic, since L ϕ L ϕ is complex eC ( eC) = ( e)C linear, and the diagram η G e G e eC q ϕ eC ϕ G H commutes. Next, we show that a (unique) holomorphic homomorphism ϕ : G H with ϕ η ϕ exists if and only if ϕ ker q e : We have C C C = eC( ( C)) = H → ◦ { } ϕ η ϕ C = ϕ η◦ q ϕ q ϕ surjectivity of q C = = e ( ) ⇐⇒ ϕ q◦ ◦η ϕ ◦ using η q q η ( C C) e = e ( = C e) ⇐⇒ ◦ϕ ◦q ϕ uniqueness◦ of ◦ϕ C C = eC ( C) ⇐⇒ L ◦ϕ L ϕ L ϕ . Theorem 4.16 ( C) = ( eC) = ( e)C ( ) ⇐⇒ The claim follows from the fact that a homomorphism ϕ with ϕ q ϕ exists C C C = C if and only if ϕ ker q e . Note that L ϕ L ϕ not only◦ shows that ϕ eC( ( C)) = H ( C) = ( eC) C is holomorphic but also that{ it is} uniquely determined by virtue of the Integrability Theorem (Theorem 4.16). Now, we prove the assertion in Lemma 4.41: If ker q η ker q , then (with the ( C) = e( ( )) notation from above)

ϕ ker q ϕ η ker q ϕ q ker q e eC( ( C)) = eC e( ( )) = ( ( )) = H ◦ ◦ { } is satisfied for all ϕ and H. Conversely, if η, G is a universal property of complexifi- ( C) cation, then this means that ϕ ker q e is satisfied for any choices of H and ϕ. eC( ( C)) = H Choose H : G /Γ with Γ : η ker q and{ let}q : G H be the canical surjection. = eC = e( ( )) H eC → 4.2 Integration and differentiation of 1-cocycles 83

By the universal property of the quotient G Ge/ ker q , there exists a unique smooth ∼= ( ) map ϕ : G H such that the diagram → η G e G e eC

q qH ϕ G H commutes. Now the condition q ker q e reads ker q Γ and, in view of H ( ( C)) = H ( C) Γ ker q , we obtain equality. { } ⊆ ( C) ⊆ 2 2 In Subsection 5.2, we introduce the unitary groups U U(n),U( ),Up(` ),U(` ) as Lie subgroups of the corresponding general linear group∈ { ∞ }

2 2 G GL(n), GL( ), GLp(` ), GL(` ) . ∈ { ∞ } The corresponding Lie algebra u of U consists of skew-hermitian operators and its universal complexiﬁcation is isomorphic to the Lie algebra g of G, viewed as a linear subspace of bounded linear operators on `2 : `2 , . In this sense, we have u = (N C) C ∼= g. The universal covering group Ue is a subgroup of Ge containing the kernel of the universal covering map q : Ge G. In view of Lemma 4.41, we conclude: → 2 2 Proposition 4.42. The general linear groups GL(n), GL( ), GLp(` ), GL(` ) are the { ∞ 2 2 } universal complexiﬁcations of the unitary groups U(n),U( ),Up(` ),U(` ) respectively. { ∞ }

Proof. Lemma 4.41 covers all cases but one, namely the case of U = U( ) = lim U(n). ∞ Here the assertion follows because it is true for all subgroups U(n): Let ϕ :U(−→) H be a smooth homomorphism into a complex Lie group H. Then we obtain,∞ for→ all n , unique holomorphic representations ϕ : GL n H with ϕ ϕ . N C,n ( ) C,n U(n) = U(n) The∈ uniqueness implies that, for all n < m, we have ϕ → ϕ so that| we obtain| C,m GL(n) = C,n a well-deﬁned homomorphism ϕ : GL H via ϕ| g ϕ g if g GL n . C ( ) C( ) = C,n( ) ( ) Since GL( ) is endowed with the direct∞ limit→ topology deﬁned by the subgroups∈ GL n , this∞ homomorphism is continuous because all ϕ are. Hence it is smooth ( ) C,n by Theorem 4.4 and even holomorphic, since L ϕ lim L ϕ is complex linear. ( C) = ( C,n) Since the restrictions of ϕ to all subgroups GL n are uniquely determined, ϕ itself C ( ) −→ C is uniquely determined. This completes the proof. 2 84 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

2 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

Let be a complex, infinite dimensional, separable Hilbert space. The aim of this sectionH is two-fold: First, we introduce the unitary subgroups of U( ) and second, we construct natural 1-cocycles of these unitary subgroups. For 1H p , the 1≤ ≤ ∞ group Up( ) consists of those unitary operators g, for which g p( ), the pth SchattenH class operators. For p = , we identify ( ) :−= ∈( B) withH the ideal of compact operators on . We∞ shall see that allB∞ theseH groupsK H carry a nat- H ural structure of a connected Banach–Lie group with Lie algebra p( ) and with an exponential function that is defined via the traditional power series.B H Moreover, the countable union U : S U n of the finite dimensional unitary subgroups ( ) = n N ( ) ∞ ∈ U(n) := U(n, C) which are naturally embedded in the Up( ) is a dense subgroup in H every Up( ). As we shall see, the direct limit topology defines on U( ) a locally exponentialH Lie group topology so that all our unitary groups belong to∞ the class of locally exponential Lie groups. We have already seen, that the full unitary group U( ) and the group U ( ) are bounded groups and, therefore, have Property (FH)H which means that they∞ H do not admit nontrivial 1-cocycles. For 1 p < , the group Up( ) naturally acts on ≤ ∞ H the Hilbert space 2( ) of Hilbert–Schmidt operators via left (right) multiplication and via conjugation.B H We will show that the corresponding continuous 1-cocycles β :Up( ) 2( ) are of the form H → B H 1 1 β(g) = gB B, β(g) = Bg− B, and β(g) = gBg− B − − − respectively, for some B ( ). In the case p > 2, we have B q( ) with 1 1 1 ∈ B H ∈ B H p + q = 2 . In partiuclar, we obtain explicit unbounded 1-cocycles of the direct limit U( ). A more general method to constuct unbounded 1-cocycles of U( ) is presented∞ later in Section7. ∞

5.1 Schatten class operators

We start with a brief review of the crucial facts on Schatten class operators on a separable, inﬁnite dimensional, complex Hilbert space . One should think of the pth H p Schatten class operators p( ) as the operator analogues of the classical ` -spaces. This point of view is basedB onH the following fundamental observations on ideals and Lie algebra ideals in ( ). We write ( ) for the closed (two-sided) ideal of com- B H K H pact operators and glf in( ) for the ideal of all ﬁnite rank operators on . An ideal or a Lie ideal is denoted byH and is always assumed to be a linear subspace.H J Theorem 5.1. Let 0 = = ( ) be a nontrivial (two-sided) ideal. Then, ∗ = , := A : A { } 6 Jand 6 B H J J |J | {| | ∈ J } ⊆ J glf in( ) ( ). H ⊆ J ⊆ K H Moreover, the ideal is uniquely determined by its subset of compact diagonal operators. J 5.1 Schatten class operators 85

Proof. The assertions, that = ∗, follow from polar decomposition: If A |J | ⊆ J J ∈ ( ), then there exists a partial isometry U such that A = U A and A = U ∗A. Here, 1 B H 2 | | | | A := (A∗A) is deﬁned via the functional calculus. We immediately obtain A | | | | ∈ J if A . Moreover, A∗ = A U ∗ = U ∗AU ∗ shows that A∗ if A . The proof ∈ J | | ∈ J ∈ J of the two inclusions can be found in [Sch60, chap. I, th. 11]. Assume that 0 is another non-trivial ideal with precisely the same compact diagonal elements. SinceJ all operators are compact, and 0 also coincide on their self-adjoint (even on their normal) elements. Again, byJ polarJ decomposition, we conclude that A = U A belongs | | to 0 for all A or A 0. This shows = 0. J ∩ J ∈ J ∈ J J J There is a natural analogous version for Lie algebra ideals in ( ) which we will B H 1 also use in this subsection. We call a Lie ideal standard if ∗ = , and if J 1 J J ∈ J J is isometrically invariant in the sense that U U − = holds for all linear isometries J J U on . We denote by slf in( ) the Lie ideal of 0-trace ﬁnite rank operators. H H 1 Theorem 5.2. Let 0 , C = = ( ) be a nontrivial Lie ideal. Then, { } · 6 J 6 B H 1 slf in( ) C + ( ). H ⊆ J ⊆ · K H If, in addition, is standard, then it is uniquely determined as a standard Lie ideal by its subset of compactJ diagonal operators.

Proof. The two inclusions are shown in [dlH72, chap. II, Prop 1A]. If 0 is another standard Lie ideal with the same compact diagonal elements, then, byJ the isometric invariance, and 0 coincide on their self-adjoint compact elements. In view of the J J1 1 decomposition A = 2 (A + A∗) + i 2 i (A A∗) and the -invariance, we conclude that − ∗ J and 0 coincide on their compact elements. Then, the above inclusion implies that J 1 1 0 = C + 0 ( ) = C + ( ) = . J · J ∩ K H · J ∩ K H J For a compact operator A , the singular values µ A are the eigenvalues ( ) ( n( ))n N of the compact positive operator∈ K H A : A A. ∈ = p ∗ | | Theorem 5.3. Let J `∞ := `∞(N, C) be a Calkin space, i.e. , a linear ideal in the ⊆ commutative Banach algebra `∞ (deﬁned by pointwise multiplication) which is invariant under the permutation action of the full permutation group S on ` . Then N ∞ : A : µ A J J = ( ) ( n( ))n N J { ∈ K H ∈ ∈ } deﬁnes a (two-sided) ideal in ( ). Every (two-sided) ideal ( ) is of the form B H J ⊆ B H = J for unique Calkin space J. J J Proof. This follows from [Cal41, Thm. 1.11].

p p For 1 p , the spaces ` := ` (N, C) are Calkin spaces so that we obtain the following≤ ideals≤ ∞ of compact operators, which are called the pth Schatten class operators:

: A : µ A `p for 1 p . p( ) = ( ) ( n( ))n N B H { ∈ K H ∈ ∈ } ≤ ≤ ∞ 2 86 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

For p < , the pth Schatten class operators become a Banach space with respect to the norm∞ 1 p 1 X p p p A p := Tr( A ) := µn(A) . k k | | n N ∈ For p = , we have ( ) := ( ) which is a Banach space for the usual oper- ∞ B∞ H K H ator norm A := A . The operators in 1( ) are also called trace-class operators. k k∞ k k B H

The following facts, equalities and estimates for A p( ) are quite useful. A nice reference is [GK69, Chap. III]. ∈ B H

1.) Tr A p P e , A pe for an arbitrary choice of a complete orthonormal ( ) = n N n n system| | e ∈of〈 .| For| p〉H 1, the function Tr is deﬁned on all positive operators ( n)n N = ∈ H 0 A 1( ) and it extends to a continuous linear monotone functional ≤ ∈ B H X Tr : 1( ) C, A Tr(A) := en, Aen , B H → 7→ n N〈 〉 ∈ which again is independent of the en. The functional Tr is cyclic in the sense that for A 1( ), B ( ), we have Tr(AB) = Tr(BA). ∈ B H ∈ B H 2.) We have A∗ p = A p and UAV p = A p for arbitrary unitary operators U, V . k k k k k k k k 3.) For B, B0 ( ), we have BAB0 p B A p B0 . ∈ B H k k ≤ k k k k k k 4.) For 1 p p0 , we have ≤ ≤ ≤ ∞ A 1 A p A p A = A , k k ≥ k k ≥ k k 0 ≥ k k∞ k k so that we obtain the chain of inclusions

1( ) p( ) p ( ) ( ) = ( ). B H ⊆ B H ⊆ B 0 H ⊆ B∞ H K H 1 1 1 5.) For p, q, r [1, ] with p + q = r and A p( ), B q( ), then AB ∈ ∞ ∈ B H ∈ B H ∈ r ( ) and we have the estimate AB r A p B q. This result is known as Bthe Hgeneralized Hölder inequality fork Schattenk ≤ k classk k operators.k The special case p = q = 2 and r = 1, shows that the product of two operators A, B 2( ) is a trace-class operator and the trace-functional Tr deﬁnes a scalar product∈ B H on 2( ) which is given by B H A, B : Tr AB . 2( ) = ( ∗) 〈 〉B H Thus, 2( ) becomes a Hilbert space which is called the Hilbert space of Hilbert– SchmidtB operatorsH .

6.) For aribitrary v, w we introduce the rank-1-operators v w∗ by (v w∗)u := ∈ H 37 ⊗ ⊗ u, w v for all u . We have (v w∗)∗ = w v∗ and v w∗ p = v w . 〈 〉· ∈ H ⊗ ⊗ k ⊗ k k k k k 37 In braket-notation, this would read v w∗ := v w . ⊗ | 〉〈 | 5.1 Schatten class operators 87

The generalized Hölder inequality yields another characterization of trace-class operators which turns out to be useful for our purposes.

1 1 1 Lemma 5.4. Let 1 r < p and q := r p . For a bounded linear operator B ( ), the following≤ are equivalent:≤ ∞ − ∈ B H B q( ) p( ) B r ( ). ∈ B H ⇐⇒ B H · ⊆ B H

Proof. Consider the ideal := B ( ) : p( ) B r ( ) of ( ) which J { ∈ B H B H · ⊆ B H } B H contains the ideal q( ) by virtue of the generalized Hölder inequality. We have to B H show that = q( ). In view of r < p, does not contain the identity opera- 1 J B H J tor and we are left to show that and q( ) have the same compact diagonal elements (Theorem 5.1). Let D J be aB compactH diagonal operator of the form D P d e e . For any diagonal∈ J operator A P a e e , we = n N n n n∗ = n n n n∗ p( ) ∈ · ⊗ P · ⊗ ∈ B H have, by deﬁnition of , that AD = n andn en en∗ r ( ). This is equivalent to P a r d r < forJ all sequences a · ⊗`p. Writing∈ B Ha d for the sequence n n n ( n)n N a d | |, the| | latter∞ condition means that we∈ have∈ a linear map · ( n n)n N ∈ `p `r , a a d. → 7→ · An application of Theorem 3.7 in combination with the fact that `p is of second (Baire) category reveals that this linear map must be continuous. Hence, there exists some p constant C > 0 such that a d r C a p holds for all a ` . Choosing N N and q r q −r p k · k ≤ k k ∈ ∈ an := dn = dn , we have | | | | 1 1 N r N r X q X r r dn = an dn n=1 | | n=1 | | | |

1 1 N p N p X p X q C an = C dn ≤ n=1 | | n=1 | | so that we obtain 1 q X q dn C n=1 | | ≤ q for all N N. We conclude that d ` with d q C. In view of the generalized Hölder inequality,∈ the condition that∈a d `r kfork all≤a `p is satisﬁed if and only if d `q which, in turn, is equivalent· to∈D .∈ ( n)n N q( ) ∈ ∈ ∈ B H 1 1 1 Lemma 5.5. Let 1 r < p and q := r p . For a bounded linear operator B ( ), we have≤ ≤ ∞ − ∈ B H 1 B C + q( ) [ p( ), B] r ( ). ∈ · B H ⇐⇒ B H ⊆ B H

Proof. Consider the standard Lie ideal := B ( ) : [ p( ), B] r ( ) J {1 ∈ B H B H ⊆ B H } which contains the standard Lie ideal 0 := C + q( ) by virtue of the generalized J · B H 2 88 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

Hölder inequality. We have to show that = 0. In view of Theorem 5.2, we have J J to show that and q( ) have the same compact diagonal elements and that = . Let DJ beB a compactH diagonal operator of the form D P d e Je 6 . ( ) = n N n n n∗ B H ∈ J ∈ · ⊗ Both assertions follow if we can show that D q( ). The proof follows the same lines as the proof of the Lemma in Chap. IV.2∈ B in [HdlH72]: The compactness of D implies that limn dn = 0. Putting f (0) := 0, we thus ﬁnd recursively for every n →∞f n f n n f n d 1 d N a number ( ) N satisfying ( ) > max( , ( 1)) and f (n) 2 n . For any∈ sequence a ∈ `p, put A : P a e e − .| We have| ≤ |A,|D ( n)n N = n n n ∗f (n) p( ) [ ] = P a d d e ∈ e∈ . In view of⊗ the estimate∈ B Hd d 1 d , this n n( f (n) n) n ∗f (n) r ( ) f (n) n 2 n − ⊗ r B H p | −q | ≥ | | implies that (andn) ` for all (an) ` so that we conclude (dn) ` and hence, that ∈ ∈ ∈ D q( ). ∈ B H Remark 5.6. If p r, then we have ≤

p( ) B p( ) r ( ) and [ p( ), B] p( ) r ( ) B H · ⊆ B H ⊆ B H B H ⊆ B H ⊆ B H for every B ( ). ∈ B H

5.2 The Banach–Lie group Up( ) H After our preceding discussion on the ideal structure of p( ) ( ), we now B H ⊂ B H discuss the topology and the Lie group structure of the unitary subgroup Up( ) := U 1 U . We continue assuming that `2 , is a complex,H ( ) ( + p( )) ( ) ∼= (N C) separable,H ∩ inﬁniteB H dimensional⊂ H Hilbert space and we ﬁx someH complete orthonormal system e of . ( n)n N ∈ H

To this end, we introduce the following -subalgebras of the C ∗-algebra ( ) of bounded linear operators on : For 1 ∗p , we consider the BanachB-algebraH H ≤ ≤ ∞ ∗ p( ) ( ) of operators of p-th Schatten class with the Schatten norm p. ForB nH ⊂N, B theH finite dimensional -algebra Mat(n, C) can be viewed naturallyk·k as a -subalgebra∈ of ( ) by virtue of∗ the embedding ∗ B H X 0 X for every X Mat(n, C). 7→ 0 0 ∈ Note that we write the corresponding operators in a block diagonal form which is based on our initial choice of a complete orthonormal system e . We define ( n)n N (n) ( ) to be the image of this embedding in ( ) and obtain∈ a strictly increasingM ⊂ B chainH of -subagebras whose union is writtenB asH : S n ( ) = n N ( ) and is endowed with∗ the direct limit topology. Clearly, everyMX ∞ (n) belongs∈ M to ∈ M p( ) for every p [1, ] because it is a finite rank operator. This yields the fol- BlowingH chain of -algebras∈ ∞ ∗

(n) (n + 1) ( ) p( ). M ⊂ M ⊂ M ∞ ⊂ B H 5.2 The Banach–Lie group Up( ) 89 H

Let be one of these subalgebras. Then does not have a multiplicative unit in A A 1 ( ) and the unitization of is given by + = C + . Then, Example 4.28 revealsB H that is a Mackey completeA CIA. ByA Proposition· A4.23, we have a natural locally convexA Lie group structure on

g 1 g 1 GL( ) := ( +)× ( 1 ) = (C + )× : A A ∩ { } × A { ∈ · A − ∈ A } and, by Corollary 4.27, the unitary group

1 U( ) := g GL( ) : g∗ = g− A { ∈ A } is a Lie subgroup. We observe that, for an element g 1 + which is invertible in 1 1 ∈ A ( ), the inverse g− is also belongs to + : If = p( ), then this follows B H A A B H from the fact that p( ) is an ideal in ( ): We have B H B H 1 1 1 1 g− = g− ( g) p( ), − − ∈ B H 1 1 so that g− + p( ). If = (n) or = ( ), then the observation follows from the block∈ diagonalB H structureA M of the correspondingA M ∞ operator g. In particular, this shows that GL( ) = GL( ) (1 + ), U(A ) = U( H) ∩(1 + A). A H ∩ A To sum it up, we obtain locally exponential Lie groups

GL( ), H 1 GLp( ) := GL( ) ( + p( )), GL(H ) := GL(H ) ∩ (1 + B (H )), GL∞(n) := GL(H ) ∩ (1 + M (∞n)) H ∩ M with Lie algebras gl( ) := ( ) H B H glp( ) := p( ), gl(H ) := B (H ), gl∞(n) := M (∞n). M In all cases, the exponential function is given by the usual power series. The corresponding unitary groups

U( ), H 1 Up( ) := U( ) ( + p( )), U(H ) := U(H ) ∩ (1 + B (H )), U∞(n) := U(H ) ∩ (1 + M (∞n)) H ∩ M are Lie subgroups and their Lie algebras are given by the corresponding Lie subalgebras of skew-hermitian operators

u( ) := X ( ) : X ∗ = X H { ∈ B H − } up( ) := u( ) p( ), u(H ) := u(H ) ∩ B (H ), u∞(n) := u(H ) ∩ M (∞n). H ∩ M 2 90 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

Remark 5.7. Note that GL S GL n and that the Lie group topology on ( ) = n N ( ) GL( ) is just the direct limit∞ topology∈ defined by the subgroups GL(n). In fact, each GL(∞n) is a Lie subgroup of GL( ): Every GL(n) is the pre-image of gl(n) under the continuous map GL( ) gl(∞ ), g g 1 (which defines a local chart). Since the finite-dimensional∞ subalgebra→ ∞gl(n)7→is a closed− subset of gl( ), we conclude that GL(n) is a closed subgroup of GL( ). We find open identity∞ neighborhoods V ∞ ⊂ gl( ) and U GL( ) such that exp V : V U is diffeomorphism with inverse map given∞ by ⊂ ∞ | →

X 1 n+1 log : exp 1 : U V, g log g ∞ ( ) g 1 n. = ( V )− ( ) = − n ( ) | → 7→ n=0 −

Hence if g = exp(X ) exp(V ) GL(n), then X = log(g) gl(n) because g 1 gl(n) so that the power∈ series converges∩ in gl(n). This proves∈ that exp(V gl(−n)) =∈ exp(V ) GL(n) and the Lie Subgroup Theorem (Theorem 4.8) reveals that∩ GL(n) is a Lie subgroup∩ of gl( ) with Lie algebra gl(n). Hence, GL( ) = lim GL(n) is a direct ∞ ∞ limit of Lie groups. Note that we have introduced the group GL(−→) as a locally exponential Lie group of invertible elements of the Mackey complete∞-algebra ( ). Another possibility would have been to introduce GL( ) as a direct∗ limit LieM group∞in a categorical sense: The direct system of the GL(n) deﬁnes∞ an object in the category of locally convex Lie groups which is determined up to a unique isomorphism by a universal property. We did not choose that way, because we preferred working with concrete operators. For a profound exposition on Lie Theory of direct limit Lie groups, the reader is referred to the work of H. Glöckner (e.g. the reference [Gloe05]).

Proposition 7.9 in [Gloe05] reveals that also the universal covering group GLf( ) = lim GLf(n) is the direct limit Lie group of the universal covering groups of the subgroups∞ 1 −→GL(n). In fact, the subgroups GLf(n) := q− (GL(n)) form a strictly increasing sequence of Lie subgroups (Corollary 4.9) whose union is GLf( ). Applying Lemma 4.17, we find that the Lie subgroups GLf(n) are the universal covering∞ groups of the GL(n) and that q : q : GL n GL n are the corresponding covering morphisms. That n = GLf(n) f( ) ( ) the requirements| of Lemma→ 4.17 are fulfilled is explained below in the proof of The- orem 5.10 and in Remark 5.11. The results in [Gloe05] reveal that the direct limit topology on GLf( ), defined by the subgroups GLf(n), is a Lie group topology with respect to which∞GLf( ) is a 1-connected Lie group with Lie algebra gl( ). Since a group carries at most∞ one structure of a 1-connected Lie group with a fixed∞ Lie algebra, the direct limit topology coincides with the underlying group topology of the universal covering group GL( ) and we obtain GLf( ) = lim GLf(n). ∞ ∞ −→ 2 For a more explicit construction of the universal covering groups GLf( ) and GLf p(` ) we refer the reader to [Ne98, Section III] and the Appendix of [Ne04b∞]. Remark 5.8. i) During our discussion, we have seen that the unitization of the - 1 ∗ algebra ( ), p( ) ( ) is given by + = C + and that an Aa ∈ {M ∞ B H } ⊂ B H A · A C element + is invertible in + if and only if it is invertible in the ∗-algebra ∈ A A 5.2 The Banach–Lie group Up( ) 91 H

38 ( ). For any element X , we conclude that the spectrum of X in B H ∈ A A coincides with the spectrum of X in the C ∗-algebra ( ). B H ii) Let be one of the Mackey complete -CIAs in the set A ∗ (n), ( ), p( ), ( ) . {M M ∞ B H B H } We then have

Sp(g∗ g) C ] , 0] for all g GL( ). ⊆ \ − ∞ ∈ A

This is clear when (n), ( ) is a C ∗-algebra. Otherwise, we use our A ∈ {M B H } previous observation that the spectrum of g∗ g in GL( ) coincides with the spec- A trum of g∗ g in ( ). Holomorphic calculus on yields a smooth function B H A U a Sp a log : := × : ( ) C ] , 0] + { ∈ A+ ⊆ \ − ∞ } → A (cf Theorem 4.22). The restriction log : U (1 + ) is a smooth map. We ∩ A → A write Herm( ) := X : X ∗ = X for the closed real subspace of Hermitian A { ∈ A } 1 operators. For every g GL( ), the operator X (g) := 2 log(g∗ g) is hermitian and we obtain a smooth∈ mapAX : GL( ) Herm( ). We deﬁne the smooth polar map A → A

p :U( ) Herm( ) GL( ), (u, X ) u exp(X ), A × A → A 7→ which has a smooth inverse given by

1 p− : GL( ) U( ) Herm( ), g (g exp( X (g)), X (g)). A → A × A 7→ − Hence, the polar map p is a diffeomorphism. 39

Remark 5.9. i) The underlying group topology of the Lie groups GLp( ) and Up( ) 40H H are metrizable and the metric is given by d(g, g0) := g g0 p. However, the 1 k − k metric d is not complete, because gn := diag( n , 1, 1, 1, 1, . . .), n N, deﬁnes a Cauchy-sequence g in GL without a limit in GL ∈. However, if ( n)n N p( ) p( ) g 1 X is a Cauchy∈ sequenceH such that g 1 is a boundedH sequence ( n = + n)n N ( n− )n N ∈ ∈ 38Recall that SP X : λ : λ 1 X / . ( ) = C +× 39 A { ∈ · − ∈ A } In general, a Mackey complete -CIA is called hermitian if every self-adjoint element a∗ = a has ∗ real spectrum. In this case, [Bil10, Cor. 7.7] shows that Sp(a∗a) R 0 holds, so that we can deﬁne analogously a smooth logarithmic function log and hence, a smooth⊆ inverse≥ of the polar map p. If

X ∗ = X is selfadjoint, then so is exp(X ) and the Spectral Mapping Theorem (Theorem 4.22 ii)) shows that the spectrum of exp(X ) is contained in R>0. Therefore, the diffeomorphic polar map shows that every element g × uniquely decomposes into a product of a unitary element and a positive element. This is called the∈polar A decomposition. 40 1 1 This follows from the fact that GLp( ) = (C + p( ))× ( + p( )) is endowed with the 1 H · B H ∩ B H 1 subspace topology of (C + p( ))× which is an open subset of the Banach space C + p( ) with 1 · B H 1 · B H norm λ + X := λ + X p. In particular, (C + p( ))× is metrizable with metric d(g, g0) = k · k | | k k · B H g g0 and the restriction of d to GLp( ) has the form d(g, g0) = g g0 p because, if g, g0 1k − k H k − k ∈ + p( ), then their difference g g0 is a p-th Schatten class operator. The topology of Up( ) is B H − H the subspace topology of GLp( ). H 2 92 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

in GL( ), then (gn) converges in GLp( ): That (gn) is a Cauchy sequence in H H GLp( ) implies that (X n) is a Cauchy sequence in p( ) with limit X p( ). For gH:= 1 + X , we have B H ∈ B H

1 1 1 1 g = + X n + (X X n) = ( + X n)( + gn− (X X n)). − − 1 For sufﬁciently large n, we have gn− (X X n) < 1 so that the operator 1 1 − + gn− (X X n), and hence g, is invertible in ( ). Therefore, we have − 1 B H g GL( ) ( + p( )) = GLp( ). In particular, we conclude that the met- ∈ H ∩ B H H ric d turns Up( ) into a complete metric space. On GLp( ), one has to take H 1 1 H the metric de(g, g0) := d(g, g0) + d(g− , g0− ) in order to obtain a complete metric space.

ii) The fact that S n is a dense subalgebra of (see e.g. ( ) = n N ( ) p( ) [GK69, Thm.M III.7.1∞]), implies∈ M that also GL( ) = GL( ) (1B+ H ( )) is a 1 ∞ H ∩ 1 M ∞ dense subgroup of GLp( ) = GL( ) ( + p( )): If g = + X GLp( ) for some X , thenH we findH a sequence∩ B XH in converging∈ H to p( ) ( n)n N ( ) ∈ B H ∈ M ∞ X in p( ) and, in particular in ( ). Since GL( ) is an open subset, we B H1 1 B H H have gn := + X n GL( ) ( + ( )) = GL( ) for sufficiently large n N. ∈ H ∩ M ∞ ∞ ∈ Since X n X in p( ), we have gn g in GLp( ) (e.g. using the metric d). The argument→ inB theH unitary case, for→ the analogousH fact that U( ) is a dense ∞ subgroup of Up( ), relies on the polar decomposition (see Remark 5.8): If u U , we first findH a sequence g in GL converging to g : u. By polar∈ p( ) ( n)n N ( ) = H ∈ ∞ decomposition, we find a unique decomposition gn = un exp(X n) with un U( ) ∈ ∞ and X n∗ = X n gl( ) = ( ). That gn g = u means that un u in Up( ) ∈ ∞ M ∞ → → H and X n 0 in p( ). This shows that U( ) is dense in Up( ). → B H ∞ H iii) For arbitrary 1 p , the groups Up( ) and GLp( ) are separable: For each ≤ ≤ ∞ H H n N, the embedding (n) , p( ) is continuous because the p-th Schatten class∈ norm p restrictedM to the→ BfiniteH dimensional subalgebra is continuous. Us- ing the metrick·kd, we also see that the embeddings GL(n) = GL( ) (1+ (n)) , 1 H ∩ M → GL( ) ( + p( )) = GLp( ) and U(n) , Up( ) are continuous. The claim nowH follows∩ fromB H the fact thatH U(n) and GL→(n) areH separable groups together with the density of U S U n and GL S GL n in U and ( ) = n N ( ) ( ) = n N ( ) p( ) ∞ ∈ ∞ ∈ H GLp( ), respectively. H Our next aim is to show that the Banach–Lie groups GLp( ),Up( ) and the direct limit Lie groups GL( ),U( ) are connected groups andH thatH their fundamental groups are isomorphic∞ to Z.∞ For detailed informations on these issues, the reader is referred to [Ne02] and to the Appendix of [Ne04b].

Theorem 5.10. Let G be one of the Lie groups in GL(n), GL( ), GLp( ) and let { ∞ H } U U(n),U( ),Up( ) be the corresponding unitary group. Then the Lie groups G and∈ U { are connected∞ andH their} fundamental groups are isomorphic to Z. A generator for π1(G) and π1(U) is given by the homotopy class of the curve γ(t) := exp(2π i t E11), · where E11 denotes the elementary diagonal operator E11 := e1 e1∗. ⊗ 5.2 The Banach–Lie group Up( ) 93 H

Proof. The chain of inclusions maps

ι1 ιn ι ιp T = U(1) GL(1) GL(n) ∞ GL( ) GLp( ) ∞ H induces, for every k N, a chain of canonical group homomorphisms ∈ πk(ι1) πk(ιn) πk(ι ) πk(ιp) πk(T) πk(GL(1)) πk(GL(n)) ∞ πk(GL( )) πk(GLp( )) ∞ H of the homotopy groups πk(G). For k = 0 and k = 1, these group homomorphisms are isomorphisms: It follows from the Corollary of Theorem 12 in [Pa66] that πk(ιp) is an isomorphism, from Theorem A.10 in [Ne04b] that πk(ι ) is an isomorphism and from the Stability Theorem for Homotopy Groups (see [Hus94∞ , Thm. 8.4.1]) that πk(ιn) is an isomorphism. Finally, that the inclusion ι : U , G is a (weak) homotopy equivalence follows from the polar decomposition (Remark→ 5.8). In particular, πk(ι) : πk(U) πk(G) is an isomorphism. A generator of the fundamental group → 2π i t π1(T) is given by the homotopy class of the curve γ(t) := e . Therefore, the homotpy class of γ embedded into G, resp. into U, yields a generator of π G π U . 1( ) ∼= 1( ) The details of the proof can be found in [Ne02] and in the Appendix of [Ne04b]. Remark 5.11. i) The groups G = GL( ) and U = U( ) are contractible as shown H H by Kuiper in [Kui65]), so that we have πk(GL( )) = πk(U( )) = 1 for all H H { } k N0. ∈ ii) Let G be one the groups as in Theorem 5.10 and consider the universal covering q : Ge G. We have seen in Proposition 4.15 that π1(G) = ker(q). Hence → ∼ ker(q) = Z and, in terms of the isomorphism ϕ : ker(q) π1(G) in Proposi- ∼ → tion 4.15, we have [γ] = ϕ(expg(2π i E11)), so that a generator of ker(q) is given · by expg(2π i E11). Note that, for k N and t [0, 1], we have · ∈ ∈ 1 2π i t γk(t) := exp(2π i t Ekk) = (1 e )Ekk · − − and [γk] = ϕ(expg(2π i Ekk)). The loops γk are all homotopic to γ1 and a corresponding homotopy is obtained as follows: There exists a unitary operator 1 u U( ) such that uE11u− = Ekk, namely the permutation operator switch- ∈ H ing e1 and ek. Using functional calculus in ( ), we find some skew hermitian element X ( ) (actually X (k)B) withH u = exp(X ) (cf. the proof of Proposition ∈3.29 B iH)). We define ∈ M 1 2π i t 1 2π i t H(t, s) := (1 e ) exp(sX )E11 exp( sX ) = exp(sX )[ (1 e )E11] exp( sX ) − − − − − − which is the required homotopy. We conclude that [γk] = [γ1] and thus expg(2π i Ekk) = expg(2π i E11) for all k N (k n if G = GL(n)). Remark 5.12.· Applying Lemma· 4.17 to the∈ unitary≤ subgroup U of G (from Theo- rem 5.10), we find that

1 U:e = q− (U) = expg(X ) : X ∗ = X Grp Ge 〈 − 〉 ⊂ is the universal covering group of U and q : U U is the corresponding covering Ue e morphism. | → 2 94 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

5.3 Derivations of Banach algebras and 1-cocycles

It is a well-known result that every derivation of a von Neumann algebra is inner. This means that every linear map ω : satisfying the productA rule ω(XY ) = ω(X )Y + X ω(Y ) is of the form ω(X )A = XA → AAX for some ﬁxed A (see e.g. [Sa66, Thm. 1], [Ka66, Thm 15] or [BrRo79, Cor.− 3.2.47]). In particular,∈ A every derivation of the type I von Neumann algebra ( ) of bounded linear operators on a B H (complex) Hilbert space is inner. For arbitrary C ∗- algebras or Banach algebras, this result is no longer true: For p [1, ], consider the two-sided ideals p( ) ( ) ∈ ∞ 1 B H ⊂ B H of pth Schatten class operators. Fix some B ( ) C + p( ) . Then ω(X ) := ∈ B H \{ · B H } XB BX is a derivation of the Banach algebra p( ) which is not inner. However, − B H it is true that every derivation of p( ) is of that form (cf. Proposition 5.23 below), which is sometimes called spatialB. TheH problem of describing non-inner derivations of operator algebras is similar to that of describing non-trivial group 1-cocycles. In fact, (continuous) derivations of Banach algebras integrate to 1-cocycles (see Re- A 1 mark 5.16 below). In the case of p( ), we obtain β(g) := gBg− B via integration of ω(X ) = XB BX and β is a continuousB H 1-cocycle corresponding− to the conjugation − action of GLp( ) on p( ). The observation that non-inner derivations integrate to non-trivial 1-cocyclesH B is ourH motivation to study derivations on general Banach modules in some more detail.

