C*-Algebras in Kirillov Theory

The Pennsylvania State University The Graduate School Eberly College of Science

C*-ALGEBRAS IN KIRILLOV THEORY

A Dissertation in Mathematics by Pichkitti Bannangkoon

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

December 2015 The dissertation of Pichkitti Bannangkoon was reviewed and approved˚ by the following:

Nigel Higson Evan Pugh Professor of Mathematics Dissertation Adviser Chair of Committee

John Roe Professor of Mathematics

Nathanial Brown Professor of Mathematics

Martin Bojowald Professor of Physics

Svetlana Katok Professor of Mathematics Chair of Graduate Program

*Signatures are on ﬁle in the Graduate School.

ii Abstract

In this dissertation, I study connections between C˚-algebra theory and the representation theory of simply connected nilpotent Lie groups, speciﬁcally the Kirillov theory. If G is a connected, simply connected, nilpotent Lie group, then Kirillov’s famous theorem gives an explicit bijection between the set of equivalence classes of unitary irreducible representations of G and the set of coadjoint orbits of G in g˚, the dual of the Lie algebra of G. Inspired by this, and by the Plancherel theorem, I introduce two new C˚- algebras. The ﬁrst is an algebra of operators on L2pGq and the second is an algebra of operators on L2pg˚q. I formulate the conjecture that they are isomorphic, prove the conjecture in the case of Heisenberg group (which is the crucial building block for general nilpotent Lie groups) and examine the prospects for the conjecture in other cases.

iii Contents

Acknowledgements vi

1 Introduction 1

2 Preliminaries 17 2.1 Harmonic analysis ...... 17 2.2 C˚-algebras ...... 33 2.3 Kirillov theory ...... 40

3 A conjectural Kirillov isomorphism in C˚-algebra theory 53 3.1 Deﬁnition of ApGq ...... 54 3.2 Deﬁnition of Apg˚q ...... 56 3.3 The conjecture ...... 58 3.4 The case of the Heisenberg group ...... 60 3.5 Other nilpotent groups ...... 68

4 Structure of ApGq for the Heisenberg group 76 4.1 The unitary dual of a product ...... 76 4.2 The diagonal in the unitary dual of a product ...... 77 4.3 The spectrum of ApGq for the Heisenberg group ...... 80

iv 5 Structure of Apg˚q for the Heisenberg group 87 5.1 Alternative deﬁnition of Apg˚q ...... 87 5.2 Representations of G ˙ g ...... 90 5.3 Spectrum of Apg˚q for the Heisenberg group ...... 94

6 Summary and directions for future work 106

Bibliography 108

v Acknowledgments

My deepest gratitude is to my advisor Professor Nigel Higson. I am so thankful to have been a student of his. His encouragement, patience and support helped me overcome many struggles I have had with this dissertation. He has been available for many discussions, constructive criticism, and guidance. Without him, my doctoral degree would not have been possible. I hope that one day I am capable of being as good an advisor to my students as Nigel has been to me. I would like to express how grateful I am to my thesis committee members as well, for both their insightful comments and feedback. I also want to thank my mathematics teachers for the wonderful courses provided. I also want to give special thanks to Becky Halpenny and other staff members of the Department of Mathematics at the Pennsylvania State University for their assistance. I am also indebted to the Development and Promotion of Science and Technology Talents Project (DPST) for the long term financial support given and for providing me the opportunity to pursue my lifetime dream. I would like to express my heartfelt gratitude to my family members for their sacri- fices. Pattraporn Wichianrat deserves recognition as well, for her support, patience and encouragement. Thanks to all of my brothers and sisters in Christ for their support and prayers. Finally, and most importantly, I would like to dedicate this dissertation to my Lord and my Savior, Jesus Christ.

vi Chapter 1

Introduction

This thesis is about representation theory, specifically about the “Kirillov theory” (also known as the “orbit method” or the “method of coadjoint orbits”). This method provides an explicit bijection between the unitary dual space G of G, i.e., the set of equivalence classes of unitary irreducible representations of G, and the coadjoint orbits g˚{ Ad˚pGq of the action of p G on the dual vector space g˚ of its Lie algebra g. The method was originally discovered in 1962 by Alexandre Aleksan- drovich Kirillov for connected and simply connected nilpotent Lie groups in his paper [16]. Since then Kirillov theory has been extended with modifica- tions to many other classes of groups. For example, it is true after modi- fication for compact Lie groups by the Cartan-Weyl highest weight theory, and for simply connected type I solvable groups by the results of Auslander and Kostant in [1] and [2]. See [30] for more details on how Kirillov theory has been developed in other settings.

1 Nilpotent Lie Groups and Algebras

A ﬁnite-dimensional real Lie algebra g is said to be nilpotent if its descending central series eventually vanishes. That is, we inductively deﬁne

gp1q “ g, gpn`1q “ rg, gpnqs, and g is nilpotent if gpn`1q “ 0 for some n. Each gpkq is an ideal in g. For the least n such that gpn`1q “ 0 but gpnq ‰ 0 we call g an n-step nilpotent Lie algebra. In other words, g is n-step nilpotent if and only if all brackets of at least n ` 1 elements of g are 0 but not all brackets of n elements are zero. Observe that an n-step nilpotent Lie algebra has a nonempty center since gpnq is central. There is another way to think about nilpotent Lie algebras. Consider the ascending central series, which is deﬁned inductively by

gp1q “ zpgq, gpiq “ tX P g : rX, gs Ď gpi´1qu, where zpgq is the center of g. Then g is an n-step nilpotent Lie algebra if and only if g “ gpnq and g ‰ gpn´1q. A nilpotent Lie group G is one whose Lie algebra g is nilpotent. There is also an equivalent deﬁnition of nilpotent Lie group. The descending central series for the group G is deﬁned by

Gp1q “ G, Gpj`1q “ rG, Gpjqs where the bracket rH,Ks is a subgroup generated by all hkh´1k´1, h P H, k P K. G is said to be nilpotent if Gpjq “ teu for some j. It is true that Gpjq are Lie subgroups of G and the Lie algebra of Gpjq is gpjq. See §1.2 in [5] for more details.