Deﬁnition 5.13. Let E be a Banach space and be a Banach algebra. A i) We say that E is a Banach left module over if there exists a bilinear map A E E, (A, v) A.v A × → 7→ such that

(M1) A.v A v for all A and v E, k k ≤ k k k k ∈ A ∈ (M2) (AB).v = A.(B.v) for all A, B and v E. ∈ A ∈ If is a unital Banach algebra with unit 1 , then we additionally require A ∈ A (M3) 1.v = v for all v E. ∈ ii) We analogously deﬁne a Banach right module over . We call E a Banach bimodule over if it is both a Banach left and a Banach rightA module over such that A A (M4) (A.v).B = A.(v.B) for all A, B and v E. ∈ A ∈ iii) For simplicity, we call E simply a Banach module (over ), if it is either a Banach left module, a Banach right module or a Banach bimoduleA over . In case that E is a Banach left module, we put v.A := 0 and, in case that E isA a Banach right module, A.v := 0 for A and v E. ∈ A ∈ 5.3 Derivations of Banach algebras and 1-cocycles 95

Remark 5.14. If E is a Banach module over , then also its topological dual space E∗ carries a canonical structure of a Banach moduleA via the relations

A.f (v):= f (v.A), f .A(v):= f (A.v), for all f E∗, v E and A . We call E∗ the dual Banach module (over ). ∈ ∈ ∈ A A Deﬁnition 5.15. Let E be a Banach module over . A linear map ω : E is called a derivation if it satisﬁes the equation A A →

ω(AB) = A.ω(B) + ω(A).B for all A, B . ∈ A It is called an inner derivation if it is of the form

ω(A) := A.v v.A − 1 for some v E. We denote by Z ( , E) the vector space of derivations and by 1 B ( , E) its∈ subspace of inner derivations.A The quotient A 1 1 1 H ( , E) := Z ( , E)/B ( , E) A A A is called the ﬁrst order Hochschild cohomology space. Remark 5.16. Suppose that is a unital Banach algebra with unit 1. Then, every derivation ω : E on a one-sidedA Banach module E is inner, since we have ω(A) = A.ω(1) or ω(A)A = →ω(1).A respectively. For derivations ω on a Banach bimodule E over , we merely observe that ω(1) = 0. In general, derivations of naturally integrate A A to group 1-cocycles of the unit group G := × : The Lie algebra representation ρ(A)v := A.v v.A integrates to a norm-continuousA ⊂ A representation π : G GL(E) of G on E that we− call the conjugation representation and which is deﬁned as→

g.v, E is a Banach left module  1 π(g)v := v.g− , E is a Banach right module  1 g.v.g− , E is a Banach bimodule. If ω is a given derivation, then the map ¨ ω g 1 , if E is a Banach left module β : G E, g β g : ( ) ( ) = − 1 1 → 7→ ω(g ).g− , else − deﬁnes a 1-cocycle corresponding to π. That β satisﬁes the 1-cocycle equation

β(gh) = π(g)β(h) + β(g), for all g, h G ∈ is easily veriﬁed using the identity gh 1 = (g 1)(h 1) + (g 1) + (h 1). If ω is continuous, then so is β. In this case,− β is the− integration− of ω−to the Lie− group G because, for each A , we have ∈ A 1 lim β(exp(tA)) = ω(A). t 0 t → 2 96 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

Conversely, starting with a continuous 1-cocycle β : G E, we observe that β is automatically smooth since the conjugation representation→ is norm-continuous d A d tA (Lemma 4.30). However, the derivative β( ) := d t t=0β(exp( )) is in general only a Lie derivation41 on E because it is a linear map satisfying| the relation

dβ([A, B]) = A.dβ(B) dβ(B).A + dβ(A).B B.dβ(A) for all A, B . − − ∈ A Typical examples of Lie derivations which are no derivations arise whenever has a nonzero trace functional, i.e. a linear map Tr : C for which Tr(AB) = ATr(BA) holds for all A, B . Then, put L(A) := Tr(A) 1A. This → is a Lie derivation, since we 42 have L([A, B]) = 0,∈ A but it is not a derivation . · Finally, if is not unital and if E is a Banach module over , then E uniquely extends A 1 A 1 v v to a Banach module over the unitization + := C via . = and resp. or v.1 := v for all v E. If E is a Banach bimodule,A then· also⊕ A every derivation ω uniquely ∈ 1 extends to a derivation of + via ω( ) := 0. Hence, we can use the same formula for β as above to obtain a 1-cocycleA of GL : 1 which integrates ω. ( ) = +× ( + ) E A A ∩ A However, if is only a one-sided Banach module, then a derivation ω extends to + if and only if it is inner. A

In the following, we specialize to Banach modules E over ( ), the von Neumann algebra of bounded linear operators on a complex, inﬁnite-dimensional,B H separable Hilbert space . Note that E is automatically a Banach module over the (nonuni- H tal) Banach subalgebras p( ) for 1 p , endowed with the pth Schatten B H ≤ ≤ ∞ norm. We will see below that the continuous derivations of p( ) are in one- to-one correspondence with the continuous 1-cocycles of the correspondingB H unitary 1 1 group Up( ). We write Z (Up( ), E) := Z (Up( ), π, E) for the vector space of all continuousH 1-cocycles correspondingH to the conjugationH representation π Up( ) 1 | H of Up( ) GL( ) on E. Accordingly, B (Up( ), E) denotes the subspace of 1- H ⊂ H 1 H coboundaries and H (Up( ), E) the quotient space. H Lemma 5.17. Let E be a Banach module over ( ). For any p [1, ], the complex linear derivative deﬁnes a linear isomorphism B H ∈ ∞

d : Z 1 U , E Z 1 , E , β dβ ( p( ) ) ( p( ) ) C H → B H 7→ from the vector space of continuous group 1-cocycles to the vector space of continuous derivations on E.

Proof. Suppose that β :Up( ) E is a continuous 1-cocycle corresponding to the conjugation representation.H Since→ the conjugation representation is norm-continuous, we conclude that β is smooth (Lemma 4.30) so that the complex linear derivative ω := 41A linear map L : E is called a Lie derivation on the Banach module E if it satisﬁes the equation L([A, B]) = [A, L(B)]A + [ →L(A), B] for all A, B , where we have put [A, L(B)] := A.L(B) L(B).A and [L(A), B] := L(A).B B.L(A). ∈ A − 42 1 1 If L was a derivation,− then L( ) = 0 and L vanishes on C . If A is not a constant multiple of the 2 identity, then we have 2 Tr(A) A = Tr(A ) 1 which is possible· only if Tr(A) = 0 and thus L(A) = 0. But L = 0 contradicts the assumption· of Tr being· nonzero. 5.3 Derivations of Banach algebras and 1-cocycles 97 dβ : E exists. The restriction of β to the compact subgroup U n yields a C p( ) ( ) B H → 1-coboundary β(g) = vn π(g)vn for some vn E and all g U(n) (Corollary 3.42). We conclude that ω A −A.v v .A for all A ∈ n Mat∈n, . In particular, the ( ) = n n ( ) ∼= ( C) restriction of ω to (n) is a derivation.− This∈ holds M for every n N so that M ∈ [ ω(AB) = A.ω(B) + ω(A).B for all A, B ( ) := (n). ∈ M ∞ n N M ∈ Since ω is a continuous linear mapping and since ( ) is a dense subalgebra of M ∞ p( ), we conclude that ω : p( ) E is a continuous derivation. Therefore, the B H B H → above map d is a well-deﬁned linear map. It is injective because Up( ) is a connected Lie group (Proposition 4.34). It is surjective because every continuousH derivation integrates to a continuous 1-cocycle via the formula from Remark 5.16.

Lemma 5.18. Let E be a Banach module over ( ) and let E∗ be the corresponding B H dual Banach module. Then, every continuous derivation ω : ( ) E∗ is inner. In 1 B∞ H → particular, we have H (U ( ), E∗) = 0 . ∞ H { }

Proof. Let ω : ( ) E∗ be a continuous derivation and let B∞ H → ¨ ω g 1 , E is a Banach right module β g : ( ) ( ) = − 1 1 ω(g ).g− , else − be the corresponding integrated 1-cocycle on U ( ). The estimate ∞ H β(g) ω g 1 2 ω k k ≤ k k k − k ≤ k k shows that β is a bounded 1-cocycle. The group U ( ) is amenable since it contains the directed union of compact subgroups U :∞ HS U n as a dense subgroup. ( ) = n N ( ) Hence, Theorem 3.40 i) shows that β is a 1-coboundary∞ ∈ which is to say that ω is an inner derivation. The second assertion thus follows from Lemma 5.17.

Lemma 5.19. Let E be a Banach module over ( ) such that any f E∗ with X .f = B H ∈ 0 = f .X for all X ( ) is zero. Then, every continuous derivation ω : ( ) E∗ is inner. ∈ B∞ H B H →

Proof. In view of Lemma 5.18, it sufﬁces to show that the restriction map

Res Z 1 E Z 1 E : ( ( ), ∗) ( ( ), ∗), ω ω ( ) B H → B∞ H 7→ |B∞ H A X is injective. Suppose that ω ( ) = 0. For ( ) and ( ), we then have |B∞ H ∈ B H ∈ B∞ H ω(A).X = ω(AX ) A.ω(X ) = 0 − and X .ω(A) = ω(XA) ω(X ).A = 0 − because the compact operators ( ) form a two-sided ideal in ( ). The assumption thus entails that ω(A)B = ∞0. H B H 2 98 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

Lemma 5.20. Let E be a Banach module over ( ). Assume that, for some p < and for all f E , we have f sup A.Bf orH f sup f .A . Then, any∞ ∗ = A p 1 = A p 1 ∈ k k k k ≤ k k k k k k ≤ k k continuous derivation ω : p( ) E∗ is inner. B H →

Proof. It sufﬁces to prove that any derivation ω : p( ) E∗ which is continuous w.r.t. the pth Schatten norm is also continuous w.r.t.B theH operator→ norm. If this is the case, then ω uniquely extends to a continuous derivation on ( ) which contains B∞ H p( ) as a dense subalgebra. The assertion then follows from Lemma 5.18. SupposeB H that, for B , we have ω B sup A.ω B . Then we obtain p( ) ( ) = A p 1 ( ) the estimate ∈ B H k k k k ≤ k k

ω(B) = sup A.ω(B) = sup ω(AB) ω(A).B k k A p 1 k k A p 1 k − k k k ≤ k k ≤

sup [ ω AB p + ω A p B ] ≤ A p 1 k k k k k k k k k k k k ≤ sup 2 ω A p B 2 ω B . ≤ A p 1 k k k k k k ≤ k k k k k k ≤ This estimate shows that ω is continuous w.r.t. the operator norm.

Remark 5.21. We will see that the condition in Lemma 5.20 is satisﬁed in all our examples below. To see this, the following observations will be helpful: Let E be a Banach module over ( ). Then, one always has B H sup A.v v and sup v.A v A p 1 k k ≤ k k A p 1 k k ≤ k k k k ≤ k k ≤ for all v E and p [1, ] by virtue of the estimate ∈ ∈ ∞

A.v A v A p v k k ≤ k k k k ≤ k k k k and of the analogous estimate v.A v A p. If, for any v E, one can ﬁnd some ﬁnite rank operator A suchk thatk ≤A.v k =k k vk (resp. v.A = ∈v ), then we have v sup A.v (resp. v supk k kv.Ak ) for allk v k E andk k all p 1, . = A p 1 = A p 1 [ ] Assumek k thatk k ≤ thek latterk holdsk fork the Banachk k ≤ k modulek E. Then∈ it also holds∈ for∞ the Banach module (E) of bounded linear endomorphisms on E, endowed with the canonical moduleB structure

(A.X )(v) := A.(X (v)) and (X .A)(v) := X (A.v), where X (E), v E and A ( ). One veriﬁes that ∈ B ∈ ∈ B H X = sup X (v) = sup sup A.(X (v)) k k v 1 k k v 1 A p 1 k k k k≤ k k≤ k k ≤ = sup sup (A.X )(v) = sup A.X . A p 1 v 1 k k A p 1 k k k k ≤ k k≤ k k ≤ Example 5.22. The following are natural examples of Banach modules over ( ): B H 5.3 Derivations of Banach algebras and 1-cocycles 99

i) The Hilbert space endowed with the natural left action H ( ) , (A, v) A.v B H × H → H 7→

is a Banach left module. Its dual ∗ is a natural Banach right module. In view of H the reﬂexivity, is the dual Banach module of ∗. For any nonzero v , the rank-1 operatorHA : 1 v v satisﬁes A.v vHand v .A v . By Remark∈ H 5.21, = v 2 ∗ = ∗ = ∗ we have k k ⊗

v = sup A.v and v∗ = sup v∗.A k k A p 1 k k k k A p 1 k k k k ≤ k k ≤ for all v and p [1, ]. ∈ H ∈ ∞ ii) The Banach space ( ) itself carries a canonical Banach bimodule structure deﬁned by the multiplicationB H

( ) ( ) ( ), (A, B) AB. B H × B H → B H 7→ The Banach space ( ) is the dual space of the trace class operators 1( ) (see [GK69, Thm. III.12.1B H ]) which carries a canonical Banach bimodule structureB H as a two sided ideal

( ) 1( ) 1( ), (A, X ) AX , B H × B H → B H 7→ 1( ) ( ) 1( ), (X , A) XA. B H × B H → B H 7→ Hence ( ) is the dual Banach module of 1( ). By Remark 5.21, we have, for all BB H ( ) and p [1, ], B H ∈ B H ∈ ∞ B = sup AB k k A p 1 k k k k ≤ and, hence,

sup BA = sup A∗B∗ A p 1 k k A p 1 k k k k ≤ k k ≤ = sup AB∗ = B∗ = B . A p 1 k k k k k k k k ≤ We now obtain the following results as corollaries:

Proposition 5.23. Every derivation of ( ) (on itself) is inner. For 1 p , all B H ≤ ≤ ∞ derivations ω of p( ) (on itself) are spatial in the sense that they are of the form ω(X ) = XB BXB forH some B ( ). − ∈ B H

Proof. Apply Lemma 5.19, Lemma 5.18 and Lemma 5.20 with E∗ = ( ). B H Finally, we explicitly describe the 1-cocycles of the following norm continuous unitary representations of the unitary group Up( ) for p [1, ]: H ∈ ∞

1.) π1 :Up( ) , (g, v) gv (identical representation), H × H → H 7→ 2 100 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles

1 2.) π1∗ :Up( ) ∗ ∗, (g, v∗) v∗ g− (dual representation), H × H → H 7→ 3.) π2 :Up( ) 2( ) 2( ), (g, A) gA (left multiplication representation), H ×B H → B H 7→ 1 4.) π20 :Up( ) 2( ) 2( ), (g, A) Ag− (right multiplication representation) H × B H → B H 7→

1 5.) π3 :Up( ) 2( ) 2( ), (g, A) gAg− (conjugation representation). H × B H → B H 7→ 1 Proposition 5.24. For all p [1, ], we have H (Up( ), π1, ) = 0 . ∈ ∞ H H { }

Proof. Apply Lemma 5.17 and Lemma 5.19 with E∗ = . H 1 Proposition 5.25. For all p [1, ], we have H (Up( ), π1∗, ∗) = 0 . ∈ ∞ H H { }

Proof. Apply Lemma 5.17 and Lemma 5.19 with E∗ = ∗. Alternativley, the result follows from the fact that the ﬁrst order cohomology spaceH for the dual representation vanishes if and only if it vanishes for the original representation.

Proposition 5.26. For 1 p , let β be a continuous 1-cocycle of the left multiplica- ≤ ≤ ∞ tion representation π2. Then, there exists some B ( ) such that for all g Up( ), ∈ B H 1∈ 1 H 1 we have β(g) = gB B. For p > 2, we have B q( ) with q deﬁned by p + q = 2 . In particular, we have− ∈ B H

¨ 1 ( ), p 2 Z (Up( ), π2, 2( )) = B H ≤ H B H ∼ q( ), p > 2. B H

Proof. Apply Lemma 5.17 and Lemma 5.19 with the dual Banach left module E∗ = to see that dβ X XB for some uniquely determined B holds for ( ) C( ) = ( ) B H ∈ B H the complex linear derivative. That B q( ) for p > 2 follows from Lemma 5.4 with r = 2. ∈ B H

Proposition 5.27. For 1 p , let β be a continuous 1-cocycle of the right mul- ≤ ≤ ∞ tiplication representation π20 . Then, there exists some B ( ) such that for all 1 ∈ B H g Up( ), we have β(g) = Bg− B. For p > 2, we have B q( ) with q de- ∈ H1 1 1 − ∈ B H ﬁned by p + q = 2 . In particular, we have ¨ 1 ( ), p 2 Z (Up( ), π20 , 2( )) = B H ≤ H B H ∼ q( ), p > 2. B H

Proof. Apply Lemma 5.17 and Lemma 5.19 with the dual Banach right module E∗ = to see that dβ X BX for some uniquely determined B holds for ( ) C( ) = ( ) B H − ∈ B H the complex linear derivative. That B q( ) for p > 2 follows from Lemma 5.4 with r = 2. ∈ B H 5.3 Derivations of Banach algebras and 1-cocycles 101

Proposition 5.28. For 1 p , let βi be a continuous 1-cocycle of the conjugation ≤ ≤ ∞ representation π3. Then there exists some B ( ) such that for all g Up( ), we 1 ∈ B H ∈ 1 H1 1 have β(g) = gBg− B. For p > 2, we have B q( ) with q deﬁned by p + q = 2 . In particular, we have− ∈ B H

¨ 1 1 ( )/C , p 2 Z (Up( ), π3, 2( )) = B H · ≤ H B H ∼ q( ), p > 2. B H

Proof. Apply Lemma 5.17 and Lemma 5.19 with the dual Banach bimodule E∗ = to see that dβ X XB BX for some B holds for the complex ( ) C( ) = ( ) linearB H derivative. The operator B −is unique up to adding∈ B H a constant multiple of the identity. That B q( ) for p > 2 follows from Lemma 5.5 with r = 2. ∈ B H

The reader should note that the preceding results show that the group Up( ) does not have property (FH). Since property (T) implies property (FH) for arbitraryH topological groups, this entails that Up( ) does not have property (T). H Corollary 5.29. For 1 p < , the group Up( ) neither has property (FH) nor has property (T). ≤ ∞ H

Remark 5.30. For p = 2, the fact that U2( ) does not have property (T) has already H been shown in [Pe17]. The question whether U2( ) has property (FH) is however stated there as an open problem (see [Pe17, 3.5]).H 102 6 Unitary Highest Weight Representations (HWR) of Lie algebras

6 Unitary Highest Weight Representations (HWR) of Lie algebras

This section provides some foundations on unitary highest weight representations of 2 gl( ), their extension to representations of the Banach-completions glp(` ) and their ∞ 2 integration to representations of the corresponding Banach–Lie groups GLp(` ).

Let g be a complex Lie algebra with Lie bracket [ , ]. A representation (ρ, V ) of the Lie algebra g on a complex vector space V is a homomorphism· · of Lie algebras ρ : g End(V ). A linear subspace W V is called g-invariant if ρ(g)W W. In case that→ 0 and V are the only g-invariant⊆ subspaces, we say that (ρ, V ) is an⊆ irreducible representation{ } of g. Representation theorists ask for a classification of all irreducible representations (up to equivalence) of a given Lie algebra g. A lot of research has been done on this problem and it turned out that one usually needs more structure both on the Lie algebra g and on the underlying vector space V in order to obtain precise and handy classification results. Therefore, we introduce in Subsection 6.1 the notion of an involutive split Lie algebra with compact roots, which can be regarded as a generalization of the Lie algebra gl( ). Any such Lie algebra g has a maximal abelian subalgebra h which decomposes g∞into eigenspaces with respect to the adjoint action of g on itself. These eigenspaces are called root spaces and the corresponding eigenvalues are called the roots of g. Moreover, there exists an increasing exhaustive S union g = j J gj of finite-dimensional Lie subalgebras gj. Hence, it is natural to ask for a classification∈ of those irreducible representations (ρ, V ) of g, which reflect the structure of the Lie algebra in the sense that h decomposes V into eigenspaces and that S V = j J Vj can be written as an exhaustive union of gj-invariant finite-dimensional ∈ subspaces Vj. Unitary highest weight representations have these properties and their isomorphism classes are classified by certain orbits of the natural action of the Weyl group of g on the dual space h∗ (Theorem 6.15). This theory is the basis of our study ofW unitary highest weight representations of g = gl( ) in Subsections 6.2 and 6.3, where we extend and integrate unitary highest weight∞ representations of gl( ) 2 ∞ to the Lie groups GL( ) and GLp(` ). This is done in Subsection 6.2 on the basis of the h-weight space decomposition∞ of unitary highest weight representations and in Subsection 6.3 on the basis of their direct limit structure.

6.1 General facts on unitary highest weight representations

Deﬁnition 6.1. Let g be a complex Lie algebra.

i) An involution on g is an antilinear involutive antiautomorphism : g g. In particular, this means that we have ∗ →

[Z1, Z2]∗ = [Z2∗, Z1∗] and (Z ∗)∗ = Z for all Z, Z1, Z2 g. ∈ 6.1 General facts on unitary highest weight representations 103

ii) A representation (ρ, V ) of g on a pre-Hilbert space V with positive definite Her- mitian form , is called unitary if 〈· ·〉 ρ(Z)v, w = v, ρ(Z ∗)w 〈 〉 〈 〉 holds for all v, w V and Z g. In this case, we call , a contravariant scalar product (or inner∈ product) on∈ V . 〈· ·〉 Remark 6.2. 1. If one starts with a real Lie algebra g, one can consider its universal complexification g which carries a natural involution which is given by C (X + i Y )∗ := X + i Y . Conversely, given a complex Lie algebra with involution , one can define− the real subalgebra g : X g : X X , the so-called real R = ∗ = ∗form. The universal complexification of the{ real∈ form g −coincides} with g. R 2. Assume that (ρ, V ) is a unitary representation of g. Let := V be the Hilbert space completion of V . For Z g, the operator ρ(Z) EndH(V ) does not automat- 43 ically extend to a linear operator∈ on . . This happens∈ if ρ(Z) is continuous p with respect to the norm v := v,Hv which is to say that the operator norm k k 〈 〉 ρ(Z) := sup v 1 ρ(Z)v is finite. If this is the case for every Z g, then we sayk thatk the representationk k≤ k k(ρ, V ) extends to . ∈ H Definition 6.3. Let g be a complex Lie algebra and h g be an abelian subalgebra. ⊆ We write h∗ := α : h C linear for the space of linear functionals on h. { → }

i) For α h∗, α ∈ g := Z g : ( H h)[H, Z] = µ(H)Z { ∈ ∀ ∈ } is the root space of the root α and

α ∆ := α h∗ 0 : g = 0 { ∈ \{ } 6 { }} is the root system of g. We say that g has a root decomposition (with respect to h) if

M α g = h g . ⊕ α ∆ ∈ We call the Lie algebra g split if it has a root decomposition with respect to some abelian subalgebra h g. ⊆ ii) Let (ρ, V ) be a representation of g. For a linear functional µ h∗, we denote by ∈ µ V := v V : ( H h)ρ(H)v = µ(H)v { ∈ ∀ ∈ } the weight space of weight µ and by

µ V := µ h∗ : V = 0 P { ∈ 6 { }} the weight set/system of the representation (ρ, V ). We call (ρ, V ) a weight representation (with respect to h) if

M µ V = V . µ V 43 ∈P Except that V is ﬁnite–dimensional where we have = V . H 104 6 Unitary Highest Weight Representations (HWR) of Lie algebras

Remark 6.4. i) Consider the adjoint representation

ad : g End(g), Z [Z, ] → 7→ · of g on itself. If h is an abelian subalgebra of g, then g has a root decomposition with respect to h if and only if the adjoint representation is a weight representation 0 (w.r.t. h) and h = g . If this is the case, then h is maximal abelian in the sense that there is no abelian subalgebra containing h.

ii) An easy computation reveals the following relation between root and weight spaces α µ µ α ρ(g )V V + . (3) 44 ⊆ µ If a representation (ρ, V ) is generated by some (nonzero) weight vector v V , then this relation shows that it is a weight representation. This follows from∈ the observation that V is spanned by all weight vectors

α α α µ Pn α g n g n 1 g 1 + i=1 i w ρ( )ρ( − ) ... ρ( )v V where n N, αi ∆(i = 1, . . . , n). ∈ ⊆ ∈ ∈ In particular, we conclude that V µ + Z[∆], hence two weights coincide up to a ﬁnite Z-linear combinationP of roots.⊆ Moreover, if (ρ, V ) is irreducible, then the argument works for every weight µ V (provided that V is nonempty). The basic idea of highest weight theory is∈ to P choose a distinguishedP weight λ

V from which one obtains all other weights in V and which determines the∈ Pisomorphism class of the irreducible representationP(ρ, V ). Note that the weight set V is an invariant of the isomorphism class of (ρ, V ). P Deﬁnition 6.5. Let g be a complex Lie algebra, h g be an abelian subalgebra and : g g be an involution on g. ⊆ ∗ → i) The Lie algebra g, resp. the triple (g, h, ), is called an involutive split Lie algebra if g has a root decomposition with respect∗ to h such that the involution is compatible with the root decomposition in the following sense: ∗

α α ( α ∆ 0 ) Z g = Z ∗ g− . ∀ ∈ ∪ { } ∈ ⇒ ∈ In particular, this implies that α ∆ = α ∆ and that H∗ h for all H h. ∈ ⇒ − ∈ ∈ ∈ ii) A subset ∆+ ∆ is called a positive system of roots if ⊂ + + + + ∆ ( ∆ ) = ∆ and R 0[∆ ] R 0[ ∆ ] = 0 . ∪ − ≥ ∩ ≥ − { } iii) Let (ρ, V ) be a representation of an involutive split Lie algebra g and let ∆+ ∆ be a positive root system. The representation (ρ, V ) is called a highest weight⊂ representation with highest weight λ h∗ (and with respect to the positive system ∆+) if V is generated by a (nonzero)∈ weight vector v V λ satisfying ∈ α ρ(g )v = 0 for all positive roots α ∆+. { } ∈ A vector v V with the last property is called primitive. ∈ 44This means that V is the smallest g-invariant subspace containing v. 6.1 General facts on unitary highest weight representations 105

Remark 6.6. i) Let g be a complex Lie algebra with involution and abelian subalgebra h such that h is closed under . We then obtain a natural∗ involutive trans- formation on h which is given by α∗ H : α H for α h . The compatibility ∗ ∗( ) = ( ∗) ∗ of the involution with the root decomposition can be reformulated∈ as h being ∗ closed under the involution and α∗ = α for every root α ∆. The reformulation is based on the following observation: For H h and Z ∈gα, we have ∈ ∈ H, Z Z, H H , Z α H Z α H Z . [ ∗] = [ ∗]∗ = ( [ ∗ ])∗ = ( ∗) ∗ = ∗( ) ∗ − − − ii) Let (g, h, ) be an involutive split Lie algebra and (ρ, V ) a representation of g that is generated∗ by some weight vector v V λ. If v is primitive with respect to some positive system ∆+ ∆, then V is spanned∈ by all weight vectors ⊂ n α α α λ P α n n 1 1 i=1 i w ρ(g− )ρ(g− − ) ... ρ(g− )v V − ∈ ⊆ + where n N, αi ∆ and i = 1, . . . , n. For every highest weight representation with highest∈ weight∈ λ, we have thus found that V v L V µ, so that = C µ V λ λ + ⊕ ∈P \{ } dim V = 1. Observe that the positive system ∆ deﬁnes a partial ordering on the + weight set V which is given by µ µ0 if and only if µ µ0 N0[∆ ]. The highest weight λ isP a maximal element with≥ respect to this partial− ∈ ordering. Conversely, λ if λ V is a maximal element, then every weight vector v V is primitive by virtue∈ Pof the relation ∈

α λ α ρ(g )v V + = 0 if α ∆+. ⊆ { } ∈ This shows that a generating weight vector v V λ is primitive with respect to ∆+ ∈ + if and only if the weight set satisﬁes V λ N0[∆ ]. P ⊆ − Let (g, h, ) be an involutive split Lie algebra. One shows that for each λ h∗ and each positive system∗ ∆+, there exists an irreducible highest weight representation∈ with respect to ∆+ and with highest weight λ which is unique up to equivalence (cf. [Ne04b, Prop. III.2 (ii)]). We denote this representation by (ρ, V )λ,∆+ . Suppose that (ρ1, V1), (ρ2, V2) are two irreducible highest weight representations having the same weight set. + Let λ be the highest weight of (ρ1, V1) and ∆ be the corresponding positive system. We have seen in Remark 6.6 ii that then, we have λ ∆+ . The highest ) V1 N0[ ] P ⊆ − weight λ occurs also as a weight of (ρ2, V2). Any nonzero λ-weight vector v2 V2 generates V because the representation is irreducible. We have λ ∈ ∆+ , 2 V2 = V1 N0[ ] P P ⊆ − + so that the last conclusion in Remark 6.6 ii) shows that v2 is also primitive for ∆ . + Hence, (ρ2, V2) is also a highest weight representation w.r.t. the positive system ∆ and with highest weight λ and, in particular, isomorphic to (ρ1, V1). Conversely, it is clear that two weight representations of g which are isomorphic must have the same weight set. This shows that two irreducible highest weight representations are isomorphic if and only if they have the same weight set. The next question is whether the highest weight λ already determines the whole weight set. This question has a positive answer if we make two more assumptions: First, we require the highest weight representation to be unitary. As shown in [Ne04b, Prop. III.2 (iv)], each unitary highest weight representation is automatically irreducible. The second assumption concerns the roots of the Lie algebra g. We require that every root is compact. 106 6 Unitary Highest Weight Representations (HWR) of Lie algebras

Deﬁnition 6.7. Let (g, h, ) be an involutive split Lie algebra with real form ∗ g : X g : X X . R = ∗ = { ∈ − } A root α ∆ is called compact if the following conditions are satisﬁed: ∈

α α There exist a nonzero element eα g such that for fα := eα∗ g− and hα := [eα, eα∗ ] h such that ∈ ∈ ∈

α α α α 1. g = C eα, g− = C fα and (thus) [g , g− ] = C hα. · · · 2. α(hα) = 2.

45 3. The endomorphisms ad(eα) and (thus) ad(fα) are locally nilpotent .

If these conditions are satisﬁed, we obtain the relations

[hα, eα] = 2eα, [hα, fα] = 2fα and [eα, fα] = hα, − α α α α revealing that the Lie subalgebra g(α) := g g− [g , g− ] is isomorphic to sl(2, C). ⊕ ⊕ The element αˇ := hα is called the coroot of α. Remark 6.8. i) If we require each root to be compact, then this has strong consequences on the structure of the Lie algebra g: In this case, [Ne04b, Thm. II.6] shows that the Lie algebra g is already locally ﬁnite46 and almost reductive47. In particular, this implies that the Lie algebra must be a directed union of ﬁnite- dimensional Lie subalgebras.

ii) The normalization α(αˇ) = 2 of the coroot αˇ of a compact root α ∆ is motivated ∈ by the fact that then, the automorphism rα GL(h∗) which is deﬁned by ∈

rα(µ) := µ µ(αˇ) α for µ h∗ − · ∈ 2 is involutive in the sense that rα = idh and that rα( α) = α. Since, in addition, ∗ α α ± ∓ rα ﬁxes the hyperplane Hα := µ h∗ : µ([g , g− ]) = 0 pointwise, it is called a root reﬂection. If each root of{ g∈is compact, then, for{ }} every root α, the corre-

sponding root reflection rα exists and we call the subgroup GL(h∗) that is W ⊆ generated by all root reflections rα the Weyl-group of the Lie algebra g. Note that + r α = rα so that, if ∆ is a positive root system, the Weyl group is generated − + W by all positive root reflections rα with α ∆ . ∈ iii) Let α, α1 ∆ be two compact roots such that α1 / α . By virtue of the relation ad g α gα∈1 gα1 α, the subspace V : L gα1+∈kα {±is invariant} under the adjoint ( ± ) ± = k Z ⊆ ∈ 45 An endomorphism A End(V ) on a complex vector space is called locally nilpotent if V = S ker An. ∈ n N 46∈i.e. every finite subset generates a finite-dimensional Lie subalgebra. 47 i.e. [g, g] is semisimple. 6.1 General facts on unitary highest weight representations 107

action of the subalgebra g α . In view of gα1 e , it is the g α -invariant sub- ( ) = C α1 ( ) space generated by the element e . Since ad e ·and ad f are locally nilpotent, α1 ( α) ( α) the subspace V is finite-dimensional so that we obtain a finite-dimensional representation ad, V of the three-dimensional simple Lie algebra g α sl 2, . ( ) ( ) ∼= ( C) Note that g α is an involutive split Lie algebra with respect to h : gα, g α ( ) = [ − ] ∼= C and the involution inherited from g. The weight set V h∗ = C of the finite- ∗ P ⊂ dimensional representation (ad, V ) are the eigenvalues of αˇ = hα on V . It is given by

V = (α1 + k α)(αˇ) : α1 + k α ∆ = α1(αˇ) + 2 k Z : α1 + k α ∆ . P { · · ∈ } { ∈ · ∈ } Standard representation theory of sl(2, C) shows that V Z and that V = P ⊆ P V (see e.g. [HiNe12, Prop. 6.2.8]). Since α1(αˇ) V , we have α1(αˇ) Z. −P ∈ P ∈ Moreover, we have α1(αˇ) = (α1 α1(αˇ) α)(αˇ) V which means that rα.α1 ∆. − − · ∈ P ∈ If all roots α ∆ are compact, then our discussion leads to ∈ ∆(αˇ) Z and .∆ = ∆ (4) ⊆ W

since the Weyl group is generated by the root reﬂections rα.

Our focus lies on unitary highest weight representations. Therefore, our next goal is to ﬁgure out for which highest weight λ and positive system ∆+, the corresponding unitary highest weight representation (ρ, V )λ,∆+ is unitary. A simple necessary crite- λ rion is that the highest weight λ is symmetric in the sense that λ∗ = λ: If v V is nonzero such that v, v = 1, then we obtain for each H h ∈ 〈 〉 ∈ λ H λ H v, v ρ H v, v v, ρ H v λ H . ∗( ) = ( ∗) = ( ∗) = ( ) = ( ) 〈 〉 〈 〉 〈 〉

Conversely, if the highest weight λ = λ∗ is symmetric, then there exists a unique nondegenerate Hermitian form , on V for which v, v = 1 for the primitive vector v 〈· ·〉 〈 〉 and such that ρ(Z ∗)w, w0 = w, ρ(Z)w0 for all w, w0 V and Z g. However, this Hermitian form〈 need not be〉 positive〈 deﬁnite.〉 For ﬁnite∈ dimensional∈ Lie algebras, one has the following characterization.

Theorem 6.9. Let g be a ﬁnite-dimensional involutive split Lie algebra with compact roots + 48 and positive system ∆ . Let λ = λ∗ be symmetric and (ρ, V )λ,∆+ be the corresponding (irreducible) highest weight module of g. Then, the following are equivalent:

i) The representation (ρ, V )λ,∆+ is unitary.

+ ii) λ is dominant integral in the sense that, for all α ∆ , we have λ(αˇ) N0. ∈ ∈ iii) The underlying vector space V is ﬁnite-dimensional.

48For details on how to choose a positive system ∆+ ∆ in the ﬁnite-dimensional case, we refer to [Ha04, Prop. 8.12, Thm. 8.14] or to [HiNe12, Thm. 6.4.13⊂ ]. 108 6 Unitary Highest Weight Representations (HWR) of Lie algebras

In this case, the corresponding weight set V is given by P V = conv( .λ) (λ Z[∆]). P W ∩ − 49 Proof. The ﬁrst part is precisely [Ne00, Thm. IX.3.8] . The deduction of the formula of the weight set V follows from several observations in [Bour90] and [Ha04]: P

(F1) We have V conv( .λ): This follows from the fact that the weight set V is P ⊆ W P the smallest ∆-saturated subset of h∗ containing λ ([Bour90, Ch. VIII, §7, no.2, Prop. 5(i)]). A subset M h∗ is called ∆-saturated if, for any µ M and any root α ∆, the element µ⊆ n α belongs to M for every n [0, µ(∈αˇ)] Z. The subset ∈M = conv( .λ) is ∆−-saturated:· Let µ conv( .λ) and∈ α ∆.∩ We have to show that µ W([0, µ(αˇ)] Z) α conv( ∈.λ). ThisW is trivial∈ for µ(αˇ) = 0, so that we assume− that µ(αˇ∩) = 0.· Then,⊆ forW every n [0, µ(αˇ)] Z, we have n 6 ∈ ∩ t := µ αˇ [0, 1] and ( ) ∈ µ n α = (1 t) µ + t (µ µ(αˇ) α) − · − · · − · = (1 t) µ + t rα(µ) conv( .λ), − · · ∈ W where rα is the root reflection from Remark 6.8. ∈ W (F2) The weight set V is invariant under the Weyl group : This is a consequence P W of the fact that V is ∆-saturated. If rα is a root reflection, then we have rα.µ = P µ µ(αˇ) α by definition. If µ belongs to any ∆-saturated subset M h∗, then so − · ⊆ does rα.µ. Since the Weyl group is generated as a group by the root reflections rα, every ∆-saturated subset is -invariant. W + + + (F3) We have Z[∆ ] R 0[∆ ] = N0[∆ ]: The proof uses the existence of a root base 50 ≥ B ∆+ . In particular,∩ this means that B is an R-linearly independent subset ⊆ 49In order to apply this theorem, we should mention that the real form g is a compact Lie algebra in R the sense of [HiNe12, Def. 12.1.1 and Prop. 12.1.4]. To this end, we note that h : h g H h : H H R = R = ∗ = ∩ { ∈ − } is a compactly embedded subalgebra of the finite-dimensional real Lie algebra g in the sense that R

ad h e ( R) Aut g is a compact subgroup. Grp ( R) 〈 〉 ⊆ This follows from the fact the each operator ad(H) with H∗ = H is diagonalizable on g with ad H eigenvalues in i R. Consequently, the operator e ( ) is also diagonalizable− in g with spectrum in ad h T := z C : z· = 1 . Since the family of operators e ( R) is commutative, hence simultaneously diagonalizable,{ ∈ | it| is contained} in the compact subgroup n with n : dim g . Identifying Aut g with T = ( ) ( R) ad h the subgroup of Aut g consisting of the g -invariant endomorphisms, we have e ( R) Aut g be- ( ) R ( R) cause ad h Aut g . We conclude that h is compactly embedded. Now, Ne00, Prop.⊆ VII.2.5 ( R) ( R) R [ ] shows that g ⊆is a compact Lie algebra. R 50The set B is deﬁned as the set of indecomposable or simple positive roots. A positive root α ∆+ is called indecomposable if it cannot be written as a sum of two positive roots. Then, one can show∈ that B is a basis of E := R[∆] as a vector space and that every positive root α can be written as a linear combination of elements of B with positive integer coefﬁcients (see [Ha04, Thm. 8.14, Thm. 8.15] or [HiNe12, Thm. 6.4.13]). 6.1 General facts on unitary highest weight representations 109

+ of h∗ and that ∆ N0[B]. It follows that ⊆ + + + N0[∆ ] = N0[B], Z[∆ ] = Z[B] and R 0[∆ ] = R 0[B]. ≥ ≥ Since B is linearly independent, we have

Z[B] R 0[B] = N0[B]. ∩ ≥ (F4) For every µ (λ Z[∆+]) the corresponding Weyl-group orbit .µ meets the cone ∈ − W + + := µ h∗ : ( α ∆ )µ(αˇ) 0 h∗. D { ∈ ∀ ∈ ≥ } ⊂ To see this, choose a linear functional f : h∗ R such that f (α) > 0 for every positive root α ∆+. That f exists follows e.g.→ from [Ha04, Thm. 8.14+8.15]. By (F2), the orbit∈ .λ is contained in the weight set V which is a finite set because V is finite-dimensional.W For a root α ∆, the orbitP .α is contained in the root set ∆ (Remark 6.8 iii)) which is finite∈ because g isW finite-dimensional. Therefore, the orbit .µ is a finite set if µ (λ Z[∆+]) and we find some W ∈ − µ0 .µ with ∈ W f (µ0) = max f ( .µ). W For any α ∆+, we obtain ∈ 0 f (µ0) f (rα.µ0) = µ0(αˇ)f (α). ≤ − + Since f (α) > 0, we conclude that µ0(αˇ) 0. This proves that µ0 . ≥ ∈ D (F5) If + + µ int := µ h∗ : ( α ∆ )µ(αˇ) N0 ∈ D { ∈ ∀ ∈ + ∈ } is a dominant integral functional and ν N0[∆ ] such that µ + ν V is a ∈ ∈ P weight, then also µ V is a weight. This is shown in [Bour90, Ch. VIII, §7, no. 2, cor. 2]. ∈ P

Now we prove the equality

V = conv( .λ) (λ Z[∆]). P W ∩ − + The inclusion is a direct consequence of (F1) and of the inclusion V λ N0[∆ ] (Remark 6.6 ii⊆)). Hence, we are left to prove the inclusion : SinceP the⊆ weight− set ⊇ + V is invariant under the Weyl group by (F2), we have .λ V λ N0[∆ ], Pwhich leads to W W ⊆ P ⊆ − + conv( .λ) λ R 0[∆ ]. W ⊆ − ≥ Using (F3), we obtain

+ + M := conv( .λ) (λ Z[∆ ]) λ N0[∆ ]. W ∩ − ⊆ − Let µ M. By (F4), we find some r such that r.µ +. Since M is -invariant, ∈ ∈ W ∈+ D51 W + we have r.µ M and thus r.µ(αˇ) Z for all α ∆ . This implies r.µ int . ∈ +∈ ∈ ∈ D Therefore, we find some ν N0[∆ ] such that λ = r.µ + ν holds. Now, (F5) shows that r.µ V . Hence, µ ∈V by (F2). This completes the proof. 51 ∈ P + ∈ P + Here we use that λ int (first statement of Theorem 6.9) and ∆ (αˇ) Z (Remark 6.8 iii)). ∈ D ⊆ 110 6 Unitary Highest Weight Representations (HWR) of Lie algebras

Remark 6.10. If g is a finite-dimensional semisimple split Lie algebra, then each irreducible highest weight representation is finite-dimensional and, conversely, every irreducible finite-dimensional representation is a highest weight representation (cf. [HiNe12, Prop. 7.3.14, Thm. 7.3.15]). Proposition 6.11. Let g be an involutive split Lie algebra such that each root α ∆ is compact. ∈

i) A highest weight representation (ρ, V )λ,∆+ is unitary if and only if λ∗ = λ and + λ(αˇ) N0 for each positive root α ∆ . ∈ ∈ ii) If (ρ, V )λ,∆+ is unitary, then the following assertions hold: (a) The corresponding weight set is given by the formula

+ V = λ := conv( .λ) (λ + Z[∆]) = conv( .λ) (λ N0[∆ ]), P P W ∩ W ∩ − where conv denotes the convex hull. In particular, the set of weights is invariant

under the natural action of the Weyl-group GL(h∗) on h∗. Wˇ ⊆ (b) Every weight µ V satisfies µ∗ = µ and µ(∆) Z. ∈ P ⊆ (c) For each Z g, the corresponding operator ρ(Z) is locally finite in the sense that every finite∈ subset of V is contained in some finite dimensional ρ(Z)- invariant subspace.

+ iii) Let λ h∗. Then, there exists a positive system ∆ ∆ such that (ρ, V )λ,∆+ is ∈ ⊂ unitary if and only if λ∗ = λ and λ(αˇ) Z for all α ∆. ∈ ∈ Proof. We only sketch the main ideas of the proof. For more details, the reader is referred to [Ne04b], in particular Theorem III.6 . L α In view of the root decomposition g = h α ∆ g , one verifies that ⊕ ∈ M g g ˇ gα [ , ] = ∆ lin . ⊕ α ∆ ∈ This shows that (ρ, V )λ,∆+ is also a highest weight representation of the subalgebra [g, g] and that g = h+[g, g]. We have already mentioned that the assumption that each root is compact implies that g is a directed union of finite-dimensional Lie algebras (Remark 6.8 i)). More precisely, by virtue of the local finiteness of the Lie algebra g, one can find a directed set (J, ) and an increasing union of finite subsets ∆j ∆ ≤S j h ˇ g⊂ with ∆j = ∆ Z[∆j] and ∆ = j J ∆j. For each , we define j := ∆j lin, j := h L gα ∩and note that we thus∈ obtain an increasing union of finite-dimensional j α ∆j ⊕ ∈ S λ Lie subalgebras with [g, g] = j J gj. Let v V be a primitive element of the highest ∈ weight representation. For each j, we denote∈ by Vj the gj-invariant subspace which is generated by v. Then, : , V is a highest weight module of g with highest (ρj = ρ gj j) j weight : and w.r.t. the| positive system + : + . Finally, note that λj = λ hj ∆j = ∆ ∆j | ∩ each root of gj is compact and that the corresponding Weyl-group j GL(h∗j ) is the W ⊆ subgroup of which is generated by the root reflections rα with α ∆j. In summary, we have increasingW unions ∈ 6.1 General facts on unitary highest weight representations 111

S [g, g] = j J gj, • ∈ S V = j J Vj, • ∈ S ∆ = j J ∆j, • ∈ S = j J j, •W ∈ W S . V = j J Vj •P ∈ P This observation reduces matters to the ﬁnite-dimensional case which has already been covered by Theorem 6.9.

i) The ﬁrst part follows from Theorem 6.9 together with the fact that the highest

weight representation (ρ, V ) = (ρ, V )λ,∆+ is unitary if and only if all the representations (ρj, Vj) are unitary (cf. [Ne04b, Prop. III.3]). ii) (a) The formula for the weight set follows from the formula in Theorem 6.9 combined with S and S . The second equality in = j J j V = j J Vj W ∈ W P ∈ P+ the weight set formula is due to V λ N0[∆ ] (cf. Remark 6.6 ii)). P ⊆ − (b) Let µ V . We have µ λ + Z[∆] and therefore, µ∗ = µ follows from ∈ P ∈ α∗ = α for every root α ∆. Since all roots are compact, we have α1(αˇ) Z ∈ ∈ for α, α1 ∆ (Remark 6.8 iii)) and thus µ(α) Z for all α ∆. ∈ ∈ ∈ (c) If Z [g, g] and F V a finite subset, we find j J such that Z gj and ∈ ⊆ ∈ ∈ F Vj. Since the highest weight representation (ρj, Vj) of gj is unitary, it is finite-dimensional⊆ by Theorem 6.9. Hence, the finite subset F is contained in the finite-dimensional ρ(Z)-invariant subspace Vj. Recall that each Vj is the gj-invariant subspace generated by the primitive element v, so that [h, gj] gj implies that each subspace Vj is also h-invariant. The assertion now follows⊆ from g = h + [g, g]. iii) This follows from Prop. III.14 and from the proof of Thm. III.16 in [Ne04b].

The explicit formula of the weight set reveals that it does not depend on the choice of the positive system. Since two unitary highest weight representations are isomorphic if and only if the corresponding weight sets coincide, we obtain the next corollary.

Corollary 6.12. Two unitary highest weight representations ρ, V and ρ, V ( )λ,∆+ ( )λ,∆Ý+ of an involutive split Lie algebra with compact roots are isomorphic.

As a consequence, we simply write (ρ, V )λ for a unitary highest weight representation with highest weight λ for the rest of the section.

Corollary 6.13. Two unitary highest weight representations (ρ, V )λ and (ρ, V )µ of an involutive split Lie algebra with compact roots are isomorphic if and only if µ .λ. ∈ W 112 6 Unitary Highest Weight Representations (HWR) of Lie algebras

Proof. Since two unitary highest weight representations are isomorphic if and only if the corresponding weight sets coincide, we have to show that λ = µ if and only if .λ = .µ. P P W W From the relation .λ λ λ+Z[∆] we infer that λ+Z[∆] = µ+Z[∆] if µ .λ. Hence, if .λ = W.µ,⊆ then P ⊆ ∈ W W W λ = conv( .λ) (λ + Z[∆]) = conv( .µ) (µ + Z[∆]) = µ. P W ∩ W ∩ P The proof of the converse relies on the following observations

+ In view of λ λ N0[∆ ] we obtain • P ⊆ − + + conv( λ) λ R [∆ ] P ⊆ − which shows that λ Ext(conv( λ)), where Ext denotes the set of extremal ∈ P points of conv( λ). P The -invariance of λ and the weight set formula in Proposition 6.11 show • thatW we have P .λ λ conv( .λ). W ⊆ P ⊆ W Building the convex hulls yields conv( λ) = conv( .λ). P W

Hence, if λ = µ, we thus ﬁnd P P 52 µ Ext(conv( µ)) = Ext(conv( λ)) = Ext(conv( .λ)) .λ , ∈ P P W ⊆ W which shows that µ .λ. ∈ W Corollary 6.14. A unitary representation (ρ, V )λ is one-dimensional if and only if λ is a ﬁxed point under the action of the Weyl group. The unitary representation corresponding to the weight λ = 0 is thus given by the one-dimensional trivial representation, i.e. V = C and ρ 0. ≡

Proof. For every highest weight representation (ρ, V )λ with primitive weight vector v V λ, we have the decomposition V v L V µ (as already stated in = C µ V λ ∈ ⊕ ∈P \{ } Remark 6.6). Hence, (ρ, V )λ is one-dimensional if and only if V = λ . In the P { } unitary case, the weight set formula V = conv( .λ) (λ + Z[∆]) implies that this is equivalent to .λ = λ. If λ = 0 andP ∆+ ∆Wis a corresponding∩ positive system, W + ⊂ α then, for any positive root α ∆ , we have ρ(g− )v = 0 because the weight α α does not occur, ρ(g )v = 0 ∈because v is primitive and ρ{(h})v = 0 because λ =−0. This shows that ρ = 0. { } { }

Putting things together, we obtain the following classiﬁcation theorem of unitary highest weight representations.

52Here, we have used the fact that for a subset E of a real (or complex) vector space V , we have the relation Ext(conv(E)) E. ⊆ 6.2 Integration of unitary highest weight representations of gl( ) 113 ∞

Theorem 6.15. (cf. [Ne04b, Thm.III.16]) Let (g, h, ) be an involutive split Lie algebra with compact roots α ∆. The subset ∗ ∈ := λ h∗ : λ∗ = λ, ( α ∆)λ(αˇ) Z h∗ I { ∈ ∀ ∈ ∈ } ⊂ is invariant under the natural action of the Weyl-group GL(h∗) on h∗ and the map W ⊆ λ (ρ, V )λ induces a bijection of the orbits / of under the Weyl-group action onto7→ the set of isomorphism classes of irreducibleI unitaryW I highest weight representations of g.

Proof. To see that is -invariant, choose λ . By Proposition 6.11 i), we have I W ∈ I λ and, since λ is -invariant, we conclude that .λ . The remainder of theP ⊆ assertion I followsP fromW the fact that, for each λ , thereW ⊆ exists I a highest weight module with highest weight λ which is unitary (Proposition∈ I 6.11 i)) and unique up to equivalence together with Corollary 6.13.

Remark 6.16. Suppose that (ρλ, Vλ) and (ρλ0 , Vλ0) are two isomorphic unitary highest weight representations of the invoutive split Lie algebra g. This means that there exists a linear bijection A : Vλ Vλ0 such that Aρλ = ρλ0 A. If vλ is a generating primitive → weight vector in Vλ, then the same holds for v0 := Avλ in V 0. After rescaling A with λ λ a suitable positive constant, we may assume that v v and observe that, λ0 V = λ Vλ λ0 k k then A is a linear isometry: [Ne04b, Prop. III.2 (v)] shows that the contravariant inner product on a unitary highest weight module is unique up to rescaling with a positive constant. From this it follows that v, w V Av, Aw V for all v, w V . This also λ = λ0 λ means that the intertwining isomorphism〈 〉 between〈 two〉 isomorphic∈ unitary highest weight modules is unique up to rescaling with a positive constant.

6.2 Integration of unitary highest weight representations of gl( ) ∞ First, we show that the direct limit Lie algebra gl( ) := lim gl(n) is an involutive split ∞ Lie algebra so that the highest weight theory applies to this−→ Lie algebra.

Even though we have already introduced gl( ) as the Lie algebra of the direct limit Lie group GL( ) in Section5, we introduce it∞ here, in more detail and independently, as a complex∞ Lie algebra: Let C(N) be the vector space consisting of all N-tuples with only finitely many nonzero entries. This vector space is spanned by the standard basis e . The endomorphism algebra End (N) carries a natural structure of a complex ( n)n N (C ) ∈ Lie algebra whose Lie bracket is given by [Z1, Z2] := Z1 Z2 Z2 Z1. Those endomorphisms whose corresponding -matrix representation w.r.t− the basis e has N N ( n)n N ∈ only finitely nonzero entries form× a Lie subalgebra of End(C(N)) which we denote by g := gl( ). A basis of this Lie algebra is given by all elementary operators Ei j with i, j N whose∞ matrix representation consists of a single nonzero entry 1 at the position ∈ (i, j). They are defined by the relation Ei j ek = δjkek for all k N and satisfy ∈ [Ei j, Ek`] = δjk Ei` δì Ek j for all i, j, k, ` N. (5) − ∈ 114 6 Unitary Highest Weight Representations (HWR) of Lie algebras

n In view of the natural embeddings gl(n) := End(C ) End(C(N)), the Lie algebra g = gl( ) is the directed union of the ﬁnite-dimensional⊂ Lie subalgebras gl(n) and we write∞gl( ) = lim gl(n). We will now explore the structure of gl( ) in more ∞ ∞ detail: −→ involution: A natural involution on gl( ) is obtained by antilinear extension of ∗ ∞ Ei∗ j := Eji. Restricting to the subalgebras gl(n), this involution coincides with the standard n n-matrix conjugate and the real form is given by × u(n) := X gl(n) : X ∗ = X . { ∈ − } Thus, the real form of gl( ) is given by ∞ u( ) := X gl( ) : X ∗ = X = lim u(n). ∞ { ∈ ∞ − } −→ root decomposition: A maximal abelian subalgebra h is given by the diagonal operators with respect to the basis e . This means that h is spanned by the ( n)n N ∈ (N) elementary operators Eii with i N. We have h = C and thus ∈ ∼ N h∗ := α : h C linear = C { → } ∼ for the space of linear functionals on h. For i N, we introduce the functional ∈ "i h∗ which is deﬁned by "i(Ej j) := δi j. Then, the relation (5) shows that, for all∈i, j N and H h, we have ∈ ∈ [H, Ei j] = ("i "j)(H)Ei j. − Therefore, the set of roots and the corresponding root systems are given by

"j "k ∆ = "i "j : i = j , with g − = C Ei j { − 6 } · and we obtain the root decomposition

M "j "k g = gl( ) = h g − . ∞ ⊕ i=j 6 In particular, all root spaces are one-dimensional and the root decomposition is α α compatible with the involution in the sense that Z g implies that Z ∗ g− . Hence, the triple (gl( ), h, ) is∗ an involutive split Lie∈ algebra. ∈ ∞ ∗ compact roots: Let α := "i "j ∆ be a root of g = gl( ). Using the fact that α α − ∈ ∞ Ei j g and Eji g− as well as (5), it is easily seen that the elements eα := Ei j, ∈ ∈ fα := Eji and hα := Eii Ej j satisfy − [hα, eα] = 2eα, [hα, fα] = 2fα and [eα, fα] = hα. − The three-dimensional subalgebra g(α) := C Ei j C Eji C (Eii Ej j) is thus isomorphic to sl 2, . Using g u , we· ﬁnd⊕ · ⊕ · − ( C) R = ( ) ∞ g α : g α g sl 2, u 2 su 2 . ( )R = ( ) R = ( C) ( ) = ( ) ∩ ∼ ∩ 6.2 Integration of unitary highest weight representations of gl( ) 115 ∞

An iterated application of the relation (5) reveals that ad(Ei j) is nilpotent be- 3 cause we have ad(Ei j) = 0. The element αˇ := hα = Eii Ej j satisfies α(αˇ) = 2 and is called the coroot of α. All these observations show− that each root α ∆ is compact in the sense of Definition 6.7. ∈ positive systems of roots: Let be a linear/total ordering on N. Then + ∆ := "i "j : i j { − ≺ } defines a positive system of roots. Conversely, given a positive system ∆+, one obtains a linear/total ordering on N via i j if and only if either i = j or, + otherwise, "i "j ∆ . This yields a 1 : 1-correspondence between the positive systems of roots− and∈ linear/total orderings on N as shown in [Ne04b, Prop. II.8]. The positive system ∆+ which corresponds to the standard linear ordering on N is called the standard positive system of roots. ≤

Weyl group: For every root α = "i "j, the corresponding root reﬂection rα GL(h∗) is given by − ∈

rα(µ) := µ µ(αˇ)α for every µ h∗. − ∈ Using αˇ = Eii Ej j, we ﬁnd, for arbitrary k N, − ∈ " , k i  j = rα("k) = "i, k = j  "k, else.

Since every functional µ h N has a unique representation µ P µ " ∗ = C = k N k k ∈ ∼ N ∈ with µk := µ(Ekk), the above relation shows that rα acts on C as a transposition exchanging the entries i and j. Therefore, the natural action of the Weyl group

GL(h∗) which is generated by all root reflections rα is isomorphic to the W ⊆ S N action of the group ( ) of finite permutations on C . ∞ S For the remainder of this section, we will write = ( ) for the Weyl group of g gl and we will identify h with N. NoteW that, if∞µ µ N, then = ( ) ∗ C = ( n)n N C µ µ∞ so that µ µ implies µ N. Applying the results from∈ the∈ previous ∗ = ( n)n N ∗ = R ∈ ∈N subsection, we find that, for any λ R satisfying λj λk Z for all j, k N, there ∈ − ∈ ∈ exists a unitary highest weight representation ρλ : gl( ) End(Vλ) on a pre-Hilbert ∞ → space Vλ which is unique up to isomorphism. This representation is locally finite and irreducible. Its weight set is determined by λ via the formula

S λ := conv( ( ).λ) (λ + Z[∆]). P ∞ ∩ N In particular, λ R and every weight µ λ satisﬁes µj µk Z for all j, k N. P ⊆ ∈ P − ∈ ∈ Two such highest weight representations ρλ, Vλ and ρλ , Vλ are isomorphic if and ( ) ( 0 0 ) S only if λ0 ( ).λ. ∈ ∞ Example 6.17. i) The case λ := 0 corresponds to the one-dimensional trivial representation ρλ = 0 on Vλ = C (as already remarked in Corollary 6.14). 116 6 Unitary Highest Weight Representations (HWR) of Lie algebras

ii) The case λ := e1 := (1, 0, 0, 0, . . .) corresponds to the irreducible identical representation (N) (N) ρλ : gl( ) C C , (X , v) X v. ∞ × → 7→ 2 Here C(N) stands for the dense subspace of ` (N, C) which is spanned by the standard orthonormal system e . For the standard positive system of roots ∆+, a ( n)n N ∈ generating primitive λ-weight vector is given by e1. This representation extends 2 to the representation ρ1 of the Banach–Lie algebra glp(` ) from Subsection 5.3. iii) The case λ := (1, 1, 0, 0, 0, . . .) corresponds to the conjugation representation −

ρλ : gl( ) sl( ) sl( ), (X , A) XA AX = [X , A] ∞ × ∞ → ∞ 7→ − of gl( ) on sl( ) := X gl( ) : Tr(X ) = 0 = ker(Tr). Since λ ∆ is a root, a corresponding∞ ∞ generating{ ∈ primitive∞ weight vector} (w.r.t. the standard∈ positive

system of roots) is given by the vector E12 = e1 e2∗ which spans the corresponding ⊗ root space. Note that we have λ = ∆. P Next, we want to apply our integration results from AppendixC to the case of unitary highest weight representations of gl( ): ∞ We start with the following almost immediate result on ﬁnite-dimensional representations of gl(n). It will be needed in the next subsection:

Lemma 6.18. Let ρ : gl(n) End(V ) be a Lie algebra representation on the ﬁnite- dimensional vector space V . Then→ the following statements hold:

i) There exists a unique smooth representation πe : GLf(n) GL(V ) with derivative → dπe = ρ.

ii) There exists a unique smooth representation π : GL(n) GL(V ) with dπ = ρ if and → only if ker(q) ker(πe), where q : GLf(n) GL(n) is the universal covering map. ⊆ → n iii) If (ρ, V ) is a weight representation with weight set V C , then the condition n P ⊆ ker(q) ker(πe) is equivalent to V Z . ⊆ P ⊆ Proof. Since V and gl(n) are ﬁnite-dimensional, the representation ρ : gl(n) End(V ) is a continuous Lie algebra representation and the assertions i) and ii) follow→ from the Integrability Theorem Theorem 4.16. In order to prove iii), suppose that (ρ, V ) is a weight representation. This means that, for any diagonal operator H gl(n) and any µ ∈ µ-weight vector v V , we have ρ(H)v = µ(H)v. If πe is the unique smooth represen- ∈ 2π i µ(H) tation integrating ρ, then we have πe(expg(2π i H))v = e v. Recall that we have expg(2π i E11) = expg(2π i Ekk) for all 1 k n and that this is a generator of ker(q) = Z ≤ ≤ 1∼ (cf. Remark 5.11). Hence, ker(q) ker(πe) is equivalent to πe(expg(2π i Ekk)) = for 2π i µ E all k n. This, in turn, happens if⊆ and only if e ( kk) = 1 for all k and all weights ≤ µ V , which amounts to saying that µk := µ(Ekk) Z for all weights µ. ∈ P ∈ 6.2 Integration of unitary highest weight representations of gl( ) 117 ∞

We want to generalize the preceding lemma to infinite-dimensional locally convex 2 vector spaces and to the infinite-dimensional Lie algebras gl( ) and glp(` ) for 1 p . The main point here is that we can no longer apply∞ our Lie theoretic tools≤ from≤ ∞ Subsection 4.1 because GL(V ) need not carry a suitable Lie group structure: As mentioned in the Introduction of [Ne06], if V is not normable, then there is no vector topology on End(V ) for which the multiplication map is continuous. Therefore, in general, End(V ) is not a CIA and there is no natural Lie group topology on GL(V ) such that End(V ) is the corresponding Lie algebra. The following weaker concept of integrability serves as a remedy. It is based on the ideas in Appendix E of [GlNe18] elaborating on the existence of smooth homomorphisms into diffeomorphism groups which are non-Lie groups. Definition 6.19. Let ρ : g End(V ) be a representation of a locally convex Lie algebra g on a locally convex→ vector space V (over K = R, C) and suppose that G is a connected locally exponential Lie group with Lie algebra L(G) = g and exponential map exp : g G. A representation→ π : G GL(V ) is said to be an integrated representation of ρ if → d 1 π(exp(tX ))v = lim [π(exp(tX ))v v] = ρ(X )v for all X g, v V. d t t 0 t t=0 → − ∈ ∈ Remark 6.20. Note that some authors additionally require the continuity of the maps

R V V, (t, v) π(exp(tX ))v × → 7→ for each X g. Then, the Uniqueness Lemma (Lemma C.4) shows that an integrated representation∈ is unique whenever it exists. But even without this additional continuity requirement, we observe a natural uniqueness behaviour of the integrated representation which is sufﬁcient for our purposes:

i) If V is a Banach space and π is a smooth representation with derivative dπ := L(π) = ρ, then π is the unique integrated representation of ρ (cf. Remark C.7). ii) Let g be a complex Lie algebra with involution and ρ be a unitary representation. Assume that W V is a ρ-invariant subspace.∗ Suppose that there exists an ⊆ integrated representation π of ρ on V and an integrated representation π0 of ρ on W. Then, Lemma C.4 shows that the restrictions of π and π0 to unitary subgroup U := exp(X ) : X ∗ = X Grp coincide on W. This means that W is a π- 〈 − 〉 invariant subspace. In particular, any two integrated representations π, π0 coincide on U and every operator π(u) with u U is unitary on the complex pre- Hilbert space V and thus uniquely extends to∈ a unitary operator on the Hilbert space completion := V of V . H iii) Suppose that ρ is a locally finite representation, i.e. for every X g and every finite subset F V , there exists a finite-dimensional ρ(X )-invariant∈ subspace P 1 n V V . For v V⊂, the sum ∞ ρ X v converges in some finite-dimensional f in n=0 n! ( ) ⊆ ∈ P 1 n subspace of V , and thus in V . If π exp X v : ∞ ρ X v defines a repre- ( ( )) = n=0 n! ( ) sentation π : G GL(V ), then we call π a locally finite integrated representation of ρ. Obviously,→ there exists at most one locally finite integrated representation. 118 6 Unitary Highest Weight Representations (HWR) of Lie algebras

Theorem 6.21. Let ρλ : gl( ) End(Vλ) be a unitary highest weight representation ∞ → of gl( ) on a pre-Hilbert space Vλ with Hilbert space completion λ := Vλ and with ∞ N 1H 1 corresponding highest weight λ R . Let p, q [1, ] such that p + q = 1. Then, the following assertions hold: ∈ ∈ ∞

i) There exists an integrated representation πλ :U( ) U(V ) of ρλ u( ) if and N ∞ → | ∞ only if λ Z . If this happens, then πλ is uniquely determined and extends to a ∈ continuous unitary representation πλ :U( ) U( λ). Moreover, it extends to a ∞ → H locally ﬁnite integrated representation πλ : GL( ) GL(Vλ) of ρ. ∞ → ii) If λ < , then the representation ρλ extends to a continuous unitary Lie al- k k∞ ∞ 2 2 gebra representation ρλ : 1(` ) ( λ) of the Banach-completion 1(` ) of 2 gl . There exists an integratedB → representation B H π :U ` U ofBρ 2 ( ) λ 1( ) ( λ) λ u1(` ) ∞ N → H | if and only if λ Z . If this happens, then πλ is uniquely determined and uniquely ∈ 2 extends to a smooth representation πλ : GL1(` ) GL( λ) with dπλ = ρλ . → H iii) If p > 1 and λ has only ﬁnitely many non-zero entries, then the representation ρλ 2 extends to a continuous unitary Lie algebra representation ρλ : p(` ) ( λ) 2 B → B H of the Banach-completion p(` ) of gl( ). In this case, there always exists an 2 integrated representation πB :U ` ∞U of ρ 2 . It is uniquely deter- λ p( ) ( λ) λ up(` ) → H | 2 mined and uniquely extends to a smooth representation πλ : GL1(` ) GL( λ) → H with dπλ = ρλ.

Proof. We apply Theorem C.24 keeping in mind that unitary highest weight representations are locally ﬁnite unitary weight representations. Also note that the weight set formula S λ := conv( ( ).λ) (λ + Z[∆]) P ∞ ∩ implies that N N λ Z λ Z , P ⊆ ⇐⇒ ∈ which is due to ∆ ZN. Moreover, for 1 q , we have ⊆ ≤ ≤ ∞

sup µ q < λ q < . µ λ k k ∞ ⇐⇒ k k ∞ ∈P If q < , then this equivalent to supp(λ) < , i.e. the highest weight λ has only ﬁnitely∞ many non-zero entries. | | ∞

Theorem 6.21 suggests a natural way to deﬁne the class of unitary highest weight 2 representations of the unitary groups U( ) and Up(` ). ∞ Deﬁnition 6.22. i) A continuous unitary representation π :U( ) U( ) is called a unitary highest weight representation (with highest weight∞ →λ HZN) if there exists a dense linear U( )-invariant subspace V such that ∈(π, V ) is the unique integrated representation∞ of a unitary highest⊆ weightH representation (ρ, V ) = (ρλ, Vλ) of the Lie algebra gl( ) (with highest weight λ). ∼ ∞ 6.2 Integration of unitary highest weight representations of gl( ) 119 ∞

2 ii) For 1 p , a norm-continuous unitary representation π :Up(` ) U( ) is called≤ a unitary≤ ∞ highest weight representation (with highest weight λ →ZN) ifH its restriction to the dense subgroup U( ) is a unitary highest weight representation∈ of U( ) (with highest weight λ). ∞ ∞ Remark 6.23. Since every unitary highest weight representation of gl( ) is irreducible, the same holds for every unitary highest weight representation of∞ the unitary 2 groups U( ) and Up(` ) (cf. Remark C.2). Moreover, Remark 6.16 shows that two unitary highest∞ weight representations of the unitary groups are unitarily equivalent if and only if the corresponding dense unitary highest weight representations of gl( ) are isomorphic. A unitary highest weight representation (of the unitary groups) with∞ highest weight λ is denoted by (πλ, λ). Note that λ is spanned by the mutually H H orthogonal h-weight vectors corresponding to the weights in λ (cf. Remark 6.6 ii)). P The following observations will be taken up later in Section8:

2 Proposition 6.24. If (πλ, λ) is a unitary highest weight representation of Up(` ) (or H N U( )) with highest weight λ Z , then (πλ∗ , λ∗) is a unitary highest weight represen- ∞ ∈ H tation with highest weight λ, i.e. we have (πλ∗ , λ∗) = (π λ, λ). − H ∼ − H− Proof. Consider the surjective antilinear isometry

λ λ∗, v v∗ := , v H → H 7→ 〈· 〉 which intertwines the representations πλ and πλ∗ . Denote by Vλ λ the dense ⊆ H unitary highest weight module with generating primitive weight vector vλ and by

Vλ∗ λ∗ its image under the intertwining isometry. Note that Vλ∗ is πλ∗ -invariant. ⊆ H The gl( )-representation ρλ∗ (Z)v∗ := ( ρ(Z ∗)v)∗ deﬁnes a unitary highest weight ∞ − representation on Vλ∗ because the vector vλ∗ is generating, primitive for the positive + system ∆ and it is a weight vector corresponding to the weight λ. Moreover, πλ∗ − − is an integrated representation of ρλ∗ . Since the deﬁnition of a unitary highest weight module is independent of the choice of the positive system (cf Corollary 6.12) and since Vλ∗ is a dense subspace of λ∗, we conclude that (πλ∗ , λ∗) = (π λ, λ). H H ∼ − H− Proposition 6.25. Let πλ, λ , πλ , λ be two unitary highest weight representa- ( ) ( 0 0 ) tions of 2 . Then the tensorH productH contains as Up(` ) (πλ, λ)b(πλ , λ ) (πλ+λ , λ+λ ) H ⊗ 0 H 0 0 H 0 a subrepresentation. In particular, we have πλ, λ πλ , λ πλ λ , λ λ if and ( )b( 0 0 ) = ( + 0 + 0 ) only if the tensor product is an irreducible representation.H ⊗ H H

Proof. There exist dense U( )-invariant subspaces Vλ,Vλ in λ and λ respectively ∞ 0 H H 0 such that πλ, πλ are the unique integrated representations of unitary highest weight 0 representations ρλ, ρλ of the Lie algebra gl( ) on Vλ and Vλ respectively. The tensor 0 ∞ 0 product Vλ Vλ is a U( )-invariant subsapce of the tensor product λ b λ . We ⊗ 0 ∞ H ⊗H 0 deﬁne a unitary Lie algebra representation ρλ ρλ of gl( ) on Vλ Vλ by ⊗ 0 ∞ ⊗ 0

(ρλ(Z) ρλ (Z))(v v0) := ρλ(Z)v v0 + v ρλ (Z)v0 for all Z g, v Vλ, v0 Vλ . ⊗ 0 ⊗ ⊗ ⊗ 0 ∈ ∈ ∈ 0 120 6 Unitary Highest Weight Representations (HWR) of Lie algebras

By construction, the representation πλ πλ is the (unique) integrated representation ⊗ 0 of ρλ ρλ . Consider the vector vλ+λ := vλ vλ , where vλ, vλ denote the generating ⊗ 0 0 ⊗ 0 0 primitive weight vectors of the highest weight modules Vλ and Vλ respectively. It is 0 clear that vλ λ is a primitive weight vector of the weight λ λ . Hence, it generates + 0 + 0 N a highest weight module Vλ λ Vλ Vλ . Since λ λ , there exists a unique + 0 0 + 0 Z ⊂ ⊗ V ∈ integrated representation πλ+λ of ρλ ρλ on λ+λ . The uniqueness implies that 0 ⊗ 0 0 πλ λ πλ πλ on Vλ λ (see Remark 6.20 ii and Lemma C.4). This means that + 0 = 0 + 0 ) V ⊗ V λ+λ is πλ πλ -invariant. Hence, λ+λ := λ+λ is an invariant subspace which is a 0 ⊗ 0 H 0 0 unitary highest weight representation with highest weight λ+λ0. If the tensor product is an irreducible representation, then we have λ λ λ λ . Conversely, if the + 0 = b 0 tensor product is a unitary highest weight representation,H H then⊗H it is irreducible (cf. Remark 6.23).

6.3 Direct limit structures of unitary highest weight representations

In this subsection, we introduce the notion of a direct limit representation. This is motivated by the observation that every generating primitive weight vector of a unitary highest weight representation of the direct limit Lie algebra gl( ) = lim gl(n) ∞ generates unitary highest weight representations of the Lie subalgebras gl(n−→). This yields an ascending and exhausting chain of subrepresentations. In this sense a unitary highest weight representation is a direct limit representation. Therefore direct limits of irreducible representations provide a more general class of irreducible Lie algebra representations including the unitary highest weight representations. More details on this subject can be found e.g. in [DP98].

Direct limit representations

Recall that we call a Lie algebra g a direct limit Lie algebra if it can be written as the union of a strictly increasing sequence g of Lie subalgebras. In this case, we ( n)n N ∈ write g = lim gn. Likewise we deﬁne a direct limit vector space V = lim Vn and a direct limit group−→G = lim Gn. −→ −→ Deﬁnition 6.26. i) Let g = lim gn be a direct limit Lie algebra. A representation (ρ, V ) is called a direct limit−→ representation of g if V = lim Vn is a direct limit vector space such that, for every n N, the subspace Vn −→V is gn-invariant. We ∈ ⊂ write (ρn, Vn) for the subrepresentation of gn on Vn and (ρ, V ) = lim(ρn, Vn) for the direct limit representation of g. −→

ii) Analogously, we deﬁne a direct limit representation (π, V ) = lim(πn, Vn) of a direct limit group G = lim Gn. −→ −→ The following two lemmata hold both for representations of direct limit Lie algebras as well as for representations of direct limit groups. The proofs are the same. Therefore, 6.3 Direct limit structures of unitary highest weight representations 121

we use the symbolic notation X = lim X n, where X is either a direct limit Lie algebra or a direct limit group. A representation−→ (ρ, V ) of X is then accordingly a Lie algebra representation or a group representation.

Lemma 6.27. Let (ρ, V ) = lim(ρn, Vn) be a direct limit representation of X = lim X n. −→ −→ i) If all representations (ρn, Vn) are irreducible, then (ρ, V ) is irreducible.

ii) If all representations (ρn, Vn) are finite-dimensional, then (ρ, V ) is locally finite in the sense that each ρ(x), with x X , is a locally finite endomorphism on V . ∈

Proof. i) Assume that W V is an invariant subspace. Then W = lim Wn for the ⊆ X n-invariant subspaces Wn := Vn W. The irreducibility of the V−→n now implies ∩ Wn 0 , Vn for all n N. If all Wn = 0 , then W = 0 . Otherwise, there ∈ {{ } } ∈ { } { } exists some n N with Wn = Vn. Hence, for all k n, we have also Wk = Vk since ∈ ≥ 0 = Vn = Wn Wk. This means that W = V . { } 6 ⊆ ii) For a given finite subset F V and x X , we find some n N such that x X n ⊂ ∈ ∈ ∈ and F Vn. Since Vn is a finite dimensional ρ(x)-invariant subspace and F was arbitrary,⊂ we conclude that ρ(x) is a locally finite endomorphism.

In the following, we tacitly assume all vector spaces to be complex. If, for each n N, there is a representation ρ , V of X , then we call the sequence : ρ , V ∈di- ( n n) n = ( n n)n N rected V S V ∈ if each (ρn, n) is isomorphic to a (real) subrepresentation of (ρn+1, n+1) viewed X X n as a representation of n n+1. This means that, for each N, there are injective n+1 V V⊂ ∈ linear maps Φn : n n+1 intertwining the representations ρn and ρn+1. We say that a directed sequence→ defines a direct limit representation if there exists a direct S limit representation (ρ0, V 0) = lim(ρn0 , Vn0) of X such that −→ (ρn0 , Vn0) = (ρn, Vn) for all n N. ∼ ∈ Lemma 6.28. Every directed sequence ρ , V defines a direct limit representa- = ( n n)n N S ∈ tion of X . If all (ρn, Vn) are irreducible and finite-dimensional representations, then the corresponding direct limit representation defined by is unique up to isomorphism. S If ρ , V is a direct sequence of irreducible, finite-dimensional represen- = ( n n)n N S ∈ tations, then we write lim(ρn, Vn) for the corresponding direct limit representation defined by . −→ S Proof. Existence: For k n, we put ≥ ¨ id , k n Φk : Vn = n = k n+1 Φk 1 ... Φn , k > n, − ◦ ◦ n+1 V V x X where Φn : n n+1 are the intertwining operators defined by . For each n, we then have → S ∈ k k ρk(x) Φn = Φn ρn(x). ◦ ◦ 122 6 Unitary Highest Weight Representations (HWR) of Lie algebras

Note that : V , Φk is a direct (countable) system in the category 0 = (( n)n N ( n)k n) VECT of (complex)S vector spaces∈ 53.≥ Let V, Φ be the direct limit cone over in the ( ( n)n N) 0 ∈ S category in VECT with injective linear maps Φn : Vn V for which → k Φn = Φk Φn for all k n ◦ ≥ and V S Φ V . The direct limit cone satisﬁes the following universal property: = n N n( n) For every cone∈ W, Ψ over there exists a unique linear map ρ : V W such ( ( n)n N) 0 ∈ S → that ρ Φn = Ψn In particular, the diagram ◦ k Φn Vn Vk

Φn Φk V Ψ Ψ n !ρ k ∃ W commutes for all k n. Let x X S X . For sufﬁciently large n , we put = n N n N ≥ ∈ ∈ ∈ Ψn := Φn ρn(x). For k n, we then have ◦ ≥ k k k Ψk Φn = Φk ρk(x) Φn = Φk Φn ρn(x) = Ψn, ◦ ◦ ◦ ◦ ◦ so that V, Ψ is another cone over . This deﬁnes a unique linear endomor- ( ( n)n N) 0 phism ρ(x) : V ∈ V such that the diagramS → !ρ(x) V ∃ V

Φn Φn

ρn(x) Vn Vn commutes for all sufﬁciently large n N. We thus obtain a direct limit representation ∈ (ρ, V ) of X and, for each n N, the map Φn is a linear bijective intertwiner of the representations , V ∈and , V . (ρ Xn Φn( n)) (ρn n) | Uniqueness: We show that two direct limit representations (ρ, V ) = lim(ρn, Vn) and

(ρ0, V 0) = lim(ρn0 , Vn0) are isomorphic if, for every n N, the respective−→ subrepresen- ∈ tations are−→ isomorphic, irreducible and finite-dimensional. Let Φn : Vn Vn0 be an 1 → intertwining isomorphism. If Ψn is another one, then Ψn Φ−n belongs to the com- ◦ mutant of the irreducible representation (ρn, Vn), hence is a multiple of the identity 53Recall that a direct countable system : V , Φk in the category is a sequence 0 = (( n)n N ( n)k n) VECT of vector spaces V together with morphismsS Φk : V ∈ V such≥ that Φn id and Φk Φk Φm if n n n k n = Vn n = m n k m n.A cone V, Φ over is a vector space→ V together with linear maps Φ : V◦ V ( ( n)n N) 0 n n ≥ ≥ k ∈ S → such that Φn = Φk Φn if k n.A direct limit cone over 0 is a cone which satisfies the above universal property. Such a direct◦ limit≥ cone exists in and canS be obtained as follows: Put V : ` V , VECT = n N n ∈ where the equivalence relation is defined as follows: If vn Vn and vm Vm, then vn vm if and only∼ ∈ k k∈ ∼ if there exists some k N with k m, k n and such that Φn(vn) = Φm(vm). The maps Φn send each v V to its equivalence∈ class v ≥ V . In≥ particular, we have V S Φ V . If all Φk are injective, n n [ n] = n N n( n) n ∈ then∈ so are the Φn. ∈ 6.3 Direct limit structures of unitary highest weight representations 123

according to Schur’s Lemma. This shows that Φn is uniquely determined up to a scalar factor in . In particular, we have Φ λ Φ for some λ . After an iterative C× n+1 Vn = n n n C× | ∈ rescaling, we assume that all λn = 1. Then,

Φ := lim Φn : V V 0, v Φn(v) if v Vn → 7→ ∈ −→ is a well-deﬁned linear isomorphism intertwining ρ and ρ0.

The preceding lemma is now used to show that one obtains all unitary highest weight representations of the direct limit Lie algebra gl( ) = lim gl(n) from the ﬁnite di- ∞ mensional unitary highest weight representations of the −→gl(n) via a direct limit construction.

N Proposition 6.29. Suppose that λ R such that λj λk Z for all k, j N. For (n) ∈n − ∈ ∈ n N, put λ := (λ1, λ2,..., λn) R , the ﬁrst n entries of λ. We write (ρλ, Vλ) for the unitary∈ highest weight representation∈ of gl( ) with highest weight λ and, accordingly, ∞ (ρλ(n) , Vλ(n) ) for the unitary highest weight representation of gl(n) with highest weight λ(n).

Then, we obtain a directed sequence : ρ n , V n and we have = ( λ( ) λ( ) )n N S ∈

ρ , V lim ρ n , V n . ( λ λ) ∼= ( λ( ) λ( ) ) −→ In particular, every unitary highest weight representation of gl( ) is a direct limit representation. ∞

λ Proof. Let v Vλ be a primitive generating vector of the unitary highest weight repre- ∈ sentation (ρλ, Vλ). In particular, v is a primitive weight vector of every Lie subalgebra n n gl(n) corresponding to the weight λ( ) R . Therefore, v generates an increasing sequence of gl(n)-invariant subspaces ∈

Vλ(n) := ρ(gl(n))v lin Vλ. 〈 〉 ⊂

By definition, Vλ(n) is a unitary highest weight module of gl(n) and, according to The- orem 6.9, irreducible and finite-dimensional. Since v is a generating vector, we have S V V n so that ρ , V is a direct limit representation. λ = n N λ( ) ( λ λ) This argument∈ reveals that is a directed sequence of irreducible, finite-dimensional S representations defining a direct limit representation lim(ρλ(n) , Vλ(n) ) which is unique up to isomorphism (Lemma 6.28) and, in particular, isomorphic−→ to (ρλ, Vλ).

Integration of direct limit representations of gl( ) ∞ The next aim is to integrate a direct limit representation of g = gl( ) to a direct limit representation of G = GL( ), using the results from AppendixC.∞ ∞ 124 6 Unitary Highest Weight Representations (HWR) of Lie algebras

Proposition 6.30. Let (ρ, V ) = lim(ρn, Vn) be a direct limit representation of the Lie algebra g = gl( ) = lim gl(n). We−→ assume that all representations (ρn, Vn) are finite- ∞ dimensional. Then, there−→ exists a (unique) locally finite integrated representation π : GL( ) GL(V ) of ρ if and only if , for each n N, there exists a (unique) ∞ → ∈ smooth representation πn : GL(n) GL(Vn) with derivative dπn = ρn. If this is the case, → then (π, V ) = lim(πn, Vn) is a direct limit representation and has continuous orbit maps with respect to−→ the finest locally convex vector topology on V .

Proof. If all representations (ρn, Vn) are finite-dimensional, then the direct limit representation is locally finite (Lemma 6.27) and the equivalence statement is a special case of Proposition C.16, since every finite-dimensional gl(n)-invariant subspace is contained in some gl(n)-invariant subspace Vk for k n. If (π, V ) exists, then ≥ X∞ 1 π exp X v ρ X n v π exp X v whenever X gl k and v V . ( ( )) = n! ( ) = k( ( )) ( ) k n=0 ∈ ∈

This shows that (π, V ) = lim(πn, Vn). The assertion that every orbit map is continuous is contained in Proposition−→C.16. Proposition 6.31. Let : ρ , V be a directed sequence of unitary highest weight = ( n n)n N ∈ n n representations of gl(n)Swith highest weights λ( ) R . Then the direct limit representa- ∈ tion (ρ, V ) := lim(ρn, Vn) deﬁned by is an irreducible, locally ﬁnite unitary represen- 54 S tation of gl( −→) . There exists an integrated unitary representation π :U( ) U(V ) of if∞ and only if , for each n , we have (n) n. If this happens,∞ then→ the ρ u( ) N λ Z representation| ∞ (π, V ) has the following∈ properties: ∈

i) It is uniquely determined.

ii) It extends to a locally ﬁnite integrated representation π : GL( ) GL(V ) of ρ. ∞ → iii) It is a direct limit representation (π, V ) = lim(πn, Vn), where the (πn, Vn) are the unique smooth representations of n with derivatives d . U( ) −→ πn = ρn u(n) | iv) It is irreducible.

v) It extends to an irreducible continuous unitary representation π :U( ) U( ) on the Hilbert space completion := V of the pre-Hilbert space V . ∞ → H H Proof. Since the unitary highest weight representations of the gl(n) are irreducible and ﬁnite-dimensional (Theorem 6.9), the directed sequence deﬁnes a direct limit rep- S resentation (ρ, V ) := lim(ρn, Vn) which is unique up to isomorphism (Lemma 6.28). Therefore, we may view−→ the Vn as gl(n)-invariant linear subspaces of V . According to [Ne04, Prop. III.2 (iv),(v)], there exists a contravariant scalar product , n on each Vn which is unique up to rescaling with a positive constant. Hence, after〈· an·〉 iterative

54 It is not necessarily a unitary highest weight representation of gl( ). For more information, see [DP98]. ∞ 6.3 Direct limit structures of unitary highest weight representations 125

V n rescaling, we may assume w.l.o.g that , n+1 = , n on n for every N. Then, 〈· ·〉 〈· ·〉 ∈ , := lim , n defines a contravariant scalar product on V , so that (ρ, V ) is a uni- 〈· ·〉 〈· ·〉 tary representation.−→ It is irreducible and locally finite since all unitary highest weight representations (ρn, Vn) are irreducible and finite-dimensional (Lemma 6.27). Suppose that all weights λ(n) are integer-valued. According to Lemma 6.18, this is equivalent to the existence of unique smooth representations πn : GL(n) GL(Vn) → with derivatives dπn = ρn. This in turn is equivalent to the existence of a (unique) locally finite integrated representation π : GL( ) GL(V ) (Proposition 6.30). Every operator π(u) with u U( ) GL( ) is unitary∞ → (Lemma C.18), hence restricting ∈ ∞ ⊂ ∞ π to the unitary subgroup U( ) yields a unitary integrated representation of ρ u( ). ∞ | ∞ Conversely, suppose that π :U( ) U(V ) is an integrated representation of ρ u( ). Then, it has the following properties:∞ → | ∞

i): That (π, V ) is uniquely determined on U( ) follows from the Uniqueness Lemma (cf. Lemma C.18). ∞

ii): Lemma C.19, applied to the unitary locally finite representation (ρ, V ) and in combination with Proposition C.16, shows that π extends to locally finite integrated representation π : GL( ) GL(V ). ∞ → iii): Applying Proposition 6.30, we find that (π, V ) = lim(πn, Vn) is a direct limit representation of GL( ) = lim GL(n). Hence, the same−→ is true for its restriction ∞ to the subgroup U( ) = lim−→U(n) and proves iii). Since all (πn, Vn) are smooth ∞ (n) n representations with derivative−→ dπn = ρn, we conclude that λ Z for all n N. ∈ ∈ iv): The irreducibility of the unitary highest weight representations (ρn, Vn) is transferred to the smooth integrated representations (πn, Vn). Therefore, the irreducibility of (π, V ) = lim(πn, Vn) follows from Lemma 6.27. −→ v): Since every operator π(u) is unitary and extends to a unitay operator on = V , we obtain a unitary representation π :U( ) U( ). The representationH (π, V ) has continuous orbit maps w.r.t. the∞ finest→ locallyH convex topology on V p (Proposition 6.30) and therefore, also w.r.t. to the norm = , of the pre-Hilbert space V . Consequently, every orbit map of π :U(k·k ) U〈·( ·〉) is the uniform limit of a sequence of continuous orbit maps and thus∞ continuous.→ H 126 7 Direct limits and HWR of U( ) ∞

7 Direct limits and HWR of U( ) ∞ In this section we are going to prove our first main theorem. Given a unitary highest N weight representation (πλ, λ) of U( ), labeled by some highest weight λ Z , then H ∞ 1 ∈ the corresponding first order cohomology space H (U( ), πλ, λ) is nontrivial if and only if λ = 0 and λ contains both infinitely many nonnegative∞ H and infinitely many nonpositive6 entries. In particular, this happens if λ has at least one but at most finitely many nonzero entries. Before proving our theorem in Subsection 7.2, we explore the structure of 1-cocycles of general direct limit groups and introduce the concept of a conditional 1-cocycle in Subsection 7.1. Note that this section is based on Sections 3 and 4 in [NeH16].

7.1 Conditional 1-cocycles

In this subsection, we set the foundation for our analysis of the 1-cocycles of the unitary groups U( ) which is based on the observation that the restriction of a continuous 1-cocycle∞ to any of the compact subgroups U(n) yields a 1-coboundary. Thus, any 1-cocycle of U( ) may be viewed as a limit of a sequence of 1-coboundaries and the question whether∞ the coboundary property is prerserved under this limit can be translated into handy necessary and sufficient criteria for the 1-cohomology space of U( ) to be trivial (cf. Proposition 7.5). Note that the core ideas of this subsection are∞ taken from Section 3 in [NeH16]. Let G be a topological group with an ascending sequence of topological subgroups G such that G S G and the group topology of G coincides with the direct ( n)n N = n N n ∈ ∈ limit topology. This is the finest topology on G for which all inclusion maps ιn : Gn G → is continuous. We call G a direct limit topological group write G := lim Gn. Definition 7.1. Let (π, ) be a continuous unitary representation−→ of G. A 1-cocycle β : G is called conditionalH if its restriction to every subgroup Gn is a 1-coboundary. In particular,→ H every 1-coboundary on G is a conditional 1-cocycle. We write 1 1 Zcond(G, π, ) for the vector space of conditional 1-cocycles and Hcond(G, π, ) for H 1 1 H the quotient Zcond(G, π, )/B (G, π, ). H H Remark 7.2. If all the Gn are compact subgroups, then [Yam98, Theorem 2] shows that the direct limit topology on G is always a group topology. If the representation (π, ) is continuous for every compact subgroup Gn (and thus continuous for G endowedH with the direct limit topology), then every continuous 1-cocycle is conditional so that 1 1 1 1 we have Z (G, π, ) = Zcond(G, π, ) and H (G, π, ) = Hcond(G, π, ). H H H H 1 We consider a conditional 1-cocycle β Zcond(G, π, ). Restricting β to the subgroup G yields a 1-coboundary by definition,∈ i.e. H for some v . The vector n β Gn = ∂vn n | ∈ HGn vn is unique up to adding a Gn-fixed vector. We shall write n := for the subspace of Gn-fixed vectors. Any conditional 1-cocycle β thus definesH aH unique sequence v with the following compatibility condition ( n n⊥)n N ∈ H ∈ vn vm m if m n. (6) − ∈ H ≤ 7.1 Conditional 1-cocycles 127

This means that we can write β(g) = limn π(g)vn vn for any g G. Note that →∞ the spaces n form a decreasing sequence of Hilbert subspaces− whose∈ intersection is given by TH G, the subspace of G-ﬁxed vectors. n N n = This motivates∈ H us toH consider the following abstract situation: Let be any ( n)n N decreasing sequence of Hilbert subspaces. We put : and H: ∈T . 0 = = n N n H H H∞ ∈ H For n N, we deﬁne Vn := n⊥ and Wn := n⊥ n 1. Note that the Wn are mutually orthogonal∈ and we have H H ∩ H −

n M Md Vn = Wk and V := ⊥ = Wk. k N k=1 ∞ H∞ ∈ We write Q W for the linear space of all sequences w with w W and n N n ( n)n N n n Qcond V for the∈ linear space of all sequences v with v ∈V and satisfying∈ (6). n N n ( n)n N n n We∈ define P W , to be the linear space∈ of all∈ antilinear functionals ( n N n C) f : P W L whose∈ restrictions to the subspaces W are continuous. n N n C n ∈ → Lemma 7.3. Every sequence w Q W defines a unique sequence v ( n)n N n N n ( n)n N Qcond ∈ ∈ ∈ ∈ ∈ n Vn via N n ∈ X vn := wk for each n N k=1 ∈ and a unique f P W , via ( n N n C) ∈ L ∈ X X f (w) := wn, w for w Wn. n N〈 〉 ∈ n N ∈ ∈ This defines linear isomorphism

Ycond Y X V W W , . n ∼= n ∼= ( n C) n N n N L n N ∈ ∈ ∈

Moreover, for v V = ⊥ , the following statements are equivalent: ∈ ∞ H∞ i) The sequence v converges in with limit v. ( n)n N ∈ H 55 P ii) The sequence w is square-summable with v ∞ w . ( n)n N = k=1 k ∈ iii) The antilinear functional f extends to a continuous antilinear functional on such that f = v, . H 〈 ·〉

Proof. The first isomorphism is clear because it has an obvious inverse given by wn := vn vn 1 with the convention v0 := 0. The expression f (w) is well-defined because − w is− only a finite sum of elements from Wn so that the right-hand side is also a finite sum. The mapping X w w f w w , w ( n)n N ( ( ) = n ) ∈ 7→ 7→ n N〈 〉 ∈ 55i.e. P w 2 < n N n ∈ k k ∞ 128 7 Direct limits and HWR of U( ) ∞ is obviuously linear and injective. If f : P W is an antilinear functional whose n N n C restriction to each W is continuous, then∈ f defines→ a sequence w Q W n ( n)n N n N n ∈ ∈ ∈ such that, for each n N, we have f (w0n) = wn, w0n for all w0n Wn. This shows the surjectivity. ∈ 〈 〉 ∈

P 2 P 2 i ii : By definition of the v , we have v ∞ w and thus v ∞ w ) = ) n = k=1 k = k=1 k ⇒since the spaces Wk are mutually orthogonal. k k k k ii iii : If the sequence w is square-summable, then the sum P w con- ) = ) ( n)n N n N n ⇒verges in . Let v be its limit.∈ For w P W , we thus obtain ∈ 0 n N n H ∈ ∈ X f (w0) = wn, w0 = v, w0 , n N〈 〉 〈 〉 ∈ showing that f extends to a continuous antilinear functional on with f = v, . H 〈 ·〉 iii i : In view of f P w , and f v, , one infers that the orthogonal ) = ) = n N n = ⇒projection of v onto the subspace∈ 〈 ·〉 W is given〈 by·〉 w . Since v V L W , n n = cn N n we find ∈ ∞ ∈ X v wn lim vn. = = n n N →∞ ∈ Lemma 7.4. The following statements are equivalent: i) There exists some n N for which n = . ∈ H H∞ ii Every sequence v Qcond V converges in . ) ( n)n N n N n ∈ ∈ ∈ H iii Every sequence w Q W is square summable. ) ( n)n N n N n ∈ ∈ ∈ iv Every antilinear functional f P W , extends to a continuous antilinear ) ( n N n C) functional on . ∈ L ∈ H Proof. The equivalences ii) iii) iv) are immediate from Lemma 7.3. Part ⇐⇒ ⇐⇒ iii) is satisfied if and only if there exists some n N such that Wk = 0 for all k > n. In view of ∈ { } n M Md ⊥ = Wk and ⊥ = Wk, n k N H k=1 H∞ ∈ the vanishing of the Wk is equivalent to i).

With the help of the previous lemmata, we obtain the following result about conditional 1-cocycles on the direct limit group G = lim Gn. −→ Proposition 7.5. Let (π, ) be a continuous unitary representation of the direct limit H G G G n n ⊥ n 1 G := lim Gn. For each n N, we put n := ,Vn := n⊥ and Wn := − ∈ H H H H ∩H −→ 7.1 Conditional 1-cocycles 129

with the convention that G0 := e . { 1} Each conditional 1-cocycle β Zcond(G, π, ) deﬁnes unique sequences ∈ H Ycond Y v V , w W ( n)n N n ( n)n N n ∈ ∈ ∈ n N ∈ n N ∈ ∈ and a unique antilinear functional f P W , such that we have the relations ( n N n C) ∈ L ∈ β(g) = π(g)vn vn for all g Gn and all n N, (7) − ∈ ∈ X β(g) = π(g)wn wn for all g G (8) n N − ∈ ∈ and 1 f (π(g)v v) = β(g− ), v for all g G and v . (9) − 〈 〉 ∈ ∈ H These relations deﬁne linear isomorphisms

Ycond Y X Z 1 G, π, V W W , . cond( ) ∼= n ∼= n ∼= ( n C) H n N n N L n N ∈ ∈ ∈ G For v ⊥, the following statements are equivalent ∈ H i) The conditional 1-cocycle β is a 1-coboundary with β = ∂v. ii The sequence v converges in with limit v ) ( n)n N ∈ H iii The sequence w is square-summable and v P w . ) ( n)n N = n N n ∈ ∈ iv) The antilinear functional f extends to a continuous antilinear functional on such H that f (v0) = v, v0 for all v0 . 〈 〉 ∈ H This leads to 1 Gn G Hcond(G, π, ) = 0 ( n N) = . H { } ⇐⇒ ∃ ∈ H H Remark 7.6. In equation (9), we have used that

1 X D := (π(G) ) lin Wn, 〈 − H〉 ⊆ n N ∈ so that the expression a(π(g)v v) makes sense for arbitrary g G and v . The − 1 ∈ ∈ H reason is that, for any g G, we have (π(g) )Wn = 0 for all but ﬁnitely many n G by construction and, clearly,∈ (π(g) 1) =− 0 . { } − H { } cond Proof. Let v Q V . This means that each v V and v v Gm ( n)n N n N n n n n m m = ∈ ∈ ∈ ∈ − ∈ H H for all m n. This means that, for g Gm, we have π(g)vn vn = π(g)vm vm for all n ≤m. Thus, the sequence v ∈ deﬁnes a map β : G− via β g− : ( n)n N ( ) = ≥ ∈ → H limn π(g)vn vn for g G. The restriction to every subgroup Gn yields a 1- coboundary→∞ − . In particular,∈ the restriction of to G is continuous. There- β Gn = ∂vn β n fore, β is continuous| with respect to the direct limit topology on G. This shows that 130 7 Direct limits and HWR of U( ) ∞

β is a conditional 1-cocycle. At the beginning of this subsection, we have seen that, conversely, every conditional 1-cocycle β can be written as β(g) = limn π(g)vn vn for some sequence v Qcond V and the above mapping deﬁnes an→∞ isomorphism− ( n)n N n N n of vector spaces Q W∈ ∈ Z 1 ∈ G, π, . n N n cond( ) The remaining assertions∈ → follow immediatelyH from Lemma 7.3 and Lemma 7.4: The n relation (8) follows immediate from β g lim π g v v and v P w . ( ) = n ( ) n n n = k=1 k The relation (9) follows from (8) together with →∞f P −w , . If the sequence = k N k G ∈ 〈 ·〉 G v converges to some v , then v ⊥ because each v ⊥ and we ( n)n N n conclude∈ that ∈ H ∈ H ∈ H

β lim ∂v ∂v = n n = →∞ is a 1-coboundary. Conversely, assume that β = ∂v is a 1-coboundary for some v . G ∈ H We may assume w.l.o.g. that v ⊥. Then, (9) shows that ∈ H 1 f (π(g)v0 v0) = β(g− ), v0 = v, π(g)v0 v0 for arbitrary v0 . − 〈 〉 〈 − 〉 ∈ H Hence, f extends to a continuous antlinear functional with f = v, . The ﬁnal asser- G tion is immediate from Lemma 7.4 with = . 〈 ·〉 H∞ H Remark 7.7. The methods used in the proof of Proposition 7.5 are not exclusively limited to direct limit groups: Let G be a topological group and suppose that G1 G ... G S G is an increasing exhaustive sequence of proper subgroups. As⊆ 2 = n N n ∈ before,⊆ a⊆ conditional 1-cocycle is a 1-cocycle whose restriction to each subgroup Gn yields a 1-coboundary. Assuming that, whenever the involved unitary representation is continuous, each conditional 1-cocycle is automatically continuous, we can argue along the same lines and obtain exactly the same statement as in Proposition 7.5. The automatic continuity assumption for conditional 1-cocycles holds in particular, if one of the proper subgroups Gn is open or if G is a Baire group (use Proposition 3.9 in the second case).

The following corollary is Proposition 2.4.1 in [BHV08] and we offer a new proof using the concept of conditional 1-cocycles. Corollary 7.8. Whenever a topological group G can be written as the countable union of an increasing sequence of proper open subgroups G , it cannot have Property (FH). ( n)n N ∈ Proof. For each n N, consider the quasi-regular representation of G on 2 ∈ 2 ` (G/Gn) = 0 . By construction, Gn acts trivially on ` (G/Gn), but there does not 6 { } exist a G-ﬁxed vector since G/Gn = . The continuity of this representation follows from the assumption that G|n is an| open∞ subgroup. The direct sum representation on : L `2 G/G then satisﬁes G 0 but Gn 0 for all n . Hence, = n N ( n) = = N HProposition∈ 7.5 together with RemarkH 7.7 show{ } thatH there6 { exists} a nontrivial∈ conditional 1-cocycle, so that G does not have Property (FH).

Corollary 7.9. Let G = lim Gn and assume that (π0, 0) is a unitary representation of G. Then, for the direct sum representation π, : H π , , we have −→ ( ) = b n N( 0 0) H ⊕ ∈ H 1 1 Hcond(G, π, ) = 0 Hcond(G, π0, 0) = 0 . H { } ⇐⇒ H { } 7.1 Conditional 1-cocycles 131

G G Gn G n Proof. For every n N, we have = 0 = 0 (cf. Lemma 7.15 in the next subsection). By∈ virtue of theH ﬁnalH assertion⇐⇒ H in PropositionH 7.5, this implies the assertion.

Problem 7.10. Is Property (FH) for a direct limit group equivalent to every conditional 1-cocycle being a 1-coboundary?

Lemma 7.3 and Lemma 7.4 also apply to unitary representations of Lie algebras and one obtains a Lie algebra analogue of Proposition 7.5: Let g be a complex Lie algebra with involution which is the union of an increasing sequence of Lie subalgebras gn ∗ which are stable under . We write g = lim gn. A homomorphism of Lie algebras ∗ ρ : g ( ) is called a unitary representation−→ of g on the complex Hilbert space if → B H H

ρ(Z)v, w = v, ρ(Z ∗)w holds for all Z g and v, w . 〈 〉 〈 〉 ∈ ∈ H We deﬁne

g gn := v : ρ(g)v = 0 and := v : ρ(gn)v = 0 . H { ∈ H { }} H { ∈ H { }} g T g Note that n . A Lie algebra 1-cocycle ω : g is a linear map with = n N H ∈ H → H ω([Z1, Z2]) = ρ(Z1)ω(Z2) ρ(Z2)ω(Z1). It is called conditional if its restriction to every Lie subalgebra gn yields− a Lie algebra 1-coboundary. This means that, for each n N, we find some vn such that ω(Z) = ρ(Z)vn for all Z gn. We obtain ∈ ∈ H gn ∈ uniqueness if we require vn ( )⊥. Hence, every conditional Lie algebra 1-cocycle ω defines a sequnence v ∈ withH v gn and satisfying the same compatibility ( n)n N n ( )⊥ ∈ ∈ H condition (6) as in the group case. Thus, we have ω(Z) = limn ρ(Z)vn for every conditional Lie algebra 1-cocycle ω. →∞ 1 1 The spaces Zcond (g, ρ, ) and Hcond (g, ρ, ) are defined the same way as in the group case. H H

Proposition 7.11. Let (ρ, ) be a unitary representation of the involutive direct limit H gn Lie algebra g := lim gn. For each n N, we put n := ,Vn := n⊥ and Wn := ∈ H H H gn gn 1 g ( )⊥ − with−→ the convention that 0 := 0 . H ∩ H 1 { } Each conditional 1-cocycle β Zcond (g, ρ, ) deﬁnes unique sequences ∈ H Ycond Y v V , w W ( n)n N n ( n)n N n ∈ ∈ ∈ n N n N ∈ ∈ and a unique antilinear functional f P W , such that we have the relations ( n N n C) ∈ L ∈ ω(Z) = ρ(Z)vn for all Z gn and all n N, (10) ∈ ∈ X ω(Z) = ρ(g)wn for all Z g (11) n N ∈ ∈ and

f (ρ(Z)v) = ω(Z ∗), v for all Z g and v . (12) 〈 〉 ∈ ∈ H 132 7 Direct limits and HWR of U( ) ∞

These relations deﬁne linear isomorphisms

Ycond Y X Z 1 g, ρ, V W W , . cond ( ) ∼= n ∼= n ∼= ( n C) H n N n N L n N ∈ ∈ ∈ g For v ( )⊥, the following statements are equivalent ∈ H i) The conditional Lie algebra 1-cocycle ω is a Lie algebra 1-coboundary with ω = ρ( )v. · ii The sequence v converges in with limit v ) ( n)n N ∈ H iii The sequence w is square-summable and v P w . ) ( n)n N = n N n ∈ ∈ iv) The antilinear functional f extends to a continuous antilinear functional on such H that f (v0) = v, v0 for all v0 . 〈 〉 ∈ H This leads to 1 gn g Hcond (g, ρ, ) = 0 ( n N) = . H { } ⇐⇒ ∃ ∈ H H Proof. The proof goes along the same lines as the proof of Proposition 7.5.

7.2 1-Cocycles of unitary HWR of U( ) ∞ The fixed point criterium in Proposition 7.5 yields nice results when applied to those unitary representation of U( ) which arise as a direct limit of finite-dimensional unitary highest weight representations∞ of the subgroups U(n). Using branching laws, one can express the fixed point condition by handy conditions on the occuring highest weights. Hence, we obtain very concrete criteria for the vanishing of the 1-cohomology spaces related to these direct limit representations. In particular, we derive a classification theorem for the unitary highest weight representations of U( ) which belong to this special class of unitary representations. Note that the basis∞ of this subsection is Section 4 in [NeH16].

n n+1 Deﬁnition 7.12. Let µ Z and µ0 Z . ∈ ∈

i) We say that the tuple µ interlaces the tuple µ0 (in symbols: µ ´ µ0) if there exist permutations σ : 1, 2, . . . , n 1, 2, . . . , n and σ0 : 1, 2, . . . , n + 1 1, 2, . . . , n + 1 such that{ } → { } { } → { }

µσ0 1 µσ(1) µσ0 2 µσ(2) ... µσ0 n µσ(n) µσ0 n 1 . 0( ) ≥ ≥ 0( ) ≥ ≥ ≥ 0( ) ≥ ≥ 0( + )

ii) We denote by (ρµ, µ) the unitary highest weight representation of gl(n) with H highest weight µ. Accordingly, we denote by (ρµ , µ ) the corresponding unitary 0 H 0 highest weight representation of gl(n+1). Recall that both µ and µ are ﬁnite- H H 0 dimensional Hilbert spaces. We write (ρµ, µ) (ρµ , µ ) if (ρµ, µ) occurs as H ⊂ 0 H 0 H a subrepresentation of (ρµ , µ ) restricted to the subalgebra gl(n). 0 H 0 7.2 1-Cocycles of unitary HWR of U( ) 133 ∞

n n+1 Theorem 7.13. Let µ Z and µ0 Z . Then (ρµ, µ) (ρµ , µ ) if and only if µ ∈ ∈ H ⊂ 0 H 0 interlaces µ0 so that we obtain the decomposition M ρµ gl n , µ ρµ, µ . ( 0 ( ) 0 ) = ( ) H µ µ H | ´ 0

Proof. This follows from [GW98, Thm. 8.1.1].

Let µ(n) be a sequence of integer-valued tuples with µ(n) n and such that ( )n N Z n n ∈1 µ( ) ´ µ( + ) for all n N. By virtue of Theorem 7.13, we obtain∈ a directed sequence ∈ of unitary ﬁnite-dimensional irreducible highest weight representations (ρµ(n) , µ(n) ) of H gl(n) with irreducible unitary direct limit (ρ, V ) = lim(ρµ(n) , µ(n) ) (cf. Lemma 6.28). H This direct limit Lie algebra representation of gl( )−→integrates to a unique continuous irreducible unitary direct limit representation ∞

(π, ) := lim(πµ(n) , µ(n) ) H H −→ of the direct limit group U( ) where denotes the Hilbert space completion of V ∞ H (cf. Proposition 6.31) and ((πµ(n) , µ(n) )) the integrated representation of (ρµ(n) , µ(n) ). H H We call the representation lim(πµ(n) , µ(n) ) a highest weight direct limit representation H of U( ). −→ ∞ n n Remark 7.14. A special case arises if, for a given µ ZN, we put µ( ) Z to be the n n 1 tuple consisting of the first n entries of µ. Then, µ(∈) ´ µ( + ) for all n∈ N. For the ∈ unitary highest weight representation (ρµ, Vµ) of gl( ) with highest weight µ and ∞ its integration (πµ, µ) to U( ), Proposition 6.29 shows that we have (ρµ, Vµ) = H ∞ lim(ρµ(n) , µ(n) ) and (πµ, µ) = lim(πµ(n) , µ(n) ) (by Proposition 6.31). H H H −→ −→ 1 The aim is now to decide when the first cohomology space H (U( ), π, ) for the ∞ H direct limit representation (π, ) = lim(πµ(n) , µ(n) ) of U( ) is trivial. To this end, we H H ∞ want to apply the fixed point criterium−→ in Proposition 7.5. We assume that U( ) ∞ = 0 . Otherwise there exists a nontrivial U( )-fixed vector so that the irre- ducibilityH { implies} that π, 1, is the trivial∞ one-dimensional representation. ( ) ∼= ( C) In this case, we have H

1 H (U( ), 1, C) = Hom (U( ), C) = 0 ∞ ∼ top.Grp. ∞ { } n because U( ) is a direct limit of compact subgroups. In terms of the weights µ( ), n the trivial case∞ occurs if and only if all µ( ) = 0.

We need the following elementary tool from unitary representation theory.

Lemma 7.15. Let (π, ) be a continuous unitary representation of a topological group G. H Assume that, for a subset J N, we have a family of G-invariant subspaces ( j)j J such that one of the following conditions⊆ is satisﬁed: H ∈

i) = b j J j H ⊕ ∈ H 134 7 Direct limits and HWR of U( ) ∞

S ii) = j J j H ∈ H

Let (π0, ) be an irreducible representation of G. Then, (π0, ) occurs as a subrepre- K K sentation in (π, ) if and only if it occurs as a subrepresentation in (π, j) for some j J. H H ∈

Proof. We denote by Pj the orthogonal projection onto the subspace j. We may H assume that is a Hilbert subspace of . If (π0, ) (π, ), then, in K ⊂ H H K ⊂ H both cases, we ﬁnd an index j J for which Pj( ) = 0 . The projection operator ∈ K 6 { } Pj : j intertwines π0 and π. Therefore, Pj( ) j is a G-invariant sub- |K K → H K ⊂ H 1 space. By Schur’s Lemma, we further conclude that (Pj )∗ Pj = c . Since the |K |K |K operator (Pj )∗ Pj is positive, the constant c is real-valued and nonnegative. The |K |K 1 case c 0 is excluded by Pj 0 . Therefore, A : Pj : Pj is a = ( ) = = pc ( ) K 6 { } |K K → K linear isometry intertwining π0 and π. This shows (π0, ) (π, j). The converse statement is trival. K ⊂ H

Lemma 7.16. Let (π, ) = lim(πµ(n) , µ(n) ) be a highest weight direct limit representa- H H 1 tion of U( ) which is not the−→ trivial representation. Then, H (U( ), π, ) = 0 if and only if∞ there exists some n such that, for all k n, we have∞ U(n) H 0 . { } N µ(k) = ∈ ≥ H { }

U( ) Proof. If the irreducible representation (π, ) is nontrivial, we have ∞ = 0 so 1 U n that H (U( ), π, ) = 0 if and only if thereH exists some n N forH which { (}) = ∞ H { } ∈ H 0 (Proposition 7.5). Applying the previous Lemma 7.15 with (π0, ) = (1, C), the one-dimensional{ } trivial representation, to the representation (π, )Krestricted to the S H subgroup U(n), we find that = k n µ(k) contains a U(n)-fixed vector if and only if H ≥ H U(n) µ(k) contains a U(n)-fixed vector for some k n. Therefore, = 0 is equivalent toH U(n) 0 for all k n. ≥ H { } µ(k) = H { } ≥ Lemma 7.17. Let k, n N with k > n and let (πµ, µ) be a unitary highest weight ∈ H k representation of the group U(k) with highest weight µ Z . ∈ The representation (πµ, µ) contains a nontrivial U(n)-fixed vector if and only if there k 1 H exists some µ0 Z − interlacing µ such that the unitary highest weight representation ∈ (πµ , µ ) of U(k 1) contains a nontrivial U(n)-fixed vector. 0 H 0 − Proof. In view of [GW98, Thm. 8.1.1], we have M πµ, µ πµ , µ ( ) = ( 0 0 ) H ∼ µ µ H 0´ as representations of the group U(k 1) and hence also of its subgroup − U(n) U(k 1). Applying Lemma 7.15 again with (π0, ) = (1, C) to the group U(n) yields⊆ the− assertion. K

The equivalence statement of the previous lemma motivates the following deﬁnition. Fix n N. ∈ 7.2 1-Cocycles of unitary HWR of U( ) 135 ∞

Deﬁnition 7.18. Assume that (P) is a statement which may be either true or false for k a given tuple µ Z of length k n. We say that (P) is interlacing-inheritable if (P) ∈ k ≥ holds for a tuple µ Z with k > n if and only if there exists a tuple µ0 ´ µ for which (P) holds. ∈

Thus, Lemma 7.17 can be reformulated as follows: The statement that the highest k weight representation (πµ, µ) of U(k) with highest weight µ Z has a nontrivial U(n)-fixed vector (where k H n), is interlacing-inheritable. ∈ ≥ Lemma 7.19. Two interlacing-inheritable statements (P) and (P’) are equivalent if and n only if they are equivalent for all tuples λ Z of length n. ∈ Proof. Assume that two interlacing-inheritable statements (P) and (P’) are equivalent n n+1 n for all tuples λ Z of length n. If µ Z satisfies (P), then there exists a µ0 Z which interlaces∈µ, satisfies (P) and also∈ (P’) by assumption. By definition of interlac-∈ ing inheritable, µ then also satisfies (P’). By symmetry, we conclude that the properties n 1 (P) and (P’) are equivalent for all tuples µ Z + of length n + 1. Iterating this argument leads to the equivalence of (P) and (P’).∈ The converse statement is trivial.

k Lemma 7.20. Let k n. For any µ Z , the property ≥ ∈

# j : µj 0 n and # j : µj 0 n (13) { ≥ } ≥ { ≤ } ≥ is interlacing-inheritable.

k Proof. Let µ Z for k > n. We may assume w.l.o.g. that µ is decreasingly ordered. ∈ The interlacing condition µ0 ´ µ then reads µ1 µ10 µ2 µ20 ... µ0k 1 µk. If ≥ ≥ ≥ ≥ ≥ − ≥ µ0 has the property (13), then the first (last) n entries of µ0 are 0 ( 0), hence so are the first (last) n entries of µ. Conversely, assume that µ has the≥ required≤ property. k 1 We construct µ0 Z − as follows: If k > 2n, choose µ0 ´ µ such that the first (last) ∈ n entries of µ0 coincide with the first (last) n entries of µ. If k 2n, consider the set ≤ J := 1, 2, . . . , n k 1, k 2, . . . , k n . Put µ0j := µj whenever the index j is smaller { }∩{ −J − − } j than the indices from and put µ0j := µj+1 whenever the index is greater than the indices from J. For j J put µj := 0 = µj. ∈ k Proposition 7.21. Let k n and µ Z . Then the unitary highest weight representa- ≥ ∈ tion (πµ, µ) (of U(k)) admits nonzero U(n)-fixed vectors if and only if the following conditionH is satisfied:

# j µj 0 n and # j µj 0 n. (14) { | ≥ } ≥ { | ≤ } ≥ Proof. In view of Lemma 7.20 and Lemma 7.17, both conditions are interlacing inheritable and, by virtue of Lemma 7.19 it just remains to check that they are equivalent n for tuples λ Z . Indeed, (14) is satisﬁed if and only if µ = 0. ∈ Putting things together, we arrive at the following conclusion: 136 7 Direct limits and HWR of U( ) ∞

Theorem 7.22. Let (π, ) = lim(πµ(n) , µ(n) ) be a highest weight direct limit represen- H H 1 tation of U( ) which is not the−→ trivial representation. Then H (U( ), π, ) = 0 if and only if ∞ ∞ H { }

(k) (k) ( n N)( k n) # j : µj 0 < n or # j : µj 0 < n. ∃ ∈ ∀ ≥ { ≥ } { ≤ } Proof. This is immediate from Lemma 7.16 and Proposition 7.21.

N Theorem 7.23. Let λ Z and (πλ, λ) be the corresponding unitary highest weight ∈ 1 H representation of U( ). Then H (U( ), πλ, λ) = 0 if and only if either ∞ ∞ H { } λ = 0 or # j λj 0 < or # j λj 0 < . { | ≤ } ∞ { | ≥ } ∞ Proof. The case λ = 0 corresponds to the one-dimensional trivial representation of U( ) so that we have ∞ 1 H (U( ), πλ, λ) = Homtop.Grp.(U( ), C) = 0 . ∞ H ∼ ∞ { } n We assume now that λ = 0. For each n N, we write µ( ) ZN for the tuple consisting 6 ∈ ∈ of the ﬁrst n entries of λ. Then, we have (πλ, λ) = lim(πµ(n) , µ(n) ) and this highest H H weight direct limit is not the trivial representation. Applying−→ Theorem 7.22, we ﬁnd 1 that H (U( ), πλ, λ) = 0 if and only if there exists some n N for which ∞ H { } ∈ (k) (k) ( k n) # j µj 0 < n or # j µj 0 < n. ∀ ≥ { | ≥ } { | ≤ } # j λj 0 < n or # j λj 0 < n. ⇐⇒ { | ≥ } { | ≤ } This completes the proof. 137

2 8 Tensor products and HWR of Up(` ) with ﬁnitely supported weights

In the preceding section, we determined for which highest weight λ ZN, the corresponding unitary highest weight representation of U( ) with highest∈ weight λ has a non-trivial first 1-cohomology space. In this section, we∞ turn to the following Banach- 2 2 completions of U( ): Let 1 p < and put ` := ` (N, C) for the complex separable Hilbert space of∞ square summable≤ ∞ sequences with standard complete orthonormal system e . The unitary group U `2 U `2 1 `2 is a Banach-Lie group ( n)n N p( ) = ( ) ( + p( )) ∈ 2 2 2 ∩ B with Banach-Lie algebra up(` ) = u(` ) p(` ), i.e. the skew hermitian operators of p-th Schatten class. Its complexification∩ B is given by the p-th Schatten operators u `2 `2 . For any z u `2 the involution z coincides with the usual p,C( ) = p( ) p,C( ) ∗ B ∈ 2 Hilbert adjoint. The inductive limit U( ) is a dense subgroup in Up(` ). Therefore, 2 ∞ we refer to the groups Up(` ) as Banach-completions of U( ) (cf. Subsection 5.2). From AppendixC, we know that, for p > 1, the unitary highest∞ weight representa- 2 tion extends to the Banach-completion Up(` ) if and only if the highest weight λ is finitely supported. In this case, we realize the unitary highest weight representations 2 as subrepresentations of multiple tensor product representations of Up(` ) which are 2 2 built from its natural action on ` and on the dual Hilbert space (` )∗. This is based on some Schur-Weyl Theory. The concrete realization enables us to construct unbounded 1-cocycles and we thus find that finitely supported highest weights produce nontrivial 1-cohomology spaces. This is different when p = 1 and the highest weight λ is not finitely supported (cf. Section9). The material of this section can be found in Sections 6 and 7 of [NeH16]. We have splitted Section 7 into these Subsections 8.1 and 8.3. The content of Section 6 is presented in this Subsection 8.2.

8.1 Realization of unitary HWR with ﬁnitely supported weight

For ﬁnitely supported weights λ ZN, the corresponding unitary highest weight rep- ∈ resentation (πλ, λ) of U( ) extends to a norm-continuous unitary representation H ∞ 2 of Up( ) (for p [1, ) and := ` (N, C)). We realize these representations as H ∈ ∞ H subrepresentations in ﬁnite tensor products of the natural representation of Up( ) H on resp. on ∗ (cf. the representations π1 and π1∗ from Subsection 5.3). H H

(N) We denote by (N0) the set of all decreasingly ordered non-negative integer valued ↓ (N) tuples with a finite number of positive entries. To any λ (N0) , we associate a Young ∈ ↓ diagram Dλ (also called Ferrers diagram) which consists of `λ := max j λj > 0 { | } rows and the j-th row has λj boxes, so that the whole diagram consists of λ := P`λ λ boxes. Conversely, any Young diagram D (with row (and column) length| | i=1 i (N) weakly decreasing) defines a unique λ (N0) by counting the row boxes. Given ∈ ↓ a Young diagram D, we obtain the conjugate (or transposed) Young diagram D0 by (N) switching rows and columns. Thus, for any λ (N0) , we define the conjugate tuple ∈ ↓ 2 138 8 Tensor products and HWR of Up(` ) with finitely supported weights

(N) λ 0 via the relation Dλ : D . Equivalently, this means that 0 (N ) 0 = λ0 ∈ ↓

λ0j := # k : λk j . { ≥ } j Note that `λ = λ1 and the entry occurs λj λj+1 times in λ0. 0 − Example 8.1. For λ := (3, 2, 2, 1) we have

Dλ = and Dλ0 =

and the conjugate tuple is given by λ0 = (4, 3, 1).

The corresponding Young tableau Tλ is obtained by ﬁlling in the boxes of the young diagram Dλ with the numbers 1, 2, . . . , λ in the following manner: The number 1 is placed in the top box of the ﬁrst column.| The| number k+1 is placed in the box directly below k, if it exists and otherwise in the top box of the next column.

Example 8.2. For λ = (3, 2, 2, 1) this yields

1 5 8 Dλ = Tλ = −→ 2 6 3 7 4

Let Sλ be the permutation group of the set 1, 2, . . . , λ . Denote by Rλ the subgroup { | |} of permutations leaving all subsets deﬁned by the rows of Tλ invariant and accordingly by Cλ the subgroup leaving all subsets deﬁned by the columns invariant.

2 b λ Put := ` (N, C) and consider the λ -fold tensor product ⊗| |. The permutation H | | H group Sλ acts unitarily on the tensor product via

λ λ b λ b λ S v v 1 πperm : λ ⊗| | ⊗| |, (σ, b|j |1 j) b|j |1 σ (j) × H → H ⊗ = 7→ ⊗ = −

b λ for σ Sλ and vj . We denote this representation by (πperm, ⊗| |). The ∈ ∈ H H group Up( ) (for p [1, ]) also acts on the tensor product space via the ten- H ∈ ∞ b λ sor product representation (id, )⊗| |. In the notation of Subsection 5.3, we have H (id, ) = (π1, ). Both representations commute, i.e. for σ Sλ and g Up( ), H H b λ b λ ∈ ∈ ∞ we have πperm(σ) id⊗| |(g) = id⊗| |(g) πperm(σ). Therefore, the linear operator ◦ ◦ X Pλ := sgn(c)πperm(cr), r Rλ,c Cλ ∈ ∈ 8.1 Realization of unitary HWR with ﬁnitely supported weight 139

b λ where sgn(c) denotes the signum of the permutation c, commutes with id⊗| | and its image b λ λ := Pλ( ⊗| |) H H is a Up( )-invariant subspace. We write πλ for the corresponding norm-continuous H unitary representation on λ. Note that Pλ is not necessarily an orthogonal projection, but the operators H 1 X 1 X P : sgn c π c and P : π r Cλ = ( ) perm( ) Rλ = perm( ) fC fR λ c Cλ λ r Rλ ∈ ∈ Q`λ Q`λ with constants f : 0 λ ! and f : λ ! , are orthogonal projections and Cλ = j=1( 0j ) Rλ = j=1( j ) we have P f f P P . Let e denote an ONB for . For n , we deﬁne λ = Cλ Rλ Cλ Rλ ( n)n N N the vectors · ∈ H ∈ ` λ ` λ (λ) q λ 0m q λ 0m e : f 0 e f P 0 e . (15) n = Cλ b m 1 i 1 n+i = Cλ Cλ (b m 1 b i 1 n+i) · ⊗ = ∧ = · ⊗ = ⊗ = λ `λ λ0m By construction, the vectors e( ) are mutually orthgonal. Since P 0 e n Rλ (b m=1 b i=1 n+i) = `λ λ0m ⊗ ⊗ 0 b m 1 b i 1en+i, we have constructed an orthonormal sequence in λ. In particular, if ⊗ = ⊗ = H λ 1, the space λ is inﬁnite dimensional. |Example| ≥ 8.3. For λH 3, 2, 2, 1 the orthonormal sequence e(λ) in is given by = ( ) ( n )n N λ ∈ H p e(λ) e e e e e e e e n = 4!3!1! ( n+1 n+2 n+3 n+4)b( n+1 n+2 n+3)b n+1. | {z } · ∧ ∧ ∧ ⊗ ∧ ∧ ⊗ =12

(N) Lemma 8.4. For any λ (N0) , the norm-continuous unitary representation (πλ, λ) ∈ ↓ H of Up( ) is a unitary highest weight representation with highest weight λ in the sense of DeﬁnitionH 6.22. In particular, it is irreducible.

Proof. We have to show that the restriction of πλ to the dense subgroup U( ) is a uni- ∞ tary highest weight representation with highest weight λ: For any n N with n `λ, n b λ b λ ∈ ≥ consider the canonical embedding (C )⊗| | , ⊗| |. The operator Pλ leaves the sub- n b λ → H b λ space (C )⊗| | invariant as well as the restriction of id⊗| | to the subgroup U(n) n b λ ⊂ Up( ). Therefore, the subspace Pλ (C )⊗| | is U(n)-invariant and, according to H [GW98, Thm. 9.3.10], this is a (ﬁnite-dimensional) unitary highest weight represen- U n (n) tation of ( ) with highest weight λ := (λ1, λ2,..., λn). This shows that πλ U( ) = λ | ∞ (N)b lim πλ(n) is the direct limit representation on Vλ := Pλ C ⊗| | which is dense in

−→λ. Applying Proposition 6.29 and Proposition 6.31 shows that (πλ U( ), λ) is the unitaryH highest weight representation of U( ) with highest weight| λ∞(seeH also Re- mark 7.14). ∞

Now, let λ Z(N) be an integer valued tuple such that all but finitely many entries are ∈ (N) zero. For any such λ Z , we define the postive and negative parts λ± := max( λ, 0) (N) ∈ ± and λ± (N0) to be those tuples for which the entries of λ± are decreasingly or- ↓ ∈ ↓ + dered. Then we have λ λ− = λ and (πλ , λ ) = (πλ , λ ) since λ± .λ±. We ± ± ∼ ± ± + P− H ↓ H ↓ ↓ ∈ W put λ := λ + λ− = ∞i 1 λi . | | | | | | = | | 2 140 8 Tensor products and HWR of Up(` ) with finitely supported weights

Lemma 8.5. For any λ Z(N), the continuous unitary representation ∈

(πλ+ , λ+ )b(πλ∗ , λ∗ ) H ⊗ − H − is a unitary highest weight representation of the group Up( ) with heighest weight λ, H i.e. we have (πλ, λ) = (πλ+ , λ+ )b(πλ∗ , λ∗ ). H ∼ H ⊗ − H − Proof. (cf. [BN12, Thm. 2.2] for the case p = ) That the tensor product repre- ∞ sentation (πλ+ , λ+ )b(πλ∗ , λ∗ ) is irreducible follows from Section 2.17 of [Ol90] H ⊗ − H − so that the assertion follows from Proposition 6.25 since (πλ∗ , λ∗ ) = (π λ , λ ) − − ∼ − − (Proposition 6.24). H − H−

Remark 8.6. Let n N. The analogue statement for tensor products of unitary highest weight representations∈ of the compact group U(n) is wrong: Take for instance n λ := (1, 1, 0, . . . , 0) = e1 e2 Z . The case λ = e1, e2 corresponds to the idential n representation− of U(n) on−C (cf.∈ Example 6.17). The tensor product representation on n n Mat n, corresponds to the conjugation representation of U n . C (C )∗ ∼= ( C) ( ) However, this⊗ representation is not irreducible since the subspace sl(n, C) of trace zero 56 matrices is a proper invariant subspace of Mat(n, C). n + If λ = (λ1,..., λn) Z is decreasingly ordered and λn < 0, then one splits λ = λ λ− + ∈ − with λ := λ+ m(1, 1, . . . , 1) and λ− := m(1, 1, . . . , 1) where m := λn . Then, the rep- m | | resentation πλ(g) := det(g)− πλ+ (g) deﬁnes a unitary highest weight representation · + on λ+ with highest weight λ = λ λ−. H −

8.2 Construction of unbounded 1-cocycles in ﬁnite tensor products

This section should be viewed as a preparation for Section 8.3. We consider a countably infinite sum of 1-coboundaries that converges pointwise and ask whether we thus obtain an unbounded 1-cocycle. To make life easier, we assume the underlying group G to be a Baire group since then, the pointwise converging sum is automatically con- 2 tinuous (cf. Proposition 3.9). Note that the unitary groups Up(` ) are completely metrizable for any p [1, ] and thus Baire groups. We derive a simple sufficient criterion for the unboundedness∈ ∞ of the sum when (π, ) is an arbitrary continuous unitary representation (cf. Lemma 8.7 below) and focusH afterwards on the case of a finite tensor product representation (cf. Proposition 8.10).

Let G be a Baire group and (π, ) be a continuous unitary representation of G. H Lemma 8.7. Let e be an orthonormal sequence in such that, for all n, we have ( n)n N ∈ H

en π(g)v v : g G, v lin . ∈ 〈 − ∈ ∈ H 〉 56In the inﬁnite-dimensional case, this situation does not occur since S sl n, is a dense subspace n N ( C) 2 ∈ in 2(` ). B 8.2 Construction of unbounded 1-cocycles in ﬁnite tensor products 141

Further, let a be a sequence in for which the sum β g : P a π g e e ( n)n N C ( ) = n N n[ ( ) n n] converges in for∈ every g G. Then β : G deﬁnes a (continuous)∈ 1-cocycle which− is a 1-coboundaryH if and only∈ if the sequence→a H is square summable. ( n)n N ∈

Proof. That β is a continuous 1-cocycle follows from Proposition 3.9. If P a 2 < n N n , then we have ∈ | | ∞ X an[π(g)en en] = π(g)v v (16) n N − − ∈ for the vector v : P a e . Conversely, assume that β is a 1-coboundary. Then, = n N n n ∈ we find some v which satisfies (16) for all g G. Let (ej)j J be a complete ∈ H ∈ ∈ orthonormal system in containing the en, i.e. we have N J. We expand v w.r.t. H ⊆ the system (ej) and obtain the coefficients bj := v, ej . For j J N, we put aj := 0. Equation (16) then leads to 〈 〉 ∈ \ X (aj bj)[π(g)ej ej] = 0 for all g G. j J − − ∈ ∈ P Therefore, we have j J (aj bj) ej, w = 0 for any ∈ − 〈 〉 w D := π(g)v v : g G, v lin. Since en D by assumption, we obtain that ∈ 〈 − ∈ ∈ H 〉 ∈ X an bn = (aj bj) ej, en = 0 − j J − 〈 〉 ∈ and we conclude that the sequence a is square integrable. ( n)n N ∈

Now, we turn to unbounded 1-cocycles of ﬁnite tensor products: Assume that we are m G given 2 continuous unitary representations (πi, i)i=1,...,m of . We form the Hilbert tensor≥ product , : N , . H (π ) = ci=1,...,m(πi i) H H Lemma 8.8. Let e(i) be an orthonormal sequence in . For simplicity we will write ( n )n N i g.e(i) instead of π g e(∈i). Assume that a is a sequenceH in such that, for all i and n i( ) n ( n)n N C all g G, ∈ X 2 ∈ 2 (i) (i) an g.e e < . n n i n N | | − H ∞ ∈ Then X (1) (m) (1) (m) β(g) := an[g.en b b g.en en b ben ] n N ⊗ · · · ⊗ − ⊗ · · · ⊗ ∈ deﬁnes a 1-cocycle w.r.t. the tensor product representation. Moreover, we have the estimate

m X 2 X X 2 2 (1) (m) (1) (m) 2 (i) (i) an g.e g.e e e m an g.e e . n b b n n b b n n n i n N | | ⊗ · · · ⊗ − ⊗ · · · ⊗ ≤ · i=1 n N | | − H ∈ ∈ (17) 2 142 8 Tensor products and HWR of Up(` ) with ﬁnitely supported weights

Proof. It is enough to verify that the sum converges for all g G (Proposition 3.9). ∈ Choose some g G and unitary operators Ui U( i). For ﬁxed j 1, . . . , m and arbitrary M > N∈ N we calculate ∈ H ∈ { } ∈ 2 M P a U e(1) g 1 .e(j) U e(m) n=N n[ 1 n b b( ) n b b m n ] M ⊗ · · · ⊗ − ⊗ · · · ⊗ P (1) (1) 1 (j) 1 (j) (m) (m) anan U1e , U1e g .e , g .e Ume , Ume = n,n =N 0 n n 1 ( ) n ( ) n j n n m M 0 20 H 0 H 0 H P 2 〈 1 (j) 〉 ···〈 − − 〉 ···〈 〉 an g .e 0. = n=N ( ) n N,M | | − −→→∞ P (1) (j) (m) We conclude that the sum ∞ a U e g 1 .e U e converges n=1 n[ 1 n b b( ) n b b m n ] for every j and g. This shows that ⊗ · · · ⊗ − ⊗ · · · ⊗

P (1) (m) (1) (m) n an[g.en g.en en en ] m N b b b b P ∈ P a ⊗g. ·e ·(1 ·)⊗ g−.e(j 1⊗) · ·g · ⊗ 1 .e(j) e(j+1) e(m) = j=1 n N n[ n b b n − b( ) n b n b b n ] ∈ ⊗ · · · ⊗ ⊗ − ⊗ ⊗ · · · ⊗ converges for every g G. The estimate (17) follows∈ from

2 (1) (m) (1) (m) g.en b b g.en en b ben 2 m ⊗ · · · ⊗ − ⊗ · · · ⊗ P g.e(1) g.e(j 1) g 1 .e(j) e(j+1) e(m) = j=1 n b b n − b( ) n b n b b n Pm ⊗ · · · ⊗j 2 ⊗ − ⊗ ⊗ · · · ⊗ m g 1 .e( ) . j=1 ( ) n j ≤ − H This completes the proof.

For our purposes, tensor products of the form (πλ, λ)b(πµ∗ , m∗) are of particular H ⊗ H interest, where (πλ, λ) and (πµ, µ) are two irreducible unitary representations H H of G. The canonical isomorphism λ b µ∗ = 2( µ, λ) induces an equivalent uni- H ⊗H ∼ B H H tary representation on 2( µ, λ) which is given by the conjugation action B H H 1 (g, A) πλ(g)Aπµ(g− ), for all g G, A 2( µ, λ). 7→ ∈ ∈ B H H In particular, if (πλ, λ) and (πµ, µ) are isomorphic unitary representations, then H H πλ bπµ∗ is equivalent to the conjugation representation on 2( λ). The following lemma⊗ should be considered as an analogue of Lemma 8.7 in theB caseH of tensor product representations.

Lemma 8.9. Let A ( µ, λ) be a bounded linear operator such that ∈ B H H 1 β(g) := πλ(g)Aπµ(g− ) A 2( µ, λ) − ∈ B H H for every g G. ∈

i) Assume that (πλ, λ) and (πµ, µ) are non-isomorphic irreducible representations. H H Then β is a 1-coboundary if and only if A 2( µ, λ). ∈ B H H ii) Assume that (πλ, λ) = (πµ, µ). Then β is a 1-coboundary if and only if A 1 H H ∈ 2( λ) + C . B H 8.2 Construction of unbounded 1-cocycles in ﬁnite tensor products 143

Proof. In both cases, β is a 1-coboundary if and only if there exists B 2( µ, λ) 1 ∈G B H H such that πλ(g)(A B)πµ(g− ) = A B for all g G. We denote by ( µ, λ) the − − ∈ B H H space of linear bounded operators intertwining the representations πµ and πλ. If the G representation are nonisomorphic, then Schur’s Lemma implies that ( µ, λ) = B H H 0 . This shows the first assertion. In the case πµ = πλ, Schur’s Lemma states that { }G 1 ( λ) = C which proves the second assertion. B H · Proposition 8.10. Let (πµ, µ) and (πλ, λ) be two infinite dimensional, irreducible representations of the Baire groupH G. AssumeH that there exists a bounded but not square summable sequence a ` , `2 , such that, for some orthonormal se- ( n)n N ∞(N C) (N C) quences e and f ∈ in∈ resp. \ , ( n)n N ( n)n N λ µ ∈ ∈ H H X 2 1 2 X 2 1 2 an πλ g en < and an πµ g fn < (18) ( ( ) ) λ ( ( ) ) µ n N | | k − kH ∞ n N | | − H ∞ ∈ ∈ holds for all g G. ∈ 1 Then we have H (G, πλ bπµ∗ , λ b µ∗) = 0 and an unbounded 1-cocycle for the tensor product representation⊗ πH, ⊗H π6 ,{ } is obtained as follows: If π π are ( λ λ)b( µ∗ m∗) µ ∼= λ isomorphic and if there is a (nonzero)H ⊗ c Hfor which P a c 2 < , then C n N n ∈ ∈ | − | ∞ X n β(g) := ( 1) [πλ(g)en bπµ∗ (g)fn∗ en b fn∗] n N − ⊗ − ⊗ ∈ defines an unbounded 1-cocycle. If πµ πλ, then X β(g) := an[πλ(g)en bπµ∗ (g)fn∗ en b fn∗] n N ⊗ − ⊗ ∈ defines an unbounded 1-cocycle, where the an are taken from (18).

1 1 Proof. First, we note that (πµ(g) )fn = (π∗ (g) )fn∗ . By Lemma 8.8, µ µ µ − H − H ∗ β g : P a π g e π g f e f deﬁnes a continuous 1-cocycle for the ( ) = n N n[ λ( ) n b µ∗ ( ) n∗ n b n∗] ∈ ⊗ − ⊗ tensor product. With respect to the conjugation action on 2( µ, λ) this 1-cocycle has the form β g π g Aπ g 1 A, where A P a Be Hf HP a , f e . ( ) = λ( ) µ( − ) = n N n n n∗ = n N n n µ n − ∈ ⊗ ∈ 〈· 〉H Since the sequence (an) is bounded but not square summable, the operator A is bounded but not Hilbert–Schmidt. If πµ and πλ are not isomorphic, then Lemma 8.9 implies that β is unbounded. If πµ = πλ are isomorphic, we may assume w.l.o.g. that πµ = πλ. ∼ 1 In this case, Lemma 8.9 implies that β is unbounded unless A 2( λ) + C . If this 1 is the case, then there exists 0 = c C such that A c is Hilbert–Schmidt∈ B H and we obtain the estimate 6 ∈ − ·

X 1 2 X 2 X 2 > (A c )fn = anen c fn an c . ∞ n N k − · k n N k − k ≥ n N | − | ∈ ∈ ∈ Hence, if there is no c for which P a c 2 < , then the operator A cannot C n N n be written as a linear combination∈ of the∈ identity| − | 1 and∞ a Hilbert–Schmidt operator and we conclude that the 1-cocycle β is unbounded. However, if such a c C exists, ∈ 2 144 8 Tensor products and HWR of Up(` ) with ﬁnitely supported weights then c is nonzero, because the sequence a is not square summable, and a ( n)n N ( n) is the sum of a constant sequence and a square∈ summable sequence. It is clear that condition (18) then holds for for the constant sequence an := c and, since c = 0, for 6 n all bounded sequences. In particular, we may choose the sequence an := ( 1) which − has the property that there is no c C for which the difference (an c) is square summable. This means that β g :∈ P 1 n π g e π g f −e f is an ( ) = n N( ) [ λ( ) n b µ∗ ( ) n∗ n b n∗] unbounded 1-cocycle. ∈ − ⊗ − ⊗

2 8.3 1-Cocycles of unitary HWR of Up(` )

The objective of this subsection is to show that, for λ 2, the ﬁrst cohomology space 1 | | ≥ H (Up( ), πλ, λ) never vanishes. H H 2p Lemma 8.11. Put q := p 2 if 2 < p < and q := if 1 p 2. For every sequence a a `q , , we− have ∞ ∞ ≤ ≤ = ( n)n N (N C) ∈ ∈ X 2 1 2 an (g )en < n N | | k − kH ∞ ∈ for every g Up( ). Moreover, this expression depends continuously on g in the Up( )- topology. ∈ H H

Proof. For A r ( ) and r [1, ], we denote the r-th Schatten norm by ∈ B H ∈ ∞ 1 ¨ r (Tr( A )) r if r < A r := | | ∞ . k k A op if r = k k ∞ We remind the reader of the generalized Hölder inequality (see [GK69, Chap. III]) 1 1 1 BC B C for B , C and , s, t 1, . r s t s( ) t ( ) r s + t [ ] k k ≤ k k · k k ∈ B H ∈ B H ≤ ∈ ∞(19) For 1 p 2, the statement is clear, since g 1 and P g 1 e 2 2( ) n N ( ) n = 1≤2 ≤ − ∈ B H ∈ k − k g 2. For p > 2, we consider the diagonal operator Aen = anen and note that kA − k and P a 2 g 1 e 2 g 1 A 2. Applying (19) with r 2, q( ) n N n ( ) n = ( ) 2 = s =∈p BandHt = q yields∈ the| assertion.| k − k k − k

(N) Lemma 8.12. Let λ (N0) such that λ 2 and deﬁne q as in Lemma 8.11. Consider ∈ ↓ | | ≥ the unitary highest weight representation (πλ, λ) of Up( ) for 1 p < . There exists an orthonormal sequence e(λ) in π Hg v v : Hg U ≤ , v ∞ ( n )n N λ( ) p( ) λ lin λ such that, for each a a ∈ `q and− each∈ g H U ∈ H, the⊆ sum H = ( n)n N p( ) ∈ ∈ ∈ H 2 P a π g 1 e(λ) converges in and P a 2 π g 1 e(λ) < . n N n[ λ( ) ] n λ n N n [ λ( ) ] n ∈ − H ∈ | | − ∞ Proof. We ﬁx some complete orthonormal system e in `2 and choose the ( n)n N = λ λ ∈ orthonormal system e(λ) Æf 1 0m e in from (15H ). The e(λ) are weight n = Cλ b m 1 i 1 n+i λ n · ⊗ = ∧ = H 2 8.3 1-Cocycles of unitary HWR of Up(` ) 145

vectors for the diagonal operators in Up( ). If t = diag(t1, t2,...) is such a diagonal H λ λ operator, then we have π t e(λ) e(λ) Q 1 Q 0m t 1 e(λ). This shows that λ( ) n n = ( m=1 i=1 n+i ) n e(λ) g v v g −v − n πλ( ) : Up( ), λ lin. To prove the second claim, we apply Lemma∈ 8.11 in− combination∈ withH Lemma∈ H 8.8 and ﬁnd that

X λ λ λ λ a 1 0m ge 1 0m e n[b m=1 b i=1 n+i b m=1 b i=1 n+i] n N ⊗ ⊗ − ⊗ ⊗ ∈ b λ exists in ⊗| |. Here, the assumption λ 2 was used. Projecting onto λ via λ λ Æf P H 1 0m e e(λ) shows that| |P ≥ a π g 1 e(λ) exists in H. Here Cλ Cλ (b m=1 b i=1 n+i) = n n N n[ λ( ) ] n λ we have· used⊗ that⊗ P intertwines the representations∈ idb λ −and π . Using theH estimate Cλ ⊗| | λ (17) in Lemma 8.8, we ﬁnally derive the estimate

2 X 2 X λ λ λ λ 2 1 (λ) 2 1 0m 1 0m an g .e fC an PC gen i en i ( ) n λ = λ λ [b m=1 b i=1 + b m=1 b i=1 + ] λ n N | | − H n N | | ⊗ ⊗ − ⊗ ⊗ H ∈ ∈ 2 X λ λ λ λ f a 2 1 0m ge 1 0m e Cλ n b m=1 b i=1 n+i b m=1 b i=1 n+i λ ≤ n N | | ⊗ ⊗ − ⊗ ⊗ H ∈ (17) X f λ 2 a 2 g 1 e 2 < . Cλ n ( ) n ≤ | | n N | | k − kH ∞ ∈ 2 Theorem 8.13. Put := ` (N, C). Let λ Z(N) be a ﬁnitely supported integer valued H ∈ sequence and (πλ, λ) be the corresponding unitary highest weight representation of H Up( ), where p [1, ). Then, H ∈ ∞ 1 H (Up( ), πλ, λ) = 0 λ 1. H H { } ⇐⇒ | | ≤ 1 If λ 2, then the spaces H (Up( ), πλ, λ) are inﬁnite-dimensional. | | ≥ H H Proof. The case λ = 0: This means that λ = 0 which corresponds to the | | one-dimensional trivial representation of Up( ) on C. Here, we have H 1 1 H (Up( ), , C) = HomGrp(Up( ), C) = 0 H H { } since Up( ) contains the increasing sequence of compact subgroups U( ) = S U nHwhich is dense in U . ∞ n N ( ) p( ) ∈ H The case λ = 1: In this case, we have λ = ek for some k N and Example 6.17 | | ± ∈ shows that (πλ, λ) is either isomorphic to the idential representation of Up( ) H H on (if λ = ek .e1) or isomorphic to the dual representation of Up( ) on H ∈ W H ∗ (Proposition 6.24). The assertion thus follows from Proposition 5.24 and HProposition 5.25.

The case λ 2: We split λ into its positive and negative part and obtain λ± |(N)| ≥ ∈ (N0) as in Subsection 8.1. Then, we have (πλ, λ) = (πλ+ , λ+ )b(πλ∗ , λ∗ ) ↓ H ∼ q H2 ⊗ − H − by Lemma 8.5. Suppose that λ− = 0. Then, for any a ` ` (where q is de- ﬁned as in Lemma 8.11), the 1-cocycle β g : P a∈ π\ g 1 e(λ) exists ( ) = n N n[ λ( ) ] n ∈ − 2 146 9 Maximal tori and general norm-continuous representations of Up(` )

(Lemma 8.12), is continuous (since Up( ) is completely metrizable and thus a Baire group) and is unbounded (LemmaH 8.7). Now, suppose that λ+ = 0. In this case we have (πλ, λ) = (πλ∗ , λ∗ ) and the assertion follows from the fact that the 1-cohomologyH space− ofH the− dual representation vanishes if and only if it vanishes for the original representation (cf. Lemma 2.4 ii)). Finally, suppose that λ± = 0. In this case, we obtain an unbounded the 1-cocycle of P λ+ λ λ+ λ the form β g 6 a π g e( ) π g e( −) e( ) e( −) for a suit- ( ) = n N n[ λ+ ( ) n b λ∗ ( )( n )∗ n b( n )∗] able a `q `2. This∈ is immediate from⊗ Proposition− −8.10 in⊗ combination with Lemma∈ 8.12\. In particular, the construction of β in all three cases shows that 1 the space H (Up( ), πλ, λ) is inﬁnite-dimensional since the quotient space `q/`2 is inﬁnite-dimensional.H H

9 Maximal tori and general norm-continuous represen- 2 tations of Up(` )

According to the results in AppendixC, it is necessary and sufficient for extending the 2 highest weight representation from U( ) to U1(` ) that λ `∞(N, Z) is bounded. In this bounded case, we study the corresponding∞ highest weight∈ representation by re- 2 2 striction to the maximal torus T1(` ) U1(` ) consisting of the diagonal operators of 2 ⊂ 2 2 U1(` ) (w.r.t. the standard complete orthonormal system of ` ). The torus T1(` ) decomposes the highest weight representation into weight spaces and from certain sum- mation properties of these characters, we derive that highest weight representations with bounded, non-finitely supported weights produce trivial first cohomology spaces. The argument is based on the following oservation for arbitray norm-continuous uni- 2 2 tary representations of U1(` ): Whenever the restriction to T1(` ) does not admit non- 2 trivial 1-cocycles, then the same holds for U1(` ) (cf. Proposition 9.13). The reason for this is detected on the Lie algebra level: The weight set of such a representation cannot contain any root of the Lie algebra gl( ) and this implies that restricting 1-cocycles to the maximal torus is an injective linear∞ mapping (cf. Lemma 9.1 below).

9.1 Restriction of 1-cocycles to a (maximal) torus

Let ρ : g End(V ) be a representation of a complex topological Lie algebra g on a complex topological→ vector space V with continuous orbit maps. Suppose that h g is an abelian topological Lie subalgebra with linear dual ⊆

h0 := µ : h C µ is linear and continuous . { → | } As usual we deﬁne

µ V := v V : ρ(H)v = µ(H)v for all H h { ∈ ∈ } 9.1 Restriction of 1-cocycles to a (maximal) torus 147 to be the µ-weight space of (ρ, V ) and deﬁne the weight set as

µ := µ h0 : V = 0 . P { ∈ 6 { }} If (ρ, V ) = (ad, g) is the adjoint representation of g on itself, then we call every nonzero weight α a root of g and write ∆ for the set of roots. We assume that g has a root decomposition with respect to h in the sense that

M α g0 := h g ⊕ α ∆ ∈ 1 is a dense Lie subalgebra of g. We denote by Z (g, ρ, V ) the vector space of continuous Lie algebra 1-cocycles ω : g V of (ρ, V ). → Lemma 9.1. Let (ρ, V ) be a representation of g such that no root occurs as a weight, i.e. ∆ = . Then the restriction map P ∩ ; 1 1 Res : Z (g, ρ, V ) Z (h, ρ h, V ), ω ω h → | 7→ | is injective.

1 Proof. Let ω Z (g, ρ, V ) such that ω(h) = 0 . We show that then ω 0. For a root α ∆, H h∈and X gα, we have { } ≡ ∈ ∈ ∈ ρ(H)ω(X ) = ρ(H)ω(X ) ρ(X ) ω(H) = ω([H, X ]) − | {z } =0 = α(H)ω(X ).

Since no root α occurs as a weight of the representation ρ by assumption, we conclude ω(X ) = 0 and hence ω gα = 0. This shows that ω vanishes on the dense Lie subalgebra L α | g0 = h α ∆ g and, since ω is continuous, on the whole Lie algebra. ⊕ ∈ We apply this lemma to the Lie algebra gl( ) which has a root decomposition with respect to the abelian Lie subalgebra h of∞ diagonal operators. It turns out that, for unitary highest weight representations, the condition that no root occurs as a weight is necessary and sufﬁcient for the restriction map Res to be injective.

N Proposition 9.2. Let λ R 0 such that λn λ1 Z for all n N. For the unitary ∈ \{ } − ∈ ∈ highest weight representation (ρλ, Vλ) of gl( ), the following are equivalent: ∞

i) Some α ∆ occurs as a weight of (ρλ, Vλ). ∈ ii) λ Z[∆]. ∈ 0 iii)V λ = 0 , i.e. there exists a nonzero v Vλ such that ρ(h)v = 0 . 6 { } ∈ { } 1 1 iv) The linear map Res : Z (gl( ), ρλ, Vλ) Z (h, ρλ h, Vλ) is not injective. ∞ → | 2 148 9 Maximal tori and general norm-continuous representations of Up(` )

If a unitary highest weight representation of gl( ) extends to one of the Banach–Lie 2 ∞ algebras glp(` ) with 1 p , then the same result holds for the extended representation. ≤ ≤ ∞

Note that we have tacitly endowed gl( ) and Vλ respectively with the discrete topology. ∞

Proof.i ) = ii) : Since = conv(S( ).λ) (λ Z[∆]), we have α λ Z[∆] which ∞ implies⇒ λ Z[∆]. P ∩ − ∈ − ∈ ii iii S V 0 ) = ) : We show that 0 conv( ( ).λ). Then, 0 which is to say = 0 . ⇒ ∈ ∞ ∈ P 6 { } Recall that all roots of gl( ) are of the form α = "i "j. In particular, they are ﬁnitely supported weights∞ whose entries add up to− zero. If λ is a Z-linear combination of such roots, then it has the same properties. In particular, there exists some n N such that λk = 0 for all k > n. For the symmetric group S S ∈ n ( ), we obtain ⊂ ∞ 1 X S λ := τ.λ conv( ( ).λ). n! ∞ τ Sn ∈ ∈ By construction, the ﬁrst n entries of λ are equal and all other entries are zero. Since n n X X λi = λi = 0, i=1 i=1 we conclude that λ = 0 and the assertion is proven.

0 iii) = iv) : For any nonzero v Vλ , the 1-coboundary ω(X ) := ρ(X )v is nonzero. ⇒ ∈ Otherwise, v would generate a one-dimensional invariant subspace and (ρλ, Vλ) would contain the one-dimensional trivial representation which is excluded by the condition that λ = 0. Hence ω = 0 but ω h = 0. 6 6 | iv) = i) : This is Lemma 9.1. ⇒ If a unitary highest weight representation of gl( ) extends to one of the Banach– 2 ∞ Lie algebra glp(` ), then the extended representation has the same weight set = S P conv( ( ).λ) (λ Z[∆]), so that the last assertion follows with the same proof. ∞ ∩ − Remark 9.3. For λ = 0, the equivalence in Proposition 9.2 does not hold: This case corresponds to the one-dimensional trivial representation whose weight set is = 0 , so that no root occurs as a weight. Hence Lemma 9.1 implies that Res is injective.P In{ } fact, if gl( ) acts trivially on C, then the corresponding 1-cocycles are linear maps ω : gl( ) ∞C satisfying ω(XY ) = ω(YX ) for all X , y gl( ). Hence they are trace functionals,∞ → i.e. ω(X ) = c Tr(X ) for some constant c ∈C. Any∞ such trace functional vanishing on the diagonal operators is identically zero.∈ However, what is true for any weight is the statement that Res is injective if and only if no root occurs as a weight. 9.2 1-Cocycles of maximal tori of Banach–Lie algebras 149

p p In the following, we abbreviate ` := ` (N, C).

Proposition 9.4. Let (πλ, λ) be the unitary highest weight representation of the unitary 2 H group Up(` ) for 1 p and with highest weight λ = 0. Then, ≤ ≤ ∞ 6 1 2 1 Res : Z (Up(` ), πλ, λ) Z (Tp, πλ, λ), ω ω `p H → H 7→ | is injective if and only if λ / Z[∆]. ∈ Proof. If :U 2 is a continuous 1-cocycle with 0, then the complex β p(` ) λ β Tp = linear derivative dβ→satisﬁes H dβ p 0. Applying the previous| Proposition 9.2, we C C ` = ﬁnd dβ 0 and thus β 0. | C = =

9.2 1-Cocycles of maximal tori of Banach–Lie algebras

Let (V, ) be a complex normed vector space and view it as an abelian Lie algebra h := V .k·k Let ρ : V ( ) be a norm-continuous Lie algebra representation of V . → B H This means that ρ is a continuous linear map for which (ρ(x))x V is a family of pairwise commuting operators. We assume that the operators ρ(x∈) are simultaneously diagonalizable, i.e. there exists a common orthonormal eigenbasis (ej)j J of to- ∈ H gether with continuous linear functionals sj : V C such that, for all j J and x V , → ∈ ∈ we have ρ(x)ej = sj(x)ej. We further assume that all functionals sj are nonzero. Then, any 1-cocycle ω : V is of the form → H X ω(x) = cjsj(x)ej (20) j J ∈ for suitable complex numbers cj. The map ω is a 1-coboundary if and only if the cj are P 2 square-summable, i.e. j J cj < . Note that the norm-continuity of the linear ∈ | | ∞ map ρ : V ( ) implies that supj J sj(x) ρ x < for all x V . Thus, we obtain a→ linear, B H continuous map ∈ | | ≤ k k k k ∞ ∈

sJ : V `∞(J, C), x sJ (x) := (sj(x))j J . (21) → 7→ ∈ 1 The problem now is to find necessary and sufficient criteria for the space H (V, ρ, ) to be trivial. It seems that this depends on “how closely” the zero functional s =H0 can be approximated by the eigenvalue functionals sj. To make this more precise, we consider the image sJ (V ) in the commutative Banach algebra `∞(J, C). For any s `∞(J, C), ∈ 2 s := s0 `∞(J, C) : s0 s ` (J, C) J { ∈ · ∈ } 2 defines an ideal in `∞(J, C) containing ` (J, C). Here, s0 s := (s0jsj)j J denotes the · ∈ multiplication in `∞(J, C). More generally, if S `∞(J, C), then ⊆ \ S := s J s S J ∈ 2 150 9 Maximal tori and general norm-continuous representations of Up(` )

2 deﬁnes an ideal in `∞(J, C)containing ` (J, C). In this notation, we have Z 1 V, ρ, (22) al g ( ) = sJ (V ) H ∼ J and the problem is to determine for which (linear) subsets S `∞(J, C), the corre- 2 ⊆ sponding ideal S equals ` (J, C). We will not solve this problem in full generality, p but we will haveJ a closer look at special cases for V = ` (N, C), 1 p . ≤ ≤ ∞

2 From now on, we assume that = ` (J, C) for some infinite set J and write ρ = H 2 b j J sj. If (ej)j J denotes the standard orthonormal system in ` (J, C), then we define ⊕ ∈ P ∈ eF := j J ej for any finite subset F J. More generally, if I J is an arbitrary subset, ∈ P ⊂ ⊆ we define eI := j I ej `∞(J, C). For s, s0 `∞(J, C), we write s := ( sj )j J and ∈ ∈ ∈ | | | | ∈ s0 s if s0j sj holds for every j J. ≤ ≤ ∈ 2 Lemma 9.5. i) For s `∞(J, C), we have s = ` (J, C) if and only if there exists some " > 0 such that∈ s "eJ , i.e. if sj J" for all j J. | | ≥ | | ≥ ∈ 1 P ii) If S is a finite subset, then S = s for s := S s S s . J J | | ∈ | | iii) If s0 s , then s s . | | ≤ | | J ⊆ J 0 iv) If S0 S, then S S . ⊆ J ⊆ J 0 Proof.i ) Suppose that " > 0 exists. Then, s is an invertible element of the Banach 1 algebra `∞(J, C). For any s0 in the ideal s, we have s0s− s and therefore 1 2 J 2 ∈ J s0 = s0s− s ` (J, C). This shows that s ` (J, C) which implies equality since ∈ 2 J ⊆ s contains ` (J, C) by construction. Conversely, if " > 0 does not exist, then J P 2 we find a countably infinite subset I J such that j I sj < . This means 2 ⊆2 ∈ | | ∞ that eI s ` (J, C) and thus eI s ` (J, C). · ∈ ∈ J \ ii) This follows from the inequality 2 2 X 1 X X X X 1 X s 2 s s 2 s 2 S 2 s 2 s , 0j S j 0j j 0j S j j J | | s S | | ≤ s S j J | | | | ≤ | | j J | | s S | | ∈ | | ∈ ∈ ∈ ∈ | | ∈

where s0 `∞(J, C). ∈ The statements iii) and iv) are immediate. 2 Lemma 9.6. Let ρ := b j J sj be a weight representation of V on = ` (J, C) such that ⊕ ∈ H supj J sj < , where sj := sup x 1 sj(x) . ∈ ∞ k k≤ | | i) Assume that there exists some " > 0 and ﬁnitely many x1, x2,..., xn V such that 1 ∈ inf sj(xk) : j J, k = 1, 2, . . . , n ". Then H (V, ρ, ) = 0 . {| | ∈ } ≥ H { } ii) Assume that there exists some " > 0 such that for all ﬁnite subsets F J, one can ⊂ pick some xF V with xF 1 such that sj(xF ) " for all j F. Then 1 H (V, ρ, ) =∈ 0 . k k ≤ | | ≥ ∈ H { } 9.2 1-Cocycles of maximal tori of Banach–Lie algebras 151

iii) If the family ( sj )j J contains a subsequence converging to zero, then 1 ∈ H (V, ρ, ) = 0 . H 6 { } iv) Assume that (V, ) is a Banach–space. Then k·k Z 1 V, ρ, . ( ) ∼= sJ (V ) H Jp 1 If there exists a p [1, ) such that sJ (V ) ` (J, C), then H (V, ρ, ) = 0 . ∈ ∞ ⊆ H 6 { } Proof. We apply Lemma 9.5:

1 i) Put s := n ( sJ (x1) + sJ (x2) + ... + sJ (xn) ). Then, s "eJ by assumption and |2 | | | | | | | ≥ thus s = ` (J, C). Now the assertion follows from J `2 J, . sJ (V ) sJ (x1),sJ (x2),...,sJ (xn) = s = ( C) J ⊆ J{ } J ii) If F J is a finite subset, then we have ω ω(xF ) since xF 1 by ⊂ P k k ≥ k k k k ≤ assumption. Further, we have ω(xF ) = j J cjsj(xF )ej and thus ∈ 2 X 2 2 X 2 2 2 X 2 ω(xF ) = cj sj(xF ) cj sj(xF ) " cj . k k j J | | | | ≥ j F | | | | ≥ j F | | ∈ ∈ ∈ 2 P 2 ω We conclude that j F cj k "2k holds for arbitrary finite subsets F J and ∈ 2 P 2| | ω≤ P ⊂ it follows that j J cj k "2k < . This shows that ω(x) = j J cjsj(x)ej = P∈ | | ≤ ∞ ∈ ρ(x)v with v := j J cj ej . ∈ ∈ H 2 P 2 iii) Let I J be an infinite subset such that C := j I sj < . Then ω(x) := P ⊆ ∈ ∞ j I sj(x)ej satisfies ∈ 2 X 2 X 2 2 2 2 ω(x) = sj(x) sj x = C x k k j I | | ≤ j J k k k k ∈ ∈ for all x V . Hence ω is a continuous 1-cocycle which is not of the form P∈ P ρ(x)v = j J vjsj(x)ej for some v = j J vj ej . ∈ ∈ ∈ H iv) If V is a Banach space, then every 1-cocycle ω is automatically continuous: Since P all functionals sj are continuous, the 1-cocycle ω(x) = j J cjsj(x)ej is totally ∈ weakly continuous with respect to the total subset E := ej : j J of (cf. Theorem 3.12). Hence, it is already continuous by virtue{ of V∈being} aH Baire space (as a Banach space). We conclude that Z 1 V, ρ, Z 1 V, ρ, . ( ) = al g ( ) = sJ (V ) H H ∼ J p Now, suppose that s V ` J, which implies p . If p > 2, put J ( ) ( C) sJ (V ) ` (J,C) 2p ⊆ J ⊇ J 1 1 1 q := p 2 > 2 and if 1 p 2, put q := . Then, we have 2 p + q and Lemma− 5.4 shows that ≤ ≤ ∞ ≤ 2 p q p s ` J, : s s ` J, for all s ` J, ` J, . ` (J,C) = 0 ∞( C) 0 ( C) ( C) = ( C) J { ∈ · ∈ ∈ } Altogether we have q 2 p ` J, ` J, . sJ (V ) ` (J,C) = ( C) ( C) J ⊇ J ) 2 152 9 Maximal tori and general norm-continuous representations of Up(` )

1 Remark 9.7. Item iii) of Lemma 9.6 shows that the first order cohomology space H vanishes only if the zero functional s = 0 is isolated from the family (sj)j J with respect ∈ to the norm of the dual V 0. However, this condition is not sufficient: Let J := N and 1 V := ` (N, C). We consider the functionals sn := en, i.e. sn(x) := xn. Then sn = 1 for all n but, since s x x `1 , , we have H1 0 by Lemma 9.6k ivk . N N( ) = (N C) = ) 1 Likewise,∈ the hypothesis that H =∈ 0 if and only if s = 60 is{ pointwise} isolated from { } 1 the family (sj)j J turns out to be false in general: Again, consider J := N, V := ` (N, C) P∈ and s x : ∞ x . Then, s 1 for all n and we have s x 0 for all n( ) = k=n k n = N n( ) 1 1 x ` (N, C). Yet, we have H k= k0 by Lemma 9.6∈ ii): If F 1, 2, . . . , n→is a finite ∈ { } ⊆ { } subset, where n = max(F), then sk(en) = 1 for all k F. However, as we will see below, the latter hypothesis might be true in more special∈ settings.

p p In the following, we take a closer look at the Banach space V := ` := ` (N, C) for 1 p < . Let e be the natural Schauder basis of `p. For a weight s `q, with ( k)k N 1 ≤ 1 ∞ ∈ ∈ p + q = 1, we call the set

supp(s) := k N : s(ek) = 0 { ∈ 6 } the support of s.

Lemma 9.8. Let ρ : s be a countable direct sum of one-dimensional weights = b n N n s `q. We assume that C⊕ :∈ sup s < and that n = n N n q ∈ ∈ k k ∞ s s for all n max supp( n) < min supp( n+1) N. ∈ 1 p 2 Then, we have H (` , ρ, ` ) = 0 . 6 { } Proof. Let x `p. Then, s x : s x , s x ,... `p because of the estimate N( ) = ( 1( ) 2( ) ) ∈ ∈ ! X X X s x p s x p s p x p N( ) p = n( ) n q k k k n N | | ≤ n N k k k supp(sn) | | ∈ ∈ ∈ p X p (sup sn q) xk , n ≤ N k k · k | | ∈ N ∈ which leads to s x C x . Now, the assertion is a direct consequence of N( ) p p Lemma 9.6 iv). k k ≤ · k k

q Proposition 9.9. Let ρ := b j J sj be a direct sum of one-dimensional weights sj ` . We ∈ assume that there exists N,⊕M N such that ∈ ∈

1.) Every sj has at most N nonzero entries, i.e. supp(sj) N for all j J. | | ≤ ∈ 1 2.) For all j J and k N with sj(ek) = 0, we have M sj(ek) M. ∈ ∈ 6 ≤ | | ≤ 1 p 2 Then, H (` , ρ, ` (J, C)) = 0 if and only if there is no subsequence of (sj)j J converging pointwise to zero. { } ∈ 2 9.3 Norm-continuous unitary representations of Up(` ) 153

Proof. Suppose first that there is such a subsequence. In this case, we may assume w.l.o.g. that J = N and that sn(x) 0 for all x and n . If sn(ek) = 0, then 1 → → ∞ 6 sn(ek) M > 0 by assumption. Since limn sn(ek) = 0, we have sn(ek) = 0 for all but| finitely| ≥ many n . This allows us to choose→∞ a subsequence s as follows: N ( nj )j N Put s : s and k ∈: max supp s . Then, we can find some sufficiently∈ large n1 = 1 1 = ( 1) N n such that min supp s > k and∈ put k : max supp s . Inductively, we thus 2 N ( n2 ) 1 2 = ( n2 ) obtain∈ a subsequence s which satisfies the condition ( nj )j N ∈ max supp s < min supp s for all j . ( nj ) ( nj 1 ) N + ∈ 1 p 2 Hence, H (` , ρ, ` (J, C)) = 0 follows from Lemma 9.8. Now, suppose that there is6 no{ subsequence} converging pointwise to zero. For p < , the spaces `p , are separable and we find a dense sequence x `p ,∞. (N C) ( k)k N (N C) ∈ For m N, we define ⊂ ∈ ¦ 1 © Um := sj : sj(xk) < for all k = 1, 2, . . . , m . | | m

If all Um were nonempty, then we could pick some sm Um for all m N and would obtain a sequence s converging pointwise to zero.∈ Indeed, for x∈ `p , and ( m)m N (N C) ∈ some xk we have the estimate ∈

sm(x) sup sj q x xk p + sm(xk) for all m N. | | ≤ j J k − k | | ∈ ∈ 1 Note that supj J sj q NM < and that sm(xk) < m for m k. This shows that ∈ ≤ ∞ | | ≥ sm(x) 0 for m which contradicts our assumption. Therefore, Um = for → → ∞ 1 ; some m N. This means that inf sj(xk) : j J, k = 1, 2, . . . , m m > 0, so that 1 p 2 H (` , ρ,∈` (J, C)) = 0 follows from{| Lemma| 9.6∈ i). } ≥ { }

2 9.3 Norm-continuous unitary representations of Up(` )

2 Let 1 p < and let ρ : glp(` ) ( ) be a norm-continuous unitary Lie algebra ≤ ∞ 2 → B H 2 representation of glp(` ) on some complex Hilbert space . We denote by h glp(` ) the abelian Lie subalgebra of diagonal operators w.r.t. theH standard orthonormal⊂ system e of `2. We then have h `p : `p , which allows us to identify h with ( n)n N = = (N C) p ∈ p∼ 2 ` via the natural embedding of ` into glp(` ) as diagonal operators. Thus, every h- weight s of ρ is a continuous linear functional on `p, so that the corresponding weight s q set := s : = 0 may be viewed as a subset of ` , where the exponent q satis- P1 1 { H 6 { }} S q ﬁes p + q = 1. Note that the Weyl group := ( ) acts on ` via permutation of the ∞ q entries, i.e. via τ.s : s 1 for s Ws ` and τ S . = ( τ (n))n N = ( n)n N ( ) − ∈ ∈ ∈ ∈ ∞ Lemma 9.10. The weight set is invariant under the action of the Weyl group, i.e. for S and s , we have s . τ ( ) τ. ∈ ∞ ∈ P ∈ P Proof. P 2 P e e Consider the unitary permutation operator τ Up(` ) deﬁned by τ n := τ(n). For any diagonal operator H : H h, the operator∈ H : Ad H P 1 H P = ( n)n N τ = Pτ = τ τ ∈ ∈ − ◦ ◦ 2 154 9 Maximal tori and general norm-continuous representations of Up(` )

H e H e is also diagonal with τ n = τ(n) n. Since the representation ρ is norm-continuous, 2 there exists a smooth representation πe : GLf p(` ) GL( ) with derivative L(π) = ρ. 2 2 → H Let q : GLf p(` ) GLp(` ) be the universal covering map. Since q is surjective, we ﬁnd some P GL→ `2 such that q P P . With Ad : Ad q, we have Ad Ad . eτ f p( ) (eτ) = τ f = Pτ = f Peτ Applying Lemma∈ C.3, we ﬁnd ◦

ρ Ad X π P ρ X π P 1 for all X gl `2 . (f Peτ ) = e(eτ) ( )e(eτ)− p( ) ∈ Now, let v be a nonzero weight vector of some weight s . Then, v : P 1 v is τ = πe(eτ)− a nonzero weight vector of the weight τ.s, because ∈ P

1 ρ(H)vτ = π(Peτ)− ρ(Hτ)v = s(Hτ)vτ = τ.s(H)vτ.

2 Lemma 9.11. If a root of glp(` ) occurs as a weight of the representation ρ, then 1 p H (h, ρ h, 0⊥) = 0 , where 0 := v : ρ(H)v = 0 for all H ` is the subspace p of 0-weight| H vectors6 { and} h := `H. { ∈ H ∈ }

Proof. Since the weight-set is invariant under the permutation action of Weyl group S P ( ) (Lemma 9.10), the fact that contains a root means that contains the subset S ∞ P P ( ).α for α := (1, 1, 0, 0, . . . , 0, . . .), i.e. all ﬁnite permutations of α. We conclude ∞ − that the restriction ρ `p contains a direct sum bτ S τ.α as a subrepresentation of 0⊥. | ⊕ ∈ ∞ H Obviously, the subset S( ).α contains a subsequence converging pointwise to zero, e.g. ∞ τn.α := (0, . . . , 0, 1, 1, 0, 0, . . . , 0, . . .)(n 1). | {z } 2n 2 − ≥ − 1 p Hence, H (` , ρ `p , 0⊥) = 0 follows from Proposition 9.9. | H 6 { } 2 Proposition 9.12. Let ρ : glp(` ) ( ) be a norm-continuous unitary Lie algebra 2 → B Hp representation of glp(` ). Then, for h := ` , we have

1 1 2 H (h, ρ h, ) = 0 = H (glp(` ), ρ, ) = 0 . | H { } ⇒ H { } 1 Proof. Assume that H (h, ρ h, ) = 0 . Then, according to Lemma 9.11, no root occurs as a weight and| H Lemma{ } 9.1 implies that the restriction map 1 1 Res : Z (glp, ρ, ) Z (h, ρ h, ) is injective. Hence, if the restriction of ω 1 2 H → | H ∈ Z (glp(` ), ρ, ) to h is a 1-coboundary, then ω must be itself a 1-coboundary. This 1H 2 shows that H (glp(` ), ρ, ) = 0 . H { } 2 Proposition 9.13. Let π :U1(` ) U( ) be a norm-continuous unitary representation 2 → H of U1(` ). Then, H1 T , , 0 H1 U 2 , , 0 . ( 1 π T1 ) = = ( 1(` ) π ) = | H { } ⇒ H { } 1 2 1 Remark 9.14. The converse implication H (U1(` )) = 0 = H (T1) = 0 is false. { } ⇒2 2 { } An example is given by the identical representation of U1(` ) on ` (cf. Subsection 5.3). Here, we have already seen that H1 U `2 H1 gl `2 0 (Proposi- ( 1( )) ∼= ( 1( )) = tion 5.24) but H1 T ` , /`2 , . The latter can be seen as{ follows:} For ( 1) ∼= ∞(N C) (N C) 2 9.3 Norm-continuous unitary representations of Up(` ) 155 the Lie algebra t of the connected Lie group , we have t `1 , and hence 1 T1 1 ∼= (N R) t `1. By Corollary 4.35 and Proposition 4.37 we thus find ( 1)C ∼= Z 1 T Z 1 t Z 1 `1 . ( 1) ∼= ( 1) ∼= ( ) Every Lie algebra 1-cocycle ω Z 1 `1 is of the form ω x P c x e (cf. equa- ( ) ( ) = n N n n n ∈ tion (20) in Subsection 9.2). The∈ coefficients cn are uniquely determined by ω and 1 2 form a bounded sequence, since cn = ω(en) ω . In view of ` ` , every bounded sequence c defines| such| k a Lie algebrak ≤ k 1-cocyclek ω and we⊂ obtain a ( n)n N linear isomorphism Z 1 `∈1 ` . ( ) ∼= ∞

Proof. It is enough to prove that the restriction map

Res : Z 1 U 2 , , Z 1 T , , , ( 1(` ) π ) ( 1 π T1 ) β β T1 H → | H 7→ | is injective: To this end, consider the complex linear Lie derivative dπ of π which is a C 2 norm-continuous unitary Lie algebra representation of gl1(` ). We claim that no root of gl `2 occurs as a weight of dπ . If this is true, then the corresponding restriction 1( ) C 1 map to h := ` on the Lie algebra level

1 2 1 Res : Z gl ` , dπ , Z h, dπ , , ω ω 1 ( 1( ) C ) ( C h ) ` H → | H 7→ | is injective by Lemma 9.1. Hence, if Z 1 U 2 , , satisﬁes 0, then β ( 1(` ) π ) β T1 = dβ 1 0 for the complex linear derivative.∈ This alreadyH implies dβ | 0 and thus C ` = C = β =|0. To show that there are no roots in the weight space of dπ , we decompose the Hilbert C space = 0 1, where H H ⊕ H : T1 v : dπ H v 0 for all H h `1 0 = = C( ) = = H H { ∈ H ∈ } and : . In view of H1 T , π , 0 , we have H1 h, ρ , 0 by 1 = 0⊥ ( 1 T1 1) = ( h 1) = H H | H { } | 2H { } Corollary 4.35, so that Lemma 9.11 shows that there is no root of gl1(` ) occuring in the weight set of dπ . C Remark 9.15. The statement in Proposition 9.13 is also true for p > 1. But we will see below, that the condition H1 T , , 0 already implies that is the triv- ( p π Tp ) = π ial representation and in this case, the| assertionH { } already follows from the fact that HomTopGrp(Up, C) = 0 . { } Theorem 9.16. For 1 < p , every norm-continuous unitary representation of 2 ≤ ∞ Up(` ) is a direct sum of (irreducible) unitary highest weight representations.

Proof. This is [Ne98, Thm. III.14].

2 Theorem 9.17. Let (π, ) be a norm-continuous unitary representation of Up(` ) for 1 < p < . Then, the followingH assertions hold ∞ 2 156 9 Maximal tori and general norm-continuous representations of Up(` )

1 2 i) We have H (Up(` ), π, ) = 0 if and only if H { } 1 G 2 2 (π, ) = ( , )b n (π1, ` )b n∗ (π1∗, (` )∗), H ∼ H ⊕ · ⊕ · 2 where n, n∗ N denote the corresponding multiplicities and (π1, ` ) denotes the ∈ 2 2 identical representation of Up(` ) on ` from Subsection 5.3. ii) We have H1 T , , 0 if and only if , 1, G is the trivial repre- ( p π Tp ) = (π ) = ( ) sentation. | H { } H H

Proof. In view of Theorem 9.16, we may assume that π, L m π , ( ) = cλ Z(N) λ ( λ λ) H ∈ · H is a direct sum of highest weight representations, where mλ denotes the multiplicity of (πλ, λ). H

i) This is a consequence of Theorem 8.13: If mλ = 0 for some highest weight λ 1 2 6 with λ 2, then H (Up(` ), πλ, λ) = 0 . If mλ = for some weight λ with λ =|1,| then ≥ this means that eitherH(π6, { )}contains a∞ subrepresentation | | H ¨ π , `2 , λ S . 1, 0, 0, 0, . . . Md 1 2 ( 2 2( )) ( ) ( ) (πλ, λ) = ( , ` )b(πλ, λ) = B 2 ∈ ∞ n N ∼ ∼ π , ` , λ S . 1, 0, 0, 0, . . . , ∈ H ⊗ H ( 20 2( )) ( ) ( ) B ∈ ∞ − where π2 :Up( ) 2( ) 2( ), (g, A) gA H × B H → B H 7→ and 1 π20 :Up( ) 2( ) 2( ), (g, A) Ag− . H × B H → B H 7→ Both representations admit nontrivial 1-cocycles (cf. Subsection 5.3), so that 1 2 H (Up(` ), π, ) = 0 in this case. If mλ = 0 whenever λ 2 and if mλ N H 6 { } 1 2 1 | | ≥2 ∈ whenever λ = 1, then H (Up(` ), π, ) = 0 since H (Up(` ), πλ, λ) = 0 whenever |λ| 1 (cf. Theorem 8.13). H { } H { } | | ≤ ii) Consider the unitary highest weight representation (πλ, λ) for some λ = 0. p Tp H 6 On the Lie algebra level, ` decomposes ( λ )⊥ into one-dimensional weight H spaces corresponding to the weights s λ 0 . The set of nonzero weights ∈ P \{ } is invariant under the action of the Weyl group S( ) (Lemma 9.10). Since all weights are ﬁnitely supported, we ﬁnd a subsequence∞ of nonzero weights with mutually disjoints supports. Then Lemma 9.8 shows that we can construct a non- p trivial 1-cocycle of ` which integrates to a nontrivial 1-cocycle of Tp. Hence, H1 T , , 0 for all 0 so that H1 T , , 0 only if 1. ( p πλ Tp λ) = λ = ( p π Tp ) = π = But the trivial| H representation6 { } does6 not admit nontrivial| 1-cocyclesH { } because there are no nonzero continuous group homomorphisms Tp C. →

2 9.4 Unitary HWR of U1(` ) with inﬁnitely supported weights

Now we turn to the case that was postponed in Section8, namely the case of a bounded but not ﬁnitely supported highest weight λ. In this case, the unitary highest weight 2 9.4 Unitary HWR of U1(` ) with inﬁnitely supported weights 157 representation of gl( ) integrates to a norm-continuous unitary highest weight rep- ∞ 2 resentation only for p = 1. As we shall see, the so obtained unitary HWR of U1(` ) does not admit unbounded 1-cocycles.

2 Proposition 9.18. Let ρ : gl1(` ) ( ) be a norm-continuous unitary representa- 2 → B H tion of the Lie algebra gl1(` ) which is generated by some weight vector corresponding to 1 2 an infinitely supported weight µ `∞(N, R). Then H (gl1(` ), ρ, ) = 0 . ∈ H { } 1 1 Proof. By Proposition 9.12, the assertion follows if H (h, ρ h, ) = 0 , where h := ` . µ Since (ρ, ) is generated by some weight vector v ,| it isH a weight{ } representation with weightH set µ+Z[∆] (cf. Remark 6.4). If ∈F H is a finite subset, then there P ⊆ ⊆ P exists a sufficiently large index n N such that µ0n = µn for all µ0 F. Since µ is ∈ ∈ infinitely supported, we may assume that µn = 0. In view of Lemma C.22, we have 6 µn µ1 Z for all n N. Hence, we find some " (0, 1] such that µn " whenever − ∈ ∈ ∈ | | ≥ µn = 0. This shows that the condition in Lemma 9.6 ii) is satisfied and we conclude 6 1 1 that H (` , ρ `1 , ) = 0 . | H { } 2 Now, we consider a norm-continuous unitary representation π :U1(` ) U( ). Any → H T1-weight of such a representation is a continuous character χ : T1 T. Its complex linear derivative µ : dχ : `1 is a weight of the Lie algebra gl →`2 and we have = C C 1( ) → i µ(x) 1 χ(exp(i x)) = e for all x t1 = ` (N, R). ∈ ∼ We say that χ is finitely supported (resp. infinitely supported) whenever so is µ.

2 Theorem 9.19. Let (π, ) be a norm-continuous unitary representation of U1(` ) which is generated by some T1-weightH vector corresponding to some infinitely supported weight 1 2 χ : T1 T. Then, H (U1(` ), π, ) = 0 . → H { } This happens if (π, ) is an irreducible norm-continuous weight representation of 2 H U1(` ) with infinitely supported weights. In particular, for every unitary highest weight 2 representation of U1(` ) with an infinitely supported highest weight λ, the first-order cohomology space is trivial.

2 Proof. We denote by v the generating weight vector and abbreviate U1 := U1(` ). If U1 Pf ix is the orthogonal projection onto the subspace , then Pf ix v = 0, since v is a T1-weight vector corresponding to a nonzero weight.H Since v generates and Pf ix commutes with π, we ﬁnd H * + 2 2 Pf ix = Pf ix π(g)v : g U1(` ) lin = Pf ix π(g)v : g U1(` ) = 0 . H 〈 ∈ 〉 | {z } ∈ { } π g P v 0 = ( ) f ix = lin

This means that U1 0 . The complex linear derivative ρ : dπ : gl : gl `2 = = C 1 = 1( ) ( ) is a norm-continuousH { } unitary Lie algebra representation and we claim that→ it isB generatedH by v. If this is true, then

1 1 H (U1, π, ) = H (gl1, ρ, ) = 0 H ∼ H { } 2 158 9 Maximal tori and general norm-continuous representations of Up(` ) follows from Proposition 9.18 combined with Corollary 4.35 and Proposition 4.37.

Denote by v the closed ρ-invariant subspace which is generated by v. This deﬁnes a H 2 norm-continuous unitary weight representation of gl1(` ) and all weights are integer- valued. Therefore, the representation ρ on v integrates to a norm-continuous unitary representation of U1 by Theorem C.24.H This integrated representation coincides with π on v by the Uniqueness Lemma (Lemma C.4). This shows that v and, conse- H H quently, also its orthogonal complement v⊥ are π-invariant subspaces. Let w v⊥. 2 H ∈ H For any X u1 := u1(` ), we have ∈ 1 w, exp X v v R w, exp tX d X v d t π( ( )) = 0 π( ( )) π( ) 1 〈 − 〉 R 〈 exp tX w, d X 〉v d t 0. = 0 π( ( )) π( ) = 〈 − 〉 Since U1 = exp(u1) Grp, we conclude that 〈 〉 w, π(g)v v = 0 for all g U1. 〈 − 〉 ∈ U1 This is only possible if w = 0 . This entails v⊥ = 0 resp. that v = as it was claimed. ∈ H { } H { } H H 159

A Automatic continuity of 1-cocycles

In this appendix, we turn to the Automatic Continuity Problem for 1-cocycles β : G . This is the question whether the 1-cocycle equation implies the continuity→ of H the map β. We shall see in Proposition A.7 that, for continuous unitary representations (π, ) of Baire groups G, the 1-cocycle β is continuous if and only if it is weakly continuousH in the sense that all maps g β(g), v are continuous. This observation often facilitates the proof of continuity7→ for 〈 a given〉 1-cocycle. Moreover, Proposition A.7 states that the continuity of β is equivalent to its local boundedness in . This is the same continuity characterization as for linear maps between Banach spaces.H From Proposition A.7, we derive that every 1-cocycle β : G , corresponding to a continuous unitary representation (π, ) of an abelian Baire→ H group G such G that = 0 , is automatically continuous (CorollaryH A.8). This appendix is self- containedH and{ } independent from the rest of the work.

Let G be a topological group57 with identity e. We call G a Baire group if it is of second Baire category, i.e. if it cannot be written as a countable union of nowhere dense subsets (cf. subsection 3.1). Let (π, ) be a continuous orthogonal / unitary representation of G on the real / complexH Hilbert space . A map β : G satisfying the relation H → H

β(gh) = β(g) + π(g)β(h), g, h G ∀ ∈ is called a 1-cocycle.

Deﬁnition A.1. A 1-cocycle β is called

continuous if the map β : G is continuous. • → H totally weakly continuous if there exists a total subset E such that for all • v E, the map g β(g), v is continuous. If E = ⊆ H, we call β weakly continuous∈ . 7→ 〈 〉 H

locally bounded if there exists an open identity neighborhood U G such that • ⊆ supu U β(u) < . ∈ k k ∞ Lemma A.2. Let G be a topological group and ψ : G V be a map with values in a normed (real or complex) vector space (V, ). Assume→ that ψ(e) = 0 and that there exists an open identity neighborhood U Gk·k with ⊆ i)C := supu U ψ(u) < ∈ k k ∞ ii) The family of maps (h ψ(hu) ψ(h))u U is equicontinuous at e. 7→ − ∈ Then ψ is continuous at e.

57If not otherwise stated, we assume G to be Hausdorff. 160 A Automatic continuity of 1-cocycles

Proof. Choose some " > 0. We have to find an open identity neighborhood W G C " ⊆ with ψ(h) < " for all h W. To this end, pick some n N such that n < 2 . Then, we findk an openk identity neighborhood∈ V U with V n ∈U. Using the equicontinuity assumption, we find an open identity neighborhood⊆ W ⊆ V such that "⊆ ψ(hu) ψ(h) ψ(u) < k − − k 2 holds for all u U and all h W. We can write ∈ n 1 ∈ n 1 1 P k k+1 1 P k+1 k ψ(h) = n k−0[ψ(h) + ψ(h ) ψ(h )] + n k−0[ψ(h ) ψ(h )] n=1 = 1 P k − k+1 1 n − = n k−0[ψ(h) + ψ(h ) ψ(h )] + n [ψ(h ) ψ(e)] = − − |{z} =0 which leads to the estimate n 1 n 1 1 X− 1 1 X− " C " " ψ h ψ h ψ hk ψ hk+1 ψ hn < < " ( ) n ( ) + ( ) ( ) + n ( ) n 2 + n 2 + 2 = k k ≤ k=0 − k k k=0 for all h W. This proves the assertion. ∈ Proposition A.3. Let G be any topological group.

a) If the 1-cocycle β is locally bounded, then β is weakly continuous. b) If the unitary representation (π, ) is norm-continuous and β is locally bounded, then it is already continuous. H

Proof. Both assertions follow from Lemma A.2. a) For v , put ψv(g) := β(g), v . Let U be an open identity neighborhood in G ∈ H 〈 〉 for which C := supu U β(u) < . Then supu U ψv(u) C v < . The estimate ∈ k k ∞ ∈ | | ≤ k k ∞

ψv(hu) ψv(h) ψv(u) = β(hu) β(h) β(u), v | − − | = |〈π(h)β(−u) β(−u), v 〉| |〈 −1 〉| 1 = β(u), π(h− )v v C π(h− )v v |〈 − 〉| ≤ − shows the equicontinuity as required in condition ii) of Lemma A.2. Hence, we obtain that ψv is continuous at e. Since v was arbitrary, all maps h β(h), v are continuous at e, hence already continuous at every point g G by7→ virtue 〈 of〉 ∈ 1 β(gh) β(g), v = β(h), π(g− )v . 〈 − 〉 〈 〉 b) Here, we apply Lemma A.2 with ψ = β. By virtue of the norm-continuity, the estimate β(hu) β(h) β(u) = π(h)β(u) β(u) k − − k k 1 − k π(h) op β(u) ≤ k − k1 k k C π(h) op ≤ k − k for all h G, u U shows that the family (h β(hu) β(h))u U is equicontinuous at e.∈ Hence,∈β is continuous at e and thus,7→ by virtue− of the 1-cocycle∈ relation, continuous at every point in G. 161

Lemma A.4. Let G be a Baire group and ψ : G R be a map with → i) ψ 0 and ψ(e) = 0 ≥ ii) For every g G, the map h ψ(gh) ψ(h) is continuous. ∈ 7→ − Then the map ψ is continuous at e.

Remark A.5. The condition ii) of Lemma A.4 is already satisﬁed if all the maps are continuous at the identity e. The relation

ψ(g g0h) ψ(g0h) = [ψ(g g0h) ψ(h)] [ψ(g0h) ψ(h)], g, g0, h G − − − − ∈ then shows that the map h ψ(gh) ψ(h) for arbitrary g is continuous at each point g0 G, hence continuous. 7→ − ∈ The proof of Lemma A.4 is divided into several steps.

Proof. Step 1: For h G we have ∈ i) ψ(h) = sup[ψ(h) ψ(gh)]. g G − ∈ This means that ψ is the pointwise supremum of a family of continuous functions. Hence, ψ is lower-semi continuous, i.e. for any r 0 the subsets g G : ψ(g) r are closed. ≥ { ∈ ≤ } Step 2: Choose some " > 0. Then [ n " o G g G : 0 < ψ g q . = ( ) 3 q " ∈ − ≤ Q 3 ∈ ≥− " For any rational number q 3 , we put ≥ − n " o Oq := g G : ψ(g) > q and Aq := g G : ψ(g) q + . { ∈ } ∈ ≤ 3

Since ψ is lower-semi continuous, the subset Oq is open and the subset Aq is closed. Thus, [ [ G = (Oq Aq) = (Oq Aq) q " ∩ q " ∩ Q 3 Q 3 ∈ ≥− ∈ ≥− is a countable union of closed subsets. Step 3: This step consists of the following lemma. Lemma A.6. Let X be a topological (Hausdorff) space, O X an open subset and A X a closed subset. If the interior of the subset O A⊆ is nonempty, then the interior⊆ of O A is also nonempty. ∩ ∩ Proof. Let U be an open nonempty subset of O A. Clearly, U A and U O implies that U O is nonempty. Hence Ue := U ∩O is a nonempty⊆ open subset⊆ of O A. ∩ ∩ ∩ 162 A Automatic continuity of 1-cocycles

Step 4: As shown in Step 2, G is a countable union of closed subsets. That G is of second Baire category means that one of these closed subsets has nonempty " interior. Therefore, we can pick some rational point q 3 for which the interior ≥ − of Oq Aq is nonempty. We thus ﬁnd a nonempty open subset U Oq Aq. For ∩ 1 " ⊆ ∩ g U and h g− U, we then have 0 < ψ(gh) q 3 . Write ∈ ∈ − ≤ ψ(h) = [ψ(h) + ψ(g) ψ(gh)] + [ψ(gh) ψ(g)]. − − Using the continuity assumption ii), we ﬁnd an open identity neighborhood V 1 " ⊆ g− U such that ψ(h) + ψ(g) ψ(gh) 3 for all h V . We conclude for all h V that | − | ≤ ∈ ∈ ψ(h) ψ(h) + ψ(g) ψ(gh) + ψ(gh) ψ(g) ≤ |" − | | − | < 3 + ψ(gh) q + ψ(g) q " |" " − | | − | 3 + 3 + 3 = ". ≤ In view of assumption i) and the fact that " > 0 was chosen arbitrarily, we conclude that ψ is continuous at e. Proposition A.7. Let G be a Baire group. Then, for the 1-cocycle β : G , the following are equivalent: → H

i) β is totally weakly continuous. ii) β is locally bounded. iii) β is continuous.

Proof.i ) = ii) : Let E be a total subset for which β is weakly continuous and D := ⇒ E lin the corresponding linear dense, subspace. All maps g w, β(g) with 〈w 〉 D are continuous and, for r 0, we find that 7→ 〈 〉 ∈ ≥ \ g G : β(g) r = g G : w, β(g) v r { ∈ k k ≤ } w D, w 1{ ∈ 〈 − 〉 ≤ } ∈ k k≤ is an intersection of closed subsets, hence closed. Hence, G S g G : = n N β(g) n is a countable union of closed subsets. Since G is of second∈ { ∈ Baire categoryk k ≤ by} assumption, we find some N N such that g G : β(g) N has nonempty interior. Thus, we find a nonempty∈ open subset{ ∈ V k G suchk ≤ that} ⊆ β(g) N for all g V . Pick some g0 V and consider the open identity k k ≤ 1∈ ∈ neighborhood U := g0− V . In view of 1 β(g− )h = β(h) β(g) β(h) + β(g) , for g, h G k − k ≤ k k k k ∈ we thus find supu U β(u) N + β(g0) < . ∈ k k ≤ k k ∞ ii) = iii) : If β is locally bounded, then it is weakly continuous (Proposition A.3). 2 ⇒We want to apply Lemma A.4 for ψ(g) := β(g) . For any g, h G, we have k k ∈ 2 2 ψ(gh) ψ(h) = β(gh) β(h) − k 1k − k 2 k 2 = β(g− ) β(h) β(h) 1 −2 − k 1 k = β(g− ) 2 Re β(g− ), β(h) . − 〈 〉 163

This expression is continuous in h for ﬁxed g G since β is weakly continuous. Now, Lemma A.4 tells us that ψ is continuous∈ at e which amounts to saying that β is continuous at e, hence continuous. iii) = i) : This is clear. ⇒ Here are some interestring consequences:

Corollary A.8. Let G be a Baire group.

G 1. If G is abelian and if (π, ) is a continuous unitary representation with = 0 , then every algebraic 1-cocycleH of (π, ) is continuous. H { } H 2. If (π, ) = j J (πj, j) is a direct sum of continuous unitary representations of G, thenH an algebraic⊕ ∈ 1-cocycleH β : G is continuous if and only if it is a direct → H sum β = j J βj of continuous 1-cocycles βj. ⊕ ∈ 1 G Proof. 1.) Consider E := π(g− )v v : v , g G . In view of = 0 , the set E is total in . Let{ β be an− algebraic∈ H1-cocycle∈ } of the representationH {(π}, ). Then, a direct calculationH using the 1-cocycle condition shows H

β(gh) = β(hg) [π(g) 1]β(h) = [π(h) 1]β(g). ⇐⇒ − − Since G was assumed to be abelian, the map

1 1 h π(g− )v v, β(h) = v, [π(g) ]β(h) 7→ 〈 − 〉 = 〈v, [π(h) −1]β(g)〉 〈 1 − 〉 = π(h− )v v, β(g) 〈 − 〉 is a continuous map.

2.) If β is continuous, then all the summands βj are continuous since the orthogonal projections onto the spaces j are continuous. We show the converse. Consider S H the set E := j J j which is total in = j J j. If v E, then v j for some index j and∈ theH map H ⊕ ∈ H ∈ ∈ H

g v, β(g) = v, βj(g) 7→ 〈 〉 〈 〉 is continuous.

The ﬁrst statement of Corollary A.8 can be generalized in the abelian case as follows.

Corollary A.9. Let G be a Baire group with a closed, abelian, normal, subgroup A Ã G. If A (π, ) is a continuous unitary representation such that = 0 , then every algebraic 1-cocycleH of (π, ) is continuous. H { } H 1 Proof. Consider the total subset E := π(a− )v v : v , a A of . Let β : G be an algebraic 1-cocycle of{ the representation− ∈(π H, )∈. Then,} byH virtue of Corollary→ H A.8, the restriction of β to A is continuous. For any Ha A and g G, the ∈ ∈ 164 B Coarsely bounded subsets of topological groups

1 1 1-cocycle condition implies (π(a) )β(g) = π(g)β(g− ag) β(a) and for v , we obtain − − ∈ H

1 1 1 1 β(g), (π(a− ) )v = β(g− ag), π(g− )v β(a), v . 〈 − 〉 〈 〉 − 〈 〉 The right-hand side is continuous in g and we conclude that β is totally weakly continuous and hence continuous.

If G is a topological group for which every algebraic 1-cocycle is automatically continuous, then, in particular, every group homomorphism G R is continuous. If G is an abelian Baire group, then this condition is already sufﬁcient.→

Corollary A.10. Let G be an abelian Baire group for which every group homomorphism G R is continuous. Then, every algebraic 1-cocycle associated to a continuous unitary representation→ is continuous.

Proof. Let (π, ) a continuous unitary representation and β : G be an algebraic G G 1-cocycle. If H = 0, then the continuity of β follows from Corollary→ HA.8. If = , then π = 1 andH β : G is a group homomorphism. For any v ,H the groupH homomorphism g β→(g) H, v is continuous since its real and imaginary∈ H part are both continuous by assumption.7→ 〈 Therefore,〉 β is a weakly continuous 1-cocycle and hence continuous as G is a Baire group. In the general case, we have G G ∼= ( )⊥ and β decomposes accordingly into two components each beingH continuousH ⊕ byH the previous arguments. Therefore, β is continuous.

B Coarsely bounded subsets of topological groups

The notion of a topological group with Rosendal’s Property (OB) was originally introduced in the paper [Ro09] as the topological version of the Bergman Property for discrete groups (see also Remark 3.20 ii)). Rosendal shows that there are many different ways to characterize Property (OB) for Polish groups ([Ro13, Thm. 1.10]). We observe in Theorem 3.13 that some of these characterizations generalize to arbitrary topological groups. In this appendix we complete the proof of Theorem 3.13 which was postponed in Subsection 3.2. The characterizations of Property (OB) suggest to view it as a boundedness concept for topological groups and also to study “subsets with Property (OB)” which we call coarsely bounded subsets. We prove a characterization theorem (Theorem B.3) for coarsely bounded subsets which contains Theorem 3.13 as a special case. The work [Hei59] of J. Hejcman deals with the notion of boundedness in general uniform spaces. As a special case, one defines a bounded subset of a topological group G to be a subset which is bounded for the left uniformity of G. Every bounded subset in the sense of Hejcman is a coarsely bounded subset. Such bounded subsets in topological groups have been characterized in [Hei59] (see also Theorem B.9). Proposition B.17) adds the characterization of bounded topological groups as those groups with Property (OB) for which every open subgroup has finite 165 index. This has been observed first by C. Rosendal in [Ro13, Prop. 1.13] for Polish groups.

Let G be a topological Hausdorff group. Lemma B.1. Let V be an exhaustive58 increasing chain of symmetric open identity ( n)n Z neighborhoods such that∈ V 3 V holds for all n . For g h G, deﬁne n n+1 Z , ⊆ ∈ ∈ n 1 δ(g, h) := inf 2 g− h Vn { | ∈ } and ¨ k « X d(g, h) := inf δ(xi 1, xi) xi G, x0 := g, xk = h . − i=1 | ∈ Then, the following assertions hold:

1 i) 2 δ(g, h) d(g, h) δ(g, h) ≤ ≤ ii) d deﬁnes a continuous, left-invariant pseudometric on G.

If, in addition, V is a neighborhood basis of the identity, then d is a left-invariant ( n)n Z metric which is compatible∈ with the group topology 59.

Proof. A proof can be found in [Bi36].

Lemma B.2. Let (E0, 0) be a semi-normed vector space. Then, there exists a tuple k·k (γ, E) consisting of a Banach space (E, ) and a linear, isometric map γ : E0 E with dense image in E. Moreover, the followingk·k assertions hold: →

i) If (γ0, E0) is another such tuple, then there exists a unique linear isometry L : E E0 → with L γ = γ0. ◦ ii) If π0 : G Iso(E0) is a linear isometric group action on E0, then there exists a → unique linear isometric action π : G Iso(E) with π γ = γ π0. If π0 is a continuous action, so is π. → ◦ ◦

Proof. If (E0, 0) is a seminormed vector space, then we obtain a normed vector space k·k by E1 := E0/N with N := x E0 : x 0 = 0 . We write p : E0 E0/N = E1 for the { ∈ k k } → canonical surjection onto E1. The norm 1 of E0 is uniquely defined by p(x) 1 = x 0. The normed vector space E1 is containedk·k as a dense subspace of somek Banach-k k k space (E, ) and we write γ1 : E1 E for the canonical isometric linear inclusion. k·k → Hence, γ := γ1 p : E0 E has all required properties. ◦ → i) If (γ0, E0) is another tuple, then L(γ(x)) := γ0(x) defines a linear isometry L : γ(E0) γ0(E0) on the dense linear subspace γ(E0) E which uniquely ex- → ⊆ tends to a linear isometry L : E E0. The latter shows that L is uniquely defined → by L γ = γ0. ◦ 58This means that G S V = n N n 59This means that the topology∈ induced by the metric d coincides with the underlying group topology. 166 B Coarsely bounded subsets of topological groups

ii) Suppose that π0 : G Iso(E0) is a linear isometric group action on E0. For → fixed g G, the map γg := γ π0(g) : E0 E is also a linear isometry with dense image∈ in E. By i), this defines◦ a unique→ linear isometry π(g) : E E with → π(g) γ = γg = γ π0(g). The uniqueness implies in particular, that π(gh) = π(g)π◦(h) so that π◦: G Iso(E) defines a linear isometric action on E which is → uniquely characterized by π γ = γ π0. Suppose that π0 has continuous orbit maps. Then, the same holds◦ for π: Since◦ π acts by isometries on E, it is enough to verify the continuity of all orbit maps g π(g)γ(x) with x E0. But this follows from 7→ ∈

π(g)γ(x) γ(x) = γ(π0(g)x x) = π0(g)x x . k − k k − k k − k The following theorem is due to Rosendal. Theorem B.3. For a subset S G of the topological Hausdorff group G, the following are equivalent: ⊆

i) Whenever G acts continuously by isometries on a metric space (X , d), we have supg,h S d(g.x, h.x) < for all x X. ∈ ∞ ∈ ii) Whenever G acts continuously by affine isometries on a Banach space (E, ), we k·k have supg,h S g.v h.v < for all v E. ∈ k − k ∞ ∈ iii) Any continuous length function ` : G R is bounded on S. → iv) The subset S has bounded diameter with respect to every continuous left-invariant pseudometric d on G. v If W is an increasing exhaustive sequence of open subsets, then S W n for ) ( k)k N k ∈ integers k, n N. ⊆ ∈ Proof.i ) = ii): This is obvious. ⇒ ii) = iii): Suppose that ` : G R is a continuous length function on G. In partic- ⇒ →1 ular, we have `(e) = 0, `(g− ) = `(g) and `(gh) `(g) + `(h) for all g, h G. From this, the reversed triangle inequality `(g) ≤`(h) `(gh) and the equal-∈ | − | ≤ ity supg G `(gh) `(g) = `(h) follow. Also note, that we have `(g) 0 for all g G. In∈ order| to− show| that ` is bounded on S, we introduce the vector≥ space ∈ X E0 := f f : G R finitely supported, such that f (h) = 0 . { | → h G } ∈ We endow E0 with the semi-norm X f 0 := sup f (h)`(gh) . g G k k h G ∈ ∈ Note that this expression is finite, as the following estimate shows: P f 0 = supg G h G f (h)(`(gh) `(g)) k k ∈ P ∈ − supg G h G f (h) `(gh) `(g) ≤ P ∈ ∈ | || − | h G f (h) `(h) < . ≤ ∈ | | ∞ 167

By construction of the semi-norm, the left-regular representation given by 1 π0(g)f := f g− is a linear isometric action on E0. It has continuous orbit maps: For fixed◦ f E0, we have ∈ P 1 π0(g0)f f 0 = supg G h G(f (g0− h) f (h))`(gh) k − k ∈ P ∈ − = supg G h G f (h)(`(g g0h) `(gh)) P ∈ ∈ 1 − h G f (h) `(h− g0h). ≤ ∈ | | Since f is finitely supported, the last expression on the right hand side is a finite

sum und thus, continuous in g0. Hence, this estimate proves that the orbit map g0 π0(g0)f is continuous in E0. Now, let (γ, E) be the tuple from Lemma B.2 together7→ with the continuous linear isometric action π on E satisfying π γ = ¨ ◦ 1 , g0 = g γ π0. We write δg for the function deﬁned by δg (g0) = and obtain ◦ 0 , else an afﬁne isometric action on E via

A(g)v := π(g)v + γ(δg δe), − which again has continuous orbit maps: Since π has continuous orbit maps, it only remains to verify that the map g γ(δg δe) is continuous. Indeed, we have 7→ −

g g g g γ(δg δe) = δg δe 0 = sup `( 0 ) `( 0) = `( ), − − g G | − | 0∈ so that the continuity of the zero-orbit of A follows from the continuity of `.

Condition ii) (applied to v = 0) now implies that supg S γ(δg δe) < and ∈ − ∞ we conclude that supg S `(g) < . ∈ ∞ iii) = iv): Suppose that d is a continuous left-invariant pseudometric on G. We ⇒ have to show that supg,h S d(g, h) < . To this end, note that `(g) := d(g, e) deﬁnes a continous length∈ function on∞G. Therefore, we have

sup d(g, h) sup (d(g, e) + d(e, h)) = 2 sup `(g) < g,h S ≤ g,h S g S ∞ ∈ ∈ ∈ by condition iii). iv) = i): If G acts continuously by isometries on metric space (X , d), then for any ⇒ x X , the expression dx (g, h) := d(g.x, h.x) deﬁnes a continous left-invariant pseudometric∈ on G. That S has bounded diameter with respect to dx translates into supg,h S d(g.x, h.x) < . iv v : Suppose∈ that W ∞is an increasing exhaustive sequence of open sub- ) = ) ( k)k N ⇒ ∈ 1 3k sets. We assume w.l.o.g. that e W1. Put Vk := (Wk Wk− ) and observe that V 3 V and that G S V∈. For k 0, choose∩ an arbitrary sequence of k k+1 = k N k open⊆ symmetric identity neighborhoods∈ satisfying≤ V 3 V . Thus, V satis- k k+1 ( k)k Z ﬁes all requirements of Lemma B.1. By condition iv),⊆ the subset S has bounded∈ diameter with respect to the pseudometric d of Lemma B.1. The estimate δ 2d shows that there exists some n N such that, for all g S, we have ≤ ∈ k n ∈ δ(g, e) = inf 2 g Vk 2 , 3n { | ∈ } ≤ which means that S Vn Wn . This shows v). ⊆ ⊆ 168 B Coarsely bounded subsets of topological groups

v) = iii): Let ` be a continuous length function on G. Put Wk := ` < k . Clearly, the ⇒ { } n Wk form an increasing exhaustive sequence of open subsets of G, so that S Wk for some integers k, n N by condition v). In view of `(gh) `(g) + `(h⊆), we conclude ∈ ≤ sup `(g) sup `(g) n sup `(g) nk < . g S g W n g W ≤ k ≤ k ≤ ∞ ∈ ∈ ∈ Definition B.4. We call a subset S G coarsely bounded if it satisfies one of the equivalent conditions of Theorem B.3. If⊆ S = G is coarsely bounded in itself, then we say that G has Property (OB). Remark B.5. Suppose that G is a metrizable group. Then, a subset S G is coarsely bounded if and only if it satisfies the (weaker) condition ⊆ iv0) The subset S has bounded diameter with respect to every left-invariant metric d compatible with the group topology.

The implication iv0) = v) is proven along the same lines as the implication iv) = v). The difference in the⇒ metrizable case is that we can pick a neighborhood basis⇒ (!) V V 3 V ( k)k 0 of the identity satisfying k k+1 and then apply the additional statement of Lemma≤ B.1. ⊆ Proposition B.6. Suppose that G is a Baire group. Then a subset S G is coarsely bounded if and only if it satisﬁes the following condition: ⊆

vi) For any continuous linear representation π : G GL(E) on a Banach space E, we → have supg S π(g) < . ∈ k k ∞ Proof. Let G be a Baire group and S be a subset of G. v) = vi): Let π : G GL(E) be a representation on a Banach space with continuous ⇒ → orbit maps. We shall prove that supg S π(g) < . For k N, we consider the closed subspace ∈ k k ∞ ∈ \ Ak := g G : π(g) k = g G : π(g)v k . { ∈ k k ≤ } v 1{ ∈ k k ≤ } k k≤ Clearly, we have G S A and, since G is a Baire group by assumption, we = k N k ∈ have A◦m = for some m N. Choose h A◦m and consider the increasing exhaus- 6 ; ∈ ∈1 tive sequence of open subsets Wk := h− A◦mAk Ak in G. By virtue of condition n ⊇ 1 n v), we ﬁnd integers k, n N such that S Wk = (h− A◦mAk) . Using the submul- tiplicativity of the operator∈ norm, this leads⊆ to

1 1 sup π(g) n π(h− ) (sup π(g) ) (sup π(g) ) n π(h− ) m k < . g S k k ≤ · · g Am k k · g Ak k k ≤ · · · ∞ ∈ ∈ ∈ vi) = ii): Let A : G E E be an afﬁne isometric group action with continuous ⇒orbits. We obtain× a unique→ linear isometric action π : G E E and a unique continuous 1-cocycle β : G E such that × → → A(g)v = π(g)v + β(g) holds for all g G and v E. ∈ ∈ 169

In order to show that every orbit of A is bounded on S, it sufﬁces to show that the 1-cocycle β is bounded on S. To this end, we introduce on the Banach space Ee := E R the operator ⊕ πe(g)(v, λ) := (π(g)v + λβ(g), λ).

This deﬁnes a linear representation πe on Ee with continuous orbit maps. Condtion vi) implies that sup β(g) sup πe(g) < g S k k ≤ g S k k ∞ ∈ ∈ and the proof is complete.

Proposition B.7. Let G be a separable topological group and S G be any subset. Then, S is coarsely bounded if and only if the following condition is satisfied:⊆ vii) For any open symmetric identity neighborhood U, there exists a finite subset F G n and some integer n N such that S (FU) . ⊆ ∈ ⊆ Proof.v ) = vii): Let U be an open symmetric identity neighborhood. Since G is separable,⇒ there exists a countable and dense subset D G and we obtain G = DU. The countable subset D can be exhausted by an increasing⊆ sequence of finite subsets Fk D so that Wk := FkU is an exhaustive increasing sequence of open subsets in⊂ G. By condition v), we find integers k, n N such that n n ∈ S (Wk) = (FkU) . This proves vii). ⊆ vii) = iii): Let ` : G R be a continuous length function. For the open symmetric n identity⇒ neighborhood→ U := g G : `(g) < 1 , we obtain S (FU) for some finite subset F and some integer{ ∈n. By virtue of}`(gh) `(g)+⊆`(h), we thus find ≤ sup ` g sup ` g n 1 max ` g < . ( ) ( ) ( + g F ( )) g S ≤ g (FU)n ≤ ∞ ∈ ∈ ∈ Lemma B.8. Let S be a subset of a topological group G.

i) Let f : G R be a (left-) uniformly continuous function and let U be an open → 1 symmetric identity neighborhood such that f (g) f (h) < 1 whenever h− g U. | − | ∈ Then supg SU n f (g) n + supg S f (g) . ∈ | | ≤ ∈ | | ii) If every (left-) uniformly continuous function f is bounded on S, then also on F S F 0 · · for arbitrary ﬁnite subsets F, F 0 G. ⊆ Proof.i ) This is a simple induction on n: For n = 0, the statement is trivial. If n > 0 n n 1 1 and g SU we ﬁnd some h SU − such that h− g U and thus f (g) f (g) ∈f (h) + f (h) 1 + f (∈h) . We conclude that ∈ | | ≤ | − | | | ≤ | | sup f (g) 1 + sup f (h) n + sup f (g) g SU n | | ≤ h SU n 1 | | ≤ g S | | ∈ ∈ − ∈ by induction hypothesis. 170 B Coarsely bounded subsets of topological groups ii) If h G and if f : G R is (left-) uniformly continuous, then also the func- ∈ → tions f1(g) := f (hg) and f2(g) := f (gh) are (left-) uniformly continuous. Here, we have used the continuity of the group conjugation. This shows that every

uniformly continuous function is bounded on hSh0 for any h F and h0 F 0. S S ∈ ∈ Since F S F 0 = h F h F hSh0 is a ﬁnite union, the same is true for the subset · · ∈ 0∈ 0 F S F 0. · · Theorem B.9. Let S G be any subset of a topological group G. Then the following are equivalent: ⊆

i) For every open symmetric identity neighborhood U, one finds a finite subset F G n and some integer n N such that S FU . ⊆ ∈ ⊆ ii) Every (left-) uniformly continuous function f : G R is bounded on S. → iii) For every open subgroup H G, there exists a finite subset F G such that S F H 1 ⊆ ⊆ ⊆ · and (F − S) H is coarsely bounded in H. ∩ Proof.i ) = ii): Let f : G R be a left uniformly continuous function. Hence, we find some⇒ open symmetric→ identity neighborhood U such that f (g) f (h) < 1 1 n | − | whenever g− h U. By condition i), we have S FU for some finite subset ∈ ⊆ F and some positive integer n. By Lemma B.8 i), we thus find supg S f (g) ∈ | | ≤ n + maxg F f (g) < . ∈ | | ∞ ii) = iii): Let H be an open subgroup of G. Since H is open, every function f : ⇒G R which is constant on all left translates gH of H is uniformly continuous. If→ all these locally constant functions are bounded on S, then S is necessarily contained in at most finitely many left translates gH. Hence, we find some finite 1 subset F with S F H. Next, we show that (F − S) H is coarsely bounded in H. Suppose that⊆` : ·H R is a continuous length function∩ on H. We obtain a uniformly continuous extension→ via ¨ `(g) , g H f (g) := 0 , g ∈/ H. ∈ Therefore, sup 1 ` g sup 1 f g < by virtue of Lemma B.8 ii . g (F S) H ( ) = g F S ( ) ) ∈ − ∩ ∈ − | | ∞ iii) = i): Let U be an open symmetric identity neighborhood in G. Consider the open⇒ subgroup H : S U n and let F be the finite subset from condition iii . = n N ) ∈ 1 1 That S F H implies S F ((F − S) H) and, since (F − S) H is coarsely ⊆ · ⊆ · ∩ 1 n ∩ bounded in H, we find some integer n with (F − S) H U . Altogether we obtain S F U n. ∩ ⊆ ⊆ · Definition B.10. We call a subset S G bounded if it satisfies one of the equivalent conditions of Theorem B.9. If S = ⊆G is bounded in itself, then we say that G is a bounded (topological) group. Remark B.11. Recall from Remark 2.10 that any continuous negative definite function p ψ : G R 0 defines a continuous length function via ` = ψ. If ` : G R 0 is a continuous→ length≥ function, then it is automatically uniformly continuous.→ This≥ is due to 171

1 the observation that ` satisﬁes the reversed triangle inequality `(g) `(h) `(g− h) for all g, h G. Using the characterizations of coarsely bounded| resp.− bounded| ≤ subsets via the∈ boundedness of continuous length functions resp. uniformly continuous functions, one immediately obtains that every bounded subset is coarsely bounded and every coarsely bounded subset belongs to the set of those subsets S G for which every continuous real-valued negative deﬁnite functionS on G has a bounded⊆ image. Remark B.12. In the following settings, the notion of a coarsely bounded subset and that of a bounded subset coincide:

i) The topological group G is connected: If U is an open symmetric identity neighborhood, then G S U n since G does not have proper open subgroups. If S is = n N a coarsely bounded subset,∈ we apply condition v) of Theorem B.3 and ﬁnd some n integer n N such that S U . In particular, the subset S is bounded. This result can be generalized∈ to topological⊆ groups with ﬁnitely many connected components (see Lemma B.15 below). ii) The topological group G is a separable and abelian: This follows directly from Proposition B.7 together with the observation that, for abelian groups, we have n n n (FU) = F U . Remark B.13. Using the characterization of (coarsely) bounded subsets in terms of uniformly continuous functions (resp. continuous length functions), one easily establishes the following hereditary properties:

i) Subsets, closures, ﬁnite unions, ﬁnite products, left- (and right-) translates of (coarsely) bounded subsets are again (coarsely) bounded.

ii) Let Φ : G G0 be a continuous homomorphism between topological groups → G, G0. If S G is (coarsely) bounded, then Φ(S) G0 is (coarsely) bounded. ⊆ ⊆ The only subtle point is to prove that finite products of bounded subsets are again bounded. Here, one can use Lemma B.8: Let S1, S2 be bounded subsets of G and f : G R be a uniformly continuous function. Choose an open symmetric identity → n neighborhood U as in part i) of Lemma B.8. We find S2 FU for some finite subset n ⊆ F and some integer n N and thus S1S2 S1 F U . By part ii) of Lemma B.8, the subset S1 F is also bounded∈ and we find ⊆ ·

sup f (g) sup f (g) n + sup f (g) < . n g S1S2 | | ≤ g S1 FU | | ≤ g S1 F | | ∞ ∈ ∈ ∈

Let H be a subgroup of G which we endow with the relative topology. If a subset S G is (coarsely) bounded, then this does not mean that S H is a (coarsely bounded)⊆ subset in H (see Remark 3.27 for a counterexample). ∩

Lemma B.14. Let G be a topological group and H G be a subgroup such that its closure H has ﬁnite index in G. If S is a (coarsely) bounded⊆ subsetof G, then S H is a (coarsely) bounded subset of H. ∩ 172 B Coarsely bounded subsets of topological groups

Proof. The proof is divided into three steps:

Step 1: Assume that H is a dense subgroup of G, i.e. that H = G. In this case, one has to show that every uniformly continuous function (continuous length function) on H is bounded on S H (cf. Theorem B.3 iii) and Theorem B.9 ii)). This follows from the fact that every∩ uniformly continuous function f : H R → uniquely extends to a uniformly continuous function f : G R. In particular, every continuous length function ` : H R continuously extends→ to a length function on G. → Step 2: Assume that H is an open subgroup of G with finite index. Let f : H R be a uniformly continuous function on H. Since H is an open subgroup, → ¨ f (g), g H fe(g) := 0, g ∈/ H ∈ is a uniformly continuous extension of f to G and we have supg S H f (g) = ∈ ∩ | | supg S fe(g) < . However, fe is not automatically a length function if f was one.∈ We| will| argue∞ similarly as in [Ro09, Prop. 4.3]: Let ` : H R be a continuous length function. Let F be a finite subset of G such that every→ g G can be uniquely decomposed as a product g = ab with a F and b H. Since∈ H is open, the function ξ : G H, g = ab b is continuous.∈ Note that∈ the function ξ has the property that, for→ any g G,7→b H, we have ξ(g b) = ξ(g)b. We put ∈ ∈ 1 1 e`(g) := sup `(ξ(g g0)ξ(g0)− ) = sup `(ξ(gab)ξ(ab)− ) g G a F,b H 0∈ ∈ ∈ 1 = sup `(ξ(ga)ξ(a)− ) < . a F ∞ ∈ Here we have used that F is finite which is to say that H has finite index. The properties of the length function ` turn e` into a continuous length function on G: 1 The condition e`(e) = 0 is clear. The condition e`(g− ) = e`(g) follows from

1 1 1 e` g− supg G ` ξ g− g0 ξ g0 − ( ) = 0 ( ( ) ( ) ) g g g ∈ 0= 00 1 supg G ` ξ g00 ξ g g00 − = 00 ( ( ) ( ) ) h 1 h ∈ `( − )=`( ) 1 = supg G `(ξ(g g00)ξ(g00)− ) 00∈ = e`(g).

The inequality e`(g1 g2) e`(g1) + e`(g2) follows from ≤ 1 e` g1 g2 supg G ` ξ g1 g2 g0 ξ g0 − ( ) = 0 ( ( ) ( ) ) g g 1 g ∈ 0= 2− 00 1 1 supg G ` ξ g1 g00 ξ g− g00 − = 00 ( ( ) ( 2 ) ) `(h1h2) `(h1)+`(h2) ∈ ≤ 1 1 1 supg G ` ξ g1 g00 ξ g00 − ` ξ g00 ξ g− g00 − 00 [ ( ( ) ( ) ) + ( ( ) ( 2 ) )] g ≤g2 g ∈ 00= 000 1 supg G `(ξ(g1 g00)ξ(g00)− ) ≤ 00∈ 1 + supg G `(ξ(g2 g000)ξ(g000)− ) 000∈ = e`(g1) + e`(g2). 173

1 Since every map g `(ξ(ga)ξ(a)− ) is continuous and F is a ﬁnite subset, we conclude that the map7→

1 1 g e`(g) = sup `(ξ(ga)ξ(a)− ) = max `(ξ(ga)ξ(a)− ) 7→ a F a F ∈ ∈ is continuous. This shows that e` is a continuous length function. For g S H, we have ∈ ∩ 1 1 `(g) = `(ξ(g)ξ(e)− ) sup `(ξ(g g0)ξ(g0)− ) = e`(g). ≤ g G 0∈ Hence, e` is bounded on S and, in particular, on S H, so that ` is bounded on S H. ∩ ∩ Step 3: Assume that H is a subgroup of G whose closure H is a ﬁnite index subgroup. In particular, H is an open subgroup and we may apply Step 2 and derive that S H is a (coarsely) bounded subset of H. Since H is a dense subgroup of H, Step∩ 1 implies that S H is a (coarsely) bounded subset of H. ∩ Lemma B.15. A subset in a topological group with ﬁnitely many connected components is coarsely bounded if and only if it is bounded.

Proof. Let G be a topological group with at most ﬁnitely many connected components.

In this case, the identity component G0 is an open, connected (normal) subgroup of finite index, so that there exists a finite subset F G such that G = F G0. Assume that ⊆ 1 · a subset S G is coarsely bounded in G. Then, the subset F − S is coarsely bounded ⊆ 1 and, for any open subgroup H, we have that (F − S) H is coarsely bounded in H ∩ by Lemma B.14 ii). Moreover, H contains the identity component G0 which leads to S G = F H. Hence, S satisfies condition iii) of Theorem B.9 and is therefore a bounded⊆ subset· in G. The converse statement is clear.

Let S be a subgroup of a topological group G. We endow S with the corresponding relative topology. If S is a bounded group [resp. a group with Property (OB)], then S is bounded [resp. coarsely bounded] as a subset of G. The converse is false in general, but there are a few exceptions:

Proposition B.16. Let G be a topological group and H G be a subgroup whose closure H is a ﬁnite index subgroup. Then H is a bounded group⊆[has Property (OB)] if and only if G is a bounded group [has Property (OB)].

Proof. If G is a bounded group [has Property (OB)], then we apply Lemma B.14 with S = G and see that also the subgroup H is bounded [has Propery (OB)]. Conversely, suppose that H is a bounded group [has Property (OB)]. In particular, this means that S := H is a [coarsely] bounded subset of G. We have G = F H = F S for some finite subset F and Remark B.13 shows that G itself is a [coarsely· ] bounded· subset. By definition, this means that G is a bounded group [has Property (OB)]. Proposition B.17. For a topological group G, the following are equivalent: 174 C Infinitesimal weight representations and their globalization

i) G is bounded. ii) Every open subgroup in G is bounded and has ﬁnite index. iii) G has Property (OB) and every open subgroup has ﬁnite index.

Proof.i ) = ii): If G is bounded, then every open subgroup has finite index. There- fore, Proposition⇒ B.16 shows that every open subgroup is bounded. ii) = iii): This follows from the fact that every bounded group has Property (OB). ⇒ iii) = i): Again, applying Proposition B.16 we find that every open subgroup has Property⇒ (OB) and has finite index. Applying Theorem B.9 iii) with S = G, we find that G is a bounded group if and only if every open subgroup has Property (OB) and has finite index.

C Inﬁnitesimal weight representations and their globalization

This appendix is based on some ideas from Appendix E in [GlNe18] and from [Ne98]: The question is whether a given representation ρ : gl( ) End(V ) of the direct limit Lie algebra gl( ) = lim gl(n) “intgrates” to a representation∞ → π : GL( ) GL(V ) of ∞ ∞ → the corresponding direct−→ limit Lie group GL( ) = lim GL(n) and whether this repre- ∞ 2 sentation extends to one of the Banach completions−→ GLp(` ) from Subsection 5.2. The subtle point here is that we do not assume GL(V ) to carry any Lie group structure, so that the results from Section4 do not apply. If ρ is a locally finite representation (in the sense that every finite subset of V generates, for each n N, a finite-dimensional gl(n)-invariant subspace), then ρ is integrable if and only if∈ its restriction to the one- dimensional Lie subalgebra gl 1 is integrable (Proposition C.16). If, in addition, ( ) ∼= C ρ is a weight representation, then this observation shows that the integrability of ρ depends on the weight set CN: In this case, ρ is integrable if and only if every weight µ is integer-valued,P ⊆ i.e. ZN (Lemma C.17, Remark C.9 and Re- mark C.8).∈ In P case ρ is a locally finiteP unitary ⊆ weight representation and integrates 2 to a representation π of GL( ), we prove that π extends to the group GLp(` ) if and ∞ 1 1 only if supµ µ q < , where q is defined by p + q = 1 (Theorem C.24). This result is applied in∈P Subsectionk k ∞ 6.2 to the special case of a unitary highest weight representation of gl( ). ∞ Let G be a connected locally exponential Lie group with Lie algebra g and exponential map exp : g G. Suppose that ρ : g End(V ) is a representation of the Lie algebra g (which is by→ definition a Lie algebra→ over the field R) on some topological vector space V over K = R, C. We assume that the topological dual V 0 separates the points 60 in V . Operators in End(V ) or in GL(V ) are not required to be continuous unless

60 The topological dual is the vector space of all linear functionals V K that are continuous w.r.t. the given vector topology on V . → 175 explicitly stated.

Deﬁnition C.1. i) Let γ : R GL(V ) be a 1-parameter group homomorphism. An operator D End(V ) is called→ the generator of γ if ∈

d 1 γ(t)v = lim [γ(t)v v] = Dv d t t 0 t t=0 → − holds for all v V . ∈ ii) A representation π : G GL(V ) is said to be an integrated representation of ρ if, for each X g, the operator→ D := ρ(X ) is the generator of the 1-parameter homomorphism∈ γ(t) := π(exp(tX )). We say that ρ is integrable (to G) if there exists an integrated representation π : G GL(V ) of ρ. We also say that π is a representation integrated to G. →

Remark C.2. Let W V be a closed subspace. Whenever π is an integrated representation of ρ and⊆ the subspace W is π-invariant, then it is also ρ-invariant. In particular, if ρ is irreducible, then so is π. However, the converse implication is false in general: Consider the vector space V := Cc∞(R) of smooth functions endowed with the topology of pointwise convergence. For G = R, the translation representation (π(t)f )(x) := f (x + t) is an integrated representation of ρ(t) := tD, where D f := f 0 is the usual derivative. Then the closed subspace W := f V : supp(f ) [0, 1] is ρ-invariant but not π-invariant. { ∈ ⊆ }

Lemma C.3. Suppose that the representation π : G GL(V ) is an integrated representation of ρ : g End(V ). If g G is such that→π(g) : V V is a continuous endomorphism, then→ we have ∈ →

1 ρ(Adg X ) = π(g)ρ(X )π(g)− (23) for each X g, where Adg := L(cg ) : g g. ∈ →

Proof. Since Adg = L(cg ) is the derivative of the conjugation cg : G G, the formula → in Theorem 4.4 shows that we have exp Adg = cg exp. The assertion now follows from the calculation ◦ ◦

d d 1 ρ(Adg X )v = π(exp(Adg (tX )))v = π(g)π(exp(tX ))π(g)− v d t d t t=0 t=0

d 1 1 = π(g) π(exp(tX ))(π(g)− v) = π(g)ρ(X )π(g)− v d t t=0 for arbitrary v V . ∈ Next, we turn to the question when the integrated representation π is unique. A sufﬁcient criterion is obtained by the so called Uniqueness Lemma which relies on our assumption that the topological dual separates the points in V : If γ, η : R V are → two differentiable curves with identical derivatives γ0 = η0 and identical initial values 176 C Inﬁnitesimal weight representations and their globalization

γ(0) = η(0), then our separation assumption allows us to conclude that γ = η. Indeed, for f V 0, the curve ξ(t) := f (γ(t) η(t)) K is differentiable with ξ0 0 and ξ(0) =∈0. Applying the Fundamental Theorem− ∈ of Calculus in K gives ≡ Z t ξ(t) = ξ0(s)ds = 0. 0

We conclude that f γ = f η for arbitrary f V 0 which leads to γ = η by separation of points in V . ◦ ◦ ∈

Lemma C.4. [Uniqueness Lemma] Let γ : R GL(V ) be a group homomorphism with generator D End(V ) and assume that the map→ ∈ R V V, (t, v) γ(t)v × → 7→ is continuous. Let W V be a D-invariant linear subspace endowed with the subspace ⊆ topology. Then, for any group homomorphism η : R GL(W) with generator D W , we have γ = η on W. → |

Proof. First we observe, that we have γ(t) D = D γ(t) since the operator γ(t) : V V is continuous. Indeed, we calculate◦ ◦ →

d d γ(t)Dv = γ(t) γ(s)v = γ(s + t)v ds ds s=0 s=0

d = γ(s)(γ(t)v) = Dγ(t)v ds s=0 for all v V . Next, we choose w W, put ξ(t) := γ( t)η(t)w and show that ξ0(t) = 0 for all t ∈ R. Then, we may conclude∈ that ξ(t) = ξ(0−) = w is constant from which the assertion∈ follows. We calculate

ξ(t + s) ξ(t) = γ( t s)η(t + s)w γ( t s)η(t)w − − − − − − +γ( t s)η(t)w γ( t)η(t)w − − − − = γ( t s)[η(s) 1](η(t)w) + [γ( s)ξ(t) ξ(t)] − − − − − and 1 ξ0(t) = lim [ξ(t + s) ξ(t)] s 0 s → − = γ( t)D η(t)w Dξ(t) = 0. | −{z } − =Dγ( t) − This proves the assertion.

Remark C.5. If necessary, one can replace the continuity condition on γ by the weaker condition that the map R W V, (t, w) γ(t)w is continuous and that γ D = D γ holds on W. × → 7→ ◦ ◦ 177

Corollary C.6. If π : G GL(V ) is an integrated representation of ρ : g End(V ) for which the maps → → R V V, (t, v) π(exp(tX ))v × → 7→ are continuous for all X g, then it is the only integrated representation of ρ. ∈

Proof. If π0 is another integrated representation, then π(exp(X )) = π0(exp(X )), for each X g, follows from the Uniqueness Lemma choosing W = V and applied to the ∈ homomorphisms γ(t) := π(exp(tX )) and η(t) := π0(exp(tX )). Since G is connected, we have G = exp(g) Grp, so that π = π0 follows. 〈 〉 Remark C.7. 1. Suppose that V is a Banach space and that π : G GL(V ) is a norm-continuous representation of G on V . In particular, π satisfies→ the continuity condition of Corollary C.6. This shows that π is the only integrated representation (in the sense of Definition C.1) of the derived representation ρ := dπ : g (V ). In this sense, Definition C.1 generalizes the classical Lie theoretic notion→ B of an integrated representation, which is a smooth representation whose derivative coincides with the given Lie algebra representation. 2. Suppose that ρ : g (V ) is a continuous Lie algebra representation on a Banach space V . Then,→ B by Theorem 4.16, there exists a smooth representation πe : Ge GL(V ) of the universal covering group Ge on V such that dπe = ρ. This is→ the only representation integrated to Ge. If there exists a representation integrated to G, say π : G GL(V ), then we have πe = π q, where q : Ge G is the universal covering map.→ Since q is a local diffeomorphism,◦ it follows→ that π is smooth with derivative dπ = ρ. In particular, π is the unique representation integrated to G in the sense of Definition C.1.

Next, we turn to the question when the integrated representation exists. A natural Lie theoretic strategy would be to ask ﬁrst whether ρ integrates to a representation πe : Ge GL(V ) of the universal covering group Ge and ask in a second step whether this representation→ factors over the universal covering map q : Ge G to a representation → π : G GL(V ) such that πe = π q. → ◦ Remark C.8. Let π : G GL(V ) and πe : Ge GL(V ) be given representations, which → → are related by πe = π q. In view of exp = q expg and the connectedness of G and Ge, we have ◦ ◦ πe = π q πe expg = π exp . ◦ ⇐⇒ ◦ ◦ Hence, π is an integrated representation of ρ if and only if so is πe. In particular, there exists an integrated representation π : G GL(V ) of ρ if and only if there exists an → integrated representation πe : Ge GL(V ) with ker(q) ker(πe). → ⊆

Weight representations

First, we introduce the algebraic concept of a weight representation of a Lie algebra: Suppose that a Lie algebra g over K = R, C contains an abelian Lie subalgebra h g. ⊆ 178 C Inﬁnitesimal weight representations and their globalization

This means that [H, H0] = 0 for all H, H0 h. Hence, restricting a Lie algebra represen- ∈ tation ρ : g End(V ) to h yields a family of pairwise commuting operators (ρ(H))H h ∈ on the K-vector→ space V . We call the representation ρ a h-weight representation (or simply a weight representation) if all operators ρ(H) are simultaneously diagonalizable in V . The corresponding simultaneous eigenvalues are called weights. A weight µ : h K is a linear map from the K-vector space h to K and the corresponding µ-eigenspace→ is the linear subspace

µ V := v V : ρ(H)v = µ(H)v for all H h . (24) { ∈ ∈ } The set µ := µ h∗ : V = 0 (25) P { ∈ 6 { }} is called the weight set of the representation ρ. Note that V µ and are deﬁned for any representation ρ. P

2 Remark C.9. Assume that g is one of the complex Lie algebras gl( ), glp(` ) from 2 2 Subsection 5.2, where 1 p and ` := ` (N, C). A natural∞ candidate for an abelian Lie subalgebra h ≤is given≤ ∞ by the set of finite rank diagonal operators with respect to some fixed complete orthonormal system e of `2. Thus, h is topologi- ( n)n N ∈ cally generated by the elemenatary rank-1 diagonal operators Enn := en en∗. A weight ⊗ µ : h C is thus uniquely determined by its values µn := µ(Enn) so that we may identify→ the weight set with a subset of CN. We write G for the connected locally exponential Lie group correspondingP to g (see Subsection 5.2) and q : Ge G for the universal covering map. Recall that ker q and that a generator of ker→q is given ( ) ∼= Z ( ) by expg(2π i E11) = expg(2π i Enn) with n N (see Remark 5.11 ii)). This means that ∈ 1 we have ker(q) ker(πe) if and only if πe(expg(2π i E11)) = . Let ρ : g End⊆(V ) be a weight representation of g. Then, the operator ρ(H) with → µ H h satisfies ρ(H)v = µ(H)v for every v V with µ . If πe : Ge GL(V ) is an in- ∈ ∈ ∈ P → tegrated representation of ρ, one would expect that also the operator πe(expg(2π i H)) is 2π i µ(H) µ diagonal and, more precisely, that πe(expg(2π i H))v = e v holds for every v V ∈ and µ . If this is the case, then we have µn µ1 Z for all weights µ and ∈ P − ∈ ∈ P n N: In view of πe(expg(2π i E11)) = πe(expg(2π i Enn)) (Remark 5.11), the eigenvalue ∈ 2π i µ1 2π i µn equation applied to H = Enn and H = E11 yields e = e . Moreover, the condition ker(q) ker(πe) is equivalent to all entries of the weights µ being integers, i.e. ZN⊆. ∈ P P ⊆

Locally ﬁnite representations

Let g be a Lie algebra over K = R, C and let ρ : g End(V ) be a representation of g on the K-vector space V . → Definition C.10. i) A linear operator D : V V is called locally finite if every finite subset F V is contained in some finite dimensional→ D-invariant linear subspace of V . ⊂ 179

ii) The Lie algebra representation ρ : g End(V ) is called locally finite if every operator ρ(X ) is locally finite. → Remark C.11. i) Unless otherwise stated, we endow the vector space V with the finest locally convex vector topology. This is the coarsest topology on V for which all seminorms p : V R are continuous. In particular every linear map f : V K is continuous.→ With the help of Zorn’s Lemma, one deduces that → the topological dual V 0 separates the points in V and that every linear subspace W V is closed, because one can write W T ker f . We shall see in = f :f W =0 ( ) Lemma⊆ C.12 below that the closedness of linear subspaces| leads to the observation that every compact subset of V is contained in some finite-dimensional subspace. Moreover, every linear operator D : V V is continuous because every seminorm of the form p D is continuous. → ◦ ii) If D : V V is a locally finite linear operator, then the expression exp(D) := P 1 n ∞ D→is well-defined and defines a locally finite linear endomorphism on V . n=0 n! This relies on the observation that, on any finite-dimensional D-invariant linear subspace, the power series exp(D) converges and coincides with the classical matrix exponential operator.

iii) Assume that gf in g is a finite dimensional subalgebra of g with basis B1, B2,..., Bn ⊆ and that all n operators ρ(B1),..., ρ(Bn) are locally finite. Then, any finite subset F V is contained in some finite-dimensional gf in-invariant subspace. This follows⊂ from the Poincaré–Birkhoff–Witt Theorem, but one can also argue directly: Consider the linear subspace

k k V B 1 B n F k k F := ρ( 1) ... ρ( n) : 1,..., n N0 lin . ∈ An induction over j = 1, . . . , n shows that VF is finite-dimensional since every operator ρ(Bj) is locally finite. To see that VF is gf in-invariant, one considers for m N the finite-dimensional subspaces ∈ k k V m B 1 B n F k k m V F ( ) := ρ( 1) ... ρ( n) : 1 + ... + n lin F ≤ ⊆ and shows by induction on m and j that ρ(Bj)VF (m) VF (m + 1). ⊆ iv) Let ρ, V be a representation of gl S gl n `2 . Since the finite ( ) ( ) = n N ( ) ( ) dimensional Lie subalgebras gl(n) are∞ spanned∈ by the⊂ elementary B rank-1 operators Ei j := ei e∗j , the previous discussion shows that ρ is locally finite if and ⊗ only if every operator ρ(Ei j) is a locally finite endomorphism on V . If this is the case, then, for every n N and every finite subset F V , there exists a finite-dimensional gl(n)-invariant∈ linear subspace containing⊂F.

Lemma C.12. Let V be a vector space (over K = R, C) endowed with the ﬁnest locally convex vector topology. Any nonempty compact subset K V is contained in some ﬁnite- dimensional subspace of V . ⊂

Proof. We argue by contradiction: Assume that K V is a nonempty compact subset which is not contained in a finite-dimensional linear⊂ subspace of V . We define a 180 C Infinitesimal weight representations and their globalization sequence k K as follows: Choose some k K and put V : k be the one- ( n)n N 0 1 = K 0 ∈ ⊆ ∈ · dimensional linear subspace generated by k0. The elements kn for n N are chosen inductively such that kn K Vn, where Vn is the finite-dimensional linear∈ subspace generated by the elements∈ k0\,..., kn 1 K. We obtain an increasing chain of finite- dimensional subspaces V and consider− ∈ its union V : S V . This is a linear ( n)n N = n N n subspace of V and therefore automatically∈ closed (Remark∞ C.11 i∈)). Hence, the subset K := K V is compact in V . We observe that the topology on V induced by V coincides∞ ∩ with∞ the finest locally∞ convex topology on V 61. On the other∞ hand, it is known that the finest locally convex vector topology on∞V coincides with the direct limit topology on V (a proof can be found e.g. in [KK63]∞). In particular, this means ∞ that a subset M V is closed if and only if every intersection M Vn is closed in the ⊆ ∞ ∩ finite-dimensional subspace Vn. This happens for the set M1 := k1, k2,... , since we { } have M1 Vn = k1,..., kn 1 by construction. In particular M1 is a compact subset in K ∩ { −M} k k K n . Likewise, the subsets n := n, n+1,... are compact in for all > 1. Since ∞ ∞ Mn V Vn, we have { } ⊆ ∞\ \ [ [ V Mn = V Mn Vn = V , ∞\ n N n N ∞\ ⊇ n N ∞ ∈ ∈ ∈ which reveals that T M . Since the M form a decreasing chain of compact n N n = n ∈ ; subsets, the compactness of K implies that Mn = for some n N. This shows that our initial sequence k cannot∞ exist and our initial; assumption∈ was wrong. ( n)n N ∈ Below, this observation is needed to show that every integrated representation of a locally finite Lie algebra representation is locally finite in the following sense:

Definition C.13. Let G be a connected locally exponential Lie group with Lie algebra g and exponential map exp : g G. Assume that ρ : g End(V ) is a locally finite representation of the Lie algebra→g. An integrated representation→ π : G GL(V ) of ρ is said to be locally finite if →

X∞ 1 π exp X v ρ X n v ( ( )) = n! ( ) n=0 holds for all v V and X g (cf. Remark C.11 ii)). ∈ ∈ Lemma C.14. Let π : G GL(V ) be an integrated representation of a locally ﬁnite representation ρ : g End→(V ). Then π has the following properties: → i) The representation π is locally ﬁnite. ii) The representation π is uniquely determined. iii) A linear subspace W V is g-invariant if and only if it is G-invariant. ⊆ 61The induced topology on V is the coarsest topology for which the inclusion map ι : V , V is continuous. The statement follows∞ from that fact that every seminorm p on V extends∞ → to a seminorm p on V . In other words, the seminorms p are precisely those of the∞ form p∞ = p ι. ∞ ∞ ◦ 181

Proof.i ) The assertion follows from the one-parameter case: Suppose that D : V V is a locally ﬁnite endomorphism. Then, D is the generator of γ(t) := exp(tD)→(cf. Remark C.11 ii)). The group homomorphism γ : R GL(V ) satisﬁes γ D = D γ and the map → ◦ ◦

R Vf in V, (t, v) γ(t)v = exp(tD)v × → 7→ is continuous for every finite-dimensional D-invariant subspace Vf in. Suppose that D is the generator of a second group homomorphism η : R GL(V ) (i.e. 1 → that limt 0 t [η(t)v v] = Dv for all v V ). For fixed v V , the orbit map t η(t→)v is continuous,− so that K := ∈η([ 1, 1])v is a compact∈ subset of V (endowed7→ with the finest locally convex vector− topology). Applying Lemma C.12

shows that K is contained in some ﬁnite-dimensional D-invariant subspace Vf in. This means that all calculations in the proof of Lemma C.4 work for γ and η with W = V (see also Remark C.5). We conclude that η = γ. We obtain the assertion i) by applying the previous discussion to D = ρ(X ), η(t) = π(exp(tX )), γ(t) = exp(tρ(X )). ii) This follows from i), the explicit formula for π(exp(X ))v and from G = exp(g) Grp (connectedness). 〈 〉 iii) Note that every linear subspace is closed w.r.t. the ﬁnest locally convex topology (Remark C.11). That a G-invariant subspace is g-invariant has already been observed in Remark C.2. Conversely, if W is a closed g-invariant subspace, then P 1 n π exp X W ∞ ρ X W W for each X g shows that W is also G- ( ( )) = n=0 n! ( ) invariant. ⊆ ∈

Lemma C.15. Let G be a connected finite-dimensional Lie group, i.e. the corresponding Lie algebra g is finite-dimensional, and let ρ : g End(V ) be a locally finite Lie algebra representation. Then ρ is integrable if and only→ if it is locally integrable in the sense that, for every finite-dimensional g-invariant linear subspace Vf in V , the restriction of ρ to the subspace Vf in is integrable. If this happens, then the⊆ (unique) integrated representation π : G GL(V ) is locally finite and has continuous orbit maps. → Note that, if G is 1-connected, then every locally finite Lie algebra representation ρ is integrable.

Proof. Assume first that ρ is locally integrable, i.e. for every finite-dimensional g- invariant linear subspace Vf in V , there exists an integrated representation πf in : ⊆ G GL(Vf in) of ρ : g End(Vf in). Recall from Remark C.7, that πf in is uniquely de- → → termined and smooth with derivative dπf in. In particular, we have πf in(exp(X ))v = P 1 n ∞n 0 n! ρ(X ) v for all v Vf in and X g, so that πf in is a locally finite integrated = ∈ ∈ representation. Let Vf in and Vf0 in be two finite-dimensional g-invariant linear subspaces with uniquely determined integrated representations πf in : G GL(Vf in) and → π0f in : G GL(Vf0 in). We obtain a third integrated representation π00f in on the finite- → dimensional g-invariant subspace Vf in + Vf0 in. The uniqueness implies that πf in =

π00f in on Vf in and π0f in = π00f in on Vf0 in. This shows that we obtain a representation 182 C Inﬁnitesimal weight representations and their globalization

π : G GL(V ) as follows: For v V , let Vf in := ρ(g)v lin be the g-invariant subspace generated→ by v which is finite-dimensional∈ since〈 ρ 〉is locally finite. Then, put π(g)v := πf in(g)v. This defines a locally finite integrated representation because all the representations πf in are locally finite integrated representations. Conversely, suppose that π : G GL(V ) is an integrated representation of ρ. Lemma C.14 shows that π is uniquely→ determined and locally finite. Every finite- dimensional g-invariant linear subspace Vf in V is also G-invariant by Lemma C.14 ⊆ iii). The restriction of π to Vf in is an integrated representation of the restriction of ρ to Vf in. This shows that ρ is locally integrable. To see that π has continuous orbit maps, pick some v V and let Vf in be a finite-dimensional π-invariant subspace containing v. The restriction∈ of π to Vf in must be smooth (Remark C.7). In particular it has continuous orbit maps. Since the embedding Vf in , V is continuous w.r.t. the finest locally convex topology on V , we obtain the continuity→ of the orbit map

G Vf in , V, g π(g)v. → → 7→ Proposition C.16. Let ρ : gl( ) End(V ) be a locally finite representation of gl( ). Then there exists a uniquely∞ determined→ locally finite integrated representation∞ πe : GLf( ) GL(V ) of ρ. Moreover, the following are equivalent: ∞ → i) We have ker(q) ker(πe). ⊆ ii) There exists a (unique) integrated representation π : GL( ) GL(V ) of ρ. The representation π is locally finite and has continuous orbit∞ maps.→ iii) The representation ρ is integrable on each finite-dimensional Lie subalgebra gl(n). iv) The representation ρ is integrable on the one-dimensional Lie subalgebra gl(1).

gl Proof. Recall from Remark 5.7 that GLf( ) = expg( ( )) Grp is the union of the ∞ 〈 ∞ 〉 gl increasing sequence of 1-connected Lie subgroups GLf(n) := expg( (n)) Grp with Lie algebras gl(n). Therefore, the restriction of ρ to each Lie subalgebra〈 gl〉 (n) is integrable to GLf(n) and we obtain uniquely determined locally finite integrated representations πen : GLf(n) GL(V ) (Lemma C.15). In particular, for all m > n, we have π π . Thus→ we obtain a representation π : lim π : GL GL V which em GLf(n) = en e = en f( ) ( ) | ∞ → is a locally finite integrated representation of ρ and thus−→ uniquely determined. i) = ii): The representation πe factors over q through a representation ⇒ π : GL( ) GL(V ), so that both representations are related by πe expg = π exp ∞ → ◦ ◦ (Remark C.8). Hence, π is an integrated representation if and only if so is πe. ii) = iii): Suppose that π : G GL(V ) exists. Then the restrictions π GL(n) to the ⇒ n → | subgroups GL( ) are integrated representations of ρ gl(n). This means that ρ gl(n) n iii | | is integrable for every N and proves ). Since all restrictions π GL(n) are locally finite and have continuous∈ orbit maps by Lemma C.15, the same| holds for the direct limit GL( ) = lim GL(n). ∞ iii) = iv): This is trivial. −→ ⇒ 183

iv) = i): We have to show that πe(expg(X ))v = v for X := 2π i E11 gl(1) and ⇒ ∈ all v V . Choose a ﬁnite-dimensional gl(1)-invariant subspace Vf in contain- ∈ ing v. Then, there exists a locally ﬁnite integrated representation πf in : GL(1) GL V of ρ : gl 1 End V . Since exp X q exp 2π i E 1, we→ ( f in) gl(1) ( ) ( f in) ( ) = (g( 11)) = obtain | →

X∞ 1 π exp X v ρ X n v π exp X v v. e(g( )) = n! ( ) = f in( ( )) = n=0

In a last step, we return to weight representations ρ : gl( ) End(V ): Recall that ∞ → the Lie algebra gl( ) is spanned by the elementary rank-1 operators Ei j := ei e∗j ∞ ⊗ and that each Ei j with i = j spans the root space corresponding to the root "i "j. For any weight space V µ, this6 means that we have −

µ µ+"i "j ρ(Ei j)V V − . ⊆

The set ∆ := α := "i "j : i = j is the set of roots of gl( ). { − 6 } ∞ Lemma C.17. Let (ρ, V ) be a weight representation of gl( ) with weight set CN. Then the following assertions hold: ∞ P ⊆

i) If each weight string in the weight set is finite, i.e. (µ + Z α) is a finite set for each weight µ and each root α ∆, then (ρ, V ) is locally· ∩ finite.P ∈ P ∈ ii) If (ρ, V ) is locally finite, then µn µ1 Z for all n N and µ . Moreover, − ∈ ∈ ∈ P if πe : GLf( ) GL(V ) is the unique locally finite integrated representation of ρ, then we have∞ → N ker(q) ker(πe) Z . ⊆ ⇐⇒ P ⊆

µ Proof. i) Let v V be a weight vector. The finiteness of the weight string µ + Z ∈ n · ("i "j) implies that for all but finitely many n N, we have ρ(Ei j) v = 0. Since − µ ∈ V = µ V , every vector v V is a finite linear combination of weight vectors. ⊕ ∈P ∈ n Therefore, if F V is a finite subset, then we also have ρ(Ei j) F = 0 for ⊂ n E n F n { } all but finitely many N, revealing that the subspace ρ( i j) : N0 lin is finite-dimensional. This∈ shows that the property that each weight string∈ is finite translates into the statement that all operators ρ(Ei j) are locally finite. Hence, (ρ, V ) is locally finite by Remark C.11 iv).

ii) Let πe : GLf( ) GL(V ) be the unique locally ﬁnite integrated representation from Proposition∞ →C.16. For each H h and each weight vector v V µ, we have ∈ ∈

X∞ 1 π exp 2π i H v ρ 2π i H n v e2π i µ(H) v. e(g( )) = n! ( ) = n=0

Hence the assertion follows from Remark C.9. 184 C Inﬁnitesimal weight representations and their globalization

Unitary representations

The purely algebraic notion of a unitary Lie algebra representation is the following: Let g be a complex Lie algebra endowed with an involution, i.e. an antilinear involutive antiautomorphism : g g. Let V be a complex Pre-Hilbert space with scalar product 62 , . A representation∗ → ρ : g End(V ) of g on V is called unitary if the scalar product〈· ·〉 is contravariant in the sense→ that

ρ(X ∗)v, w = v, ρ(X )w for all v, w V and X g. 〈 〉 〈 〉 ∈ ∈ p Recall that := , deﬁnes a norm on V and that the completion := V of V with respectk·k to this〈· norm·〉 is a Hilbert space. This allows us to view VHas a dense linear subspace of Hilbert space . In particular, the topological dual V 0 separates the points in V . If an endomorphismHD : V V is continuous, then it extends uniquely to a continuous operator D : on→ the Hilbert space completion. Hence, if every ρ(X ) is a continuous linearH endomorphism → H on V , then ρ extends to a representation ρ : g ( ) on . → B H H Now assume that G is a connected locally exponential Lie group whose Lie algebra g is complex and carries an involution . 63 We call the subgroup ∗ U := exp(X ) : X ∗ = X Grp G 〈 − 〉 ⊆ the corresponding unitary group.

Lemma C.18. Suppose that the representation π : G GL(V ) integrates the unitary representation ρ : g End(V ). Then π is uniquely determined→ on the unitary group U → and every operator π(u) with u U is unitary so that π U extends to a unitary representation π : U U( ) of U on the∈ Hilbert space completion| = V. → H H Proof. Choose arbitrary v, w V , X g and consider the curve ∈ ∈ ξ(t) := π(exp(tX ∗))v, π(exp( tX ))w . 〈 − 〉 We shall prove that ξ0(t) = 0 for all t and conclude that ξ(t) = ξ(0) = v, w . If X ∗ = X , then this shows that π(exp(tX )) is a unitary operator. Therefore, we〈 calculate〉 − 1 ξ0(t) = lim [ξ(t + s) ξ(t)] s 0 s → − 1 1 = lim π(exp((t + s)X ∗))v, [π(exp( sX )) ]π(exp( tX ))w s 0 s → 〈 − − − 〉 1 1 + lim [π(exp(sX ∗)) ]π(exp(tX ∗))v, π(exp( tX ))w s 0 s → 〈 − − 〉 62Some authors prefer the term inner product instead of scalar product. 63 A priori, g is a Lie algebra over R. When we say that g is complex, we mean that the scalar multiplication R g g extends to a scalar multiplication C g g for which the Lie bracket [ , ] : g g g is complex-bilinear.× → × → · · × → 185

= π(exp(tX ∗))v, ρ(X )π(exp( tX ))w + ρ(X ∗)π(exp(tX ∗))v, π(exp( tX ))w = 0, −〈 − 〉 〈 − 〉 since ρ is unitary. In view of U = exp(X ) : X ∗ = X Grp, we conclude that every operator π(u) : V V is unitary and,〈 in particular, a− continuous〉 endomorphism. This shows that π U extends→ to a unitary representation on . We are left to show the | H uniqueness assertion: If X ∗ = X , then the separately continuous map − R V V, (t, v) π(exp(tX ))v × → 7→ is already (jointly) continuous because the unitary operators γ(t) := π(exp(tX )) are uniformly bounded in V (and in ) 64. Therefore, the Uniqueness Lemma (Lemma C.4 H applied with W = V ) shows that π U is uniquely determined. | Lemma C.19. Let πe : Ge GL(V ) be an integrated representation of the unitary Lie algebra representation ρ : →g End(V ). Suppose that → ker(q) U:e = expg(X ) : X ∗ = X Grp Ge. ⊆ 〈 − 〉 ⊆ Then, the following are equivalent:

i) There exists an integrated unitary representation π U : U U(V ) of ρ u, where | → | u := X g : X ∗ = X . { ∈ − } ii) We have ker(q) ker(πe). ⊆ iii) There exists an integrated representation π : G GL(V ) of ρ such that πe = π q. → ◦ In this case, the representation π U is uniquely determined and extends to a unitary rep- | resentation π U : U U( ) on the Hilbert space completion = V. | → H H Proof.i ii If π exists, then we have π π q since π is uniquely de- ) = ) U e Ue = U e termined⇒ on Ue (Lemma| C.18). Now, our| assumption| ◦ ker(q) Ue shows that ⊆ πe(ker(q)) = π U q(ker(q)) = 1 . | ◦ { } ii) = iii): The representation πe factors over q through a representation ⇒π : G GL(V ) which is an integrated representation of ρ, as already observed in Remark→ C.8. iii) = i): Just restrict π to the subgroup U. ⇒ The last assertion has already been stated in Lemma C.18.

64 This is easliy seen: For ﬁxed v0 V and t0 R and given " > 0, consider the subset ∈ ∈ " " U := t R : π(exp(tX ))v0 π(exp(t0X ))v0 < v V : v v0 < { ∈ k − k 2} × { ∈ k − k 2} which is open in R V . For any (t, v) U, we have × ∈ π(exp(tX ))v π(exp(t0X ))v0 v v0 + π(exp(tX ))v0 π(exp(t0X ))v0 < ". k − k ≤ k − k k − k This shows that every point (t0, v0) R V is a continuity point and this implies the continuity of the map. ∈ × 186 C Inﬁnitesimal weight representations and their globalization

2 In the following, we assume that g is one of the complex Lie algebras gl( ), glp(` ) ∞ with 1 p from Subsection 5.2 and write u := X g : X ∗ = X for the corresponding≤ Lie≤ subalgebra ∞ of skew-hermitian operators.{ We∈ denote by−G the} correspond- 2 2 ing Lie group GL( ), GLp(` ), by U the corresponding unitary group U( ),Up(` ) ∞ ∞ and by Ge, Ue the corresponding universal covering groups. Since g is a -invariant sub- 2 ∗ algebra of the C ∗-algebra (` ), it carries a natural structure of a complex Lie algebra with involution. Let ρ : gB End(V ) be a unitary (weight) representation of g. For µ µ → 0 µ, µ0 and nonzero weight vectors v V and v0 V , the relations ∈ P ∈ ∈ µn v, v = ρ(Enn v), v = v, ρ(Enn)v = µn v, v , 〈 〉 〈 〉 〈 〉 〈 〉 µ v, v ρ E v , v v, ρ E v µ v, v n 0 = ( nn ) 0 = ( nn) 0 = 0n 0 〈 〉 〈 〉 〈 〉 〈 〉 imply that µn = µn and v, v0 = 0 if µ = µ0. This reveals that the entries of all weights are real valued and that〈 the〉 weight spaces6 are mutually orthogonal.

µ Lemma C.20. Let ρ : g End(V ) be a unitary representation such that µ V is a dense linear subset of the→ pre-Hilbert space V and for which there exists an⊕ integrated∈P representation πe : Ge GL(V ). Then, we have → N ker(q) ker(πe) Z . ⊆ ⇐⇒ P ⊆

Proof. Pick some H∗ = H h and consider the 1-parameter homomorphism γ(t) := exp 2 i tH . For every∈ t, we have 2 i tH 2 i tH and the operator t πe(g( π )) ( π )∗ = π γ( ) is thus unitary by Lemma C.18. In particular, the separately− continuous mapping (t, v) γ(t)v is already (jointly) continuous because the unitary operators γ(t) are µ uniformly7→ bounded in V . Applying the Uniqueness Lemma with W = V and η(t) := 2π i tµ(H) 1 µ e V µ , we obtain γ(t) = η(t) on V which proves that, for every µ-eigenvector v V µ, we· have ∈ 2π i µ(H) πe(expg(2π i H))v = e v. Plugging in H = Enn and taking into account that the unitary operator πe(expg(2π i Enn)) is a continuous linear endomorphism on V , we ﬁnd that the condition ZN is P ⊆ equivalent to ker(q) ker(πe) (cf. Remark C.9). ⊆ Lemma C.21. Let ρ : gl( ) End(V ) be a unitary weight representation with weight ∞ → set and assume that there exists an integrated representation πe : GLf( ) GL(V ). Then,P the following relations hold: ∞ →

i) For X gl( ) and u U( ), we have ∈ ∞ ∈ ∞ ρ(X ∗) = ρ(X ) = ρ(AduX ) . k k k k k k ii) For every H h, we have ∈ ρ(H) = sup µ(H) . k k µ | | ∈P iii) For every n N, we have ∈ ρ(Enn) = ρ(E11) = sup µ . k k k k µ k k∞ ∈P 187

1 1 iv) For 1 p, q with p + q = 1 and X gl( ), we have ≤ ≤ ∞ ∈ ∞ ρ(X ) 2(sup µ q) X p . k k ≤ µ k k k k ∈P Proof.i ) The ﬁrst equality follows from

ρ(X ∗) = sup ρ(X ∗)v, w : v , w 1 k k = sup{|〈v, ρ(X )w 〉|: kv k, kw k ≤1 }= ρ(X ) . {|〈 〉| k k k k ≤ } k k To prove the second equality, we choose some ue Ue( ) with u = q(ue). For Ad : Ad q, we then have Ad X Ad X and equation∈ ∞ 175 23 reveals that f = f ue = u ρ Ad X ◦ π u ρ X π u 1. The assertion now follows from the fact that the (f ue ) = e(e) ( )e(e)− operator πe(ue) is unitary (Lemma C.18). ii) Since (ρ, V ) is a weight representation, the operator ρ(H) acts as a diagonal operator on V with eigenvalues µ(H) for µ . Thus, the relation in ii) is clear. ∈ P iii) Let n N. Basic linear algebra tells us that there exists some unitary u U( ) ∈ 1 ∈ ∞ (namely a permutation operator) such that Enn = uE11u− = Adu E11. Using i), we ﬁnd Enn = E11 and therefore k k k k sup µ = sup sup µn = sup sup µn = sup ρ(Enn) = ρ(E11) . µ k k∞ µ n | | n µ | | n k k k k ∈P ∈P ∈P iv) For H h, the Hölder inequality shows that ∈ ρ(H) = sup µ(H) (sup µ q) H p . k k µ | | ≤ µ k k · k k ∈P ∈P

For any hermitian operator X ∗ = X gl( ), we ﬁnd some u U( ) and H h ∈ ∞ ∈ ∞ ∈ for which X = Adu H. Hence,

ρ(X ) (sup µ q) H p = (sup µ q) X p . k k ≤ µ k k · k k µ k k · k k ∈P ∈P For arbitrary X gl( ), the desired relation now follows from ∈ ∞ 1 1 X = (X + X ∗) + i (X X ∗). 2 2 i − Lemma C.22. For a unitary weight representation ρ : gl( ) End(V ) with weight set , the following statements are equivalent: ∞ → P i) ρ(X ) : V V is continuous for every X gl( ). → ∈ ∞ ii) supµ µ < . ∈P k k∞ ∞ iii) ρ(E11) < and (ρ, V ) is locally finite. k k ∞ If this is the case, then µn µ1 Z holds for every weight µ . − ∈ ∈ P Proof.i ) = ii) This follows from Lemma C.21 iii). ⇒ 188 C Infinitesimal weight representations and their globalization ii) = iii) The uniform boundedness of the weights µ implies in particular that ⇒each weight string (µ+Z α) is finite. Therefore,∈ P the representation (ρ, V ) is locally finite (Lemma C.17· i)∩P) and the first part follows again from Lemma C.21 iii). iii) = i) If (ρ, V ) is locally finite, then there exists an integrated representation ⇒ πe : GLf( ) GL(V ) (Proposition C.16) so that Lemma C.21 is applicable. The continuity∞ of→ every operator ρ(X ) follows from the estimate in Lemma C.21 iv) with q = and p = 1 and using ρ(E11) = supµ µ . ∞ k k ∈P k k∞ The last statement is a consequence of Lemma C.17 because (ρ, V ) is locally finite.

1 1 Proposition C.23. For 1 p, q with p + q = 1, the following statements are equivalent: ≤ ≤ ∞

i) supµ µ q < . ∈P k k ∞ ii) The representation ρ : gl( ) End(V ) extends to a continuous unitary Lie al- ∞ 2→ gebra representation ρ : glp(` ) ( ), where denotes the Hilbert space completion of V . → B H H

N q If this is the case for p > 1, then Z and supµ supp(µ) ρ , where supp(µ) denotes the number of nonzero entriesP ⊆ of µ and ρ∈P :| sup | ≤ kρkX is the| operator| = X p 1 ( ) norm. In particular, every weight µ is integer-valuedk k and hask k ﬁnite≤ k support.k

Proof.i ) = ii) The condition supµ µ q < entails supµ µ < so that (ρ, V )⇒is locally ﬁnite, according∈P tok Lemmak ∞C.22. Hence,∈P Lemmak k∞ C.21∞ applies and the continuity assertion follows from the estimate in part iv) of Lemma C.21. For X gl( ), we have ρ(X ∗) = ρ(X )∗ by the unitarity of the weight representation∈(ρ, V∞), so that the unitarity of the extension follows by approximation. 2 ii) = i) The continuity of the linear map ρ : glp(` ) ( ) means that ρ := ⇒sup ρ X < . For µ N and H →h such B H that H 1, wek k ﬁnd X p 1 ( ) R p thatk k ≤ k k ∞ ∈ P ⊆ ∈ k k ≤

µ(H) sup µ0(H) = ρ(H) ρ . | | ≤ µ | | k k ≤ k k 0∈P q This shows that µ ` (N, R) with µ q ρ . Since µ was arbitrary we conclude ∈ k k ≤ k k

sup µ q ρ . µ k k ≤ k k ∈P

Assume that conditions i) and ii) are satisfied for p > 1. We have seen at the beginning of the proof that (ρ, V ) is locally finite and Lemma C.17 ii) entails that µj µn Z for every j n and every µ . In view of > µ q P µ q, it follows− that∈µ has = q = j N j 6 ∈ P ∞ k k ∈ | | finite support. In particular, there exists an n N with µn = 0, hence µj = µj µn Z 1 q ∈ − ∈ for all j = n. Moreover, we have supp(µ) µ q ρ and hence that 6 | | ≤ k k ≤ k k q µ ZN and supp(µ) ρ for every weight µ . ∈ | | ≤ k k ∈ P 189

Theorem C.24. Let ρ : gl( ) End(V ) be a locally ﬁnite unitary weight representation of gl( ) on a pre-Hilbert∞ space→ V with Hilbert space completion := V and with ∞ N 1 H1 corresponding weight set R . Let p, q [1, ] such that p + q = 1. Then, the following assertions hold:P ⊆ ∈ ∞

i) There exists an integrated representation V of if and only if π :U( ) U( ) ρ u( ) ∞ ZN. If this happens, then π is uniquely determined∞ → and extends| to a continuous Punitary ⊆ representation π :U( ) U( ). Moreover, it extends to an integrated representation π : GL( ) ∞GL(V→) of ρHwith continuous orbit maps and for which P 1∞ →n π(exp(X ))v = ∞n 0 n! ρ(X ) v holds for every X gl( ) and every v V. = ∈ ∞ ∈ ii) If supµ µ < , then the representation ρ extends to a continuous unitary ∈P k k∞ ∞ 2 2 Lie algebra representation ρ : gl1(` ) ( ) of the Banach-completion gl1(` ) 2 of gl . There exists an integrated representation→ B H π :U ` U of ρ 2 ( ) 1( ) ( ) u1(` ) if and∞ only if ZN. If this happens, then π is uniquely determined→ H and uniquely| P ⊆ 2 extends to a smooth representation π : GL1(` ) GL( ) with dπ = ρ . → H iii) If p > 1 and supµ µ q < , then the representation ρ extends to a continuous ∈P k k ∞ 2 unitary Lie algebra representation ρ : glp(` ) ( ) of the Banach-completion 2 → B H glp(` ) of gl( ). In this case, there always exists an integrated representation 2 π :U ` ∞U of ρ 2 . It is uniquely determined and uniquely extends to a p( ) ( ) up(` ) → H | 2 smooth representation π : GL1(` ) GL( ) with dπ = ρ. → H Proof.i ) Since (ρ, V ) is locally finite, we may apply Lemma C.19 and Lemma C.20 with the integrated representation πe from Proposition C.16. The local finiteness of the integrated representation π : GL( ) GL(V ), expressed in terms of the ∞ → converging power series, is inherited from πe. The continuity of its orbit maps is stated in Proposition C.16. In particular, the restriction of π to the subgroup U( ) has continous orbit maps and therefore extends to a continuous unitary representation∞ π :U( ) U( ). This proves i). ∞ → H 2 ii) + iii) In both cases, we obtain a continuous extension ρ : glp(` ) ( ) by Proposition C.23. The Integrability Theorem (Theorem 4.16) shows→ B thatH there 2 exists a unique smooth representation πe : GLf p(` ) GL( ) with dπe = ρ. Applying Lemma C.19 and Lemma C.20 yields the assertion→ H because ZN P ⊆ automatically holds in the case p > 1 and supµ µ q < . The representation 2 ∈P k k ∞ π : GLp(` ) GL( ) inherits the smoothness from πe and satisfies dπ = dπe = ρ. → H 2 In particular, its restriction to the unitary subgroup Up(` ) gives a smooth unitary representation. 190 References

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Notation

α: roots of a Lie algebra β: group 1-cocycle γ: GNS-mapping with total image in a Hilbert space λ: highest weight µ: weights, measure π: (unitary) group representation ρ: Lie algebra representation ϕ: positive definite function Φ: group homomorphism χ: group character ψ: negative definite function Ψ: Lie algebra homomorphism ω: Lie algebra 1-cocycle : real or complex Hilbert space H : A (Lie) ideal in ( ) J B H ( ) = gl( ): bounded linear operators on a Hilbert space B H H H ( ) = ( ) = gl ( ): compact operators on a Hilbert space K H B∞ H ∞ H H p( ) = glp( ): operators of pth Schatten-class on a (separable) Hilbert space B H H H g: Lie algebra; X for elements of a real Lie algebra, Z for elements of a complex Lie algebra h: abelian subalgebra in g A: affine group action G: (topological) group, always Hausdorff, elements g, h, neutral element e `: length function on a group s, f : linear functional on some vector space U, V, W: open identity neighborhoods in a topoplogical group