2 The Lie algebra nn of strictly upper triangular nˆn matices is an pn´1q- step nilpotent Lie algebra of dimension npn ´ 1q{2 and its center is one- dimensional. We write the corresponding Lie group with Lie algebra nn as

Nn. Every connected, simply connected nilpotent Lie group has a faithful embedding as a closed subgroup of Nn for some n. See §1.2 in [5] for more details. If g is nilpotent, so are all subalgebras and quotient algebras of g. But it is false that if h is an ideal of g such that h and g{h are nilpotent, then g is nilpotent. Every connected nilpotent Lie group is unimodular (a left Haar measure is also right-invariant). If a nilpotent Lie group G is both connected and simply connected, then the exponential map exp : g Ñ G

8 Xj exp X “ j! j“0 ÿ is an analytic diﬀeomorphism taking Lebesgue measure on g to a left- invariant Haar measure on G. See §1.2 in [5] for more details. For any matrix Lie group G with Lie algebra g, the low order terms of the Campbell-Baker-Hausdorﬀ formula are explicitly given by

1 1 1 logpexp X ¨ exp Y q “ X ` Y ` rX,Y s ` rX, rX,Y ss ´ rY, rX,Y ss 2 12 12 1 1 ´ rY, rX, rX,Y sss ´ rX, rY, rX,Y sss 48 48 ` (commutators in ﬁve or more terms), where X,Y P g. Note that this series ends after a ﬁnite number of steps for a nilpotent Lie group. See Chapter 3 in [14] for more details. Another important example in the theory of nilpotent Lie groups and algebras is the p2n`1q-dimensional Heisenberg algebra, which is denoted by

3 hn. It is the Lie algebra with vector space basis tX1,...,Xn,Y1,...,Yn,Zu whose pairwise brackets are all zero except for

rXi,Yis “ Z, 1 ď i ď n.

n The matrix realization for Heisenberg algebra is to let zZ` i“1pxiXi`yiYiq correspond to pn ` 2q ˆ pn ` 2q matrix ř

0 x1 . . . xn z . y1 ¨ . ˛ . . . ˚ ‹ ˚ . yn‹ ˚ ‹ ˚0 0 ‹ ˚ ‹ ˝ ‚ Note that the Heisenberg algebra is a two-step nilpotent Lie algebra and n3 “ h1. Heisenberg Lie algebra got its name because it has a structure reﬂecting Heisenberg’s canonical commutation relations in quantum mechanics (see [9] for more details). In [15], Roger Howe explains the role of Heisenberg group in harmonic analysis. In 1958, Dixmier [7] listed all nilpotent Lie algebras (up to isomorphism) of dimension ď 5. We give this list below and write only the nonzero brackets rxi, xjs for a basis x1, . . . , xn where i ă j.

• Dimension 1: only the Abelian one;

• Dimension 2: only the Abelian one;

• Dimension 3: Only the Hesienberg Lie algebra h1 with the bracket

rx1, x2s “ x3;

• Dimension 4: Only k3 with the brackets rx1, x2s “ x3, rx1, x3s “ x4;

• Dimension 5: there is six algebras

– g5,1 with the brackets rx1, x2s “ x5, rx3, x4s “ x5;

4 – g5,2 with the brackets rx1, x2s “ x4, rx1, x3s “ x5;

– g5,3 with the brackets rx1, x2s “ x4, rx1, x4s “ x5, rx2, x3s “ x5;

– g5,4 with the brackets rx1, x2s “ x3, rx1, x3s “ x4, rx2, x3s “ x5;

– g5,5 with the brackets rx1, x2s “ x3, rx1, x3s “ x4, rx1, x4s “ x5;

– g5,6 with the brackets rx1, x2s “ x3, rx1, x3s “ x4, rx1, x4s “

x5, rx2, x3s “ x5;

Nielsen [23] extended the list to dimension ď 6 in 1983. There are uncount- ably many isomorphism classes for dimension ě 7; see Skjelbred [28]. The complete classiﬁcation of nilpotent Lie algebras of dimension 7 over algebraically closed ﬁelds and R can be found in Gong [13]. Seeley provided all 7-dimensional nilpotent Lie algebras over C in [27].

Representation Theory

A unitary representation of a topological group G is a continuous homomorphism π from G into the group UpHπq of unitary operators on some nonzero Hilbert space Hπ (the unitary group is equipped with the strong operator topology). So it is a map π : G Ñ UpHπq satisfying

πpxyq “ πpxqπpyq, πpx´1q “ πpxq´1 “ πpxq˚, and x ÞÑ πpxqu is continous from G to Hπ for any u P Hπ. The dimension of Hπ is called the dimension or degree of π. Basic examples of unitary representation are the left and right regular representations. Let G be a locally compact group endowed with a left Haar measure ds. For every s P G, let λpsq be the operator in L2pGq deﬁned by

pλpsqfqpxq “ fps´1xq,

5 where f P L2pGq, x P G. λ is called the left regular representation of G in L2pGq. Similarly the right regular representation ρ is deﬁned by

1 pρpsqfqpxq “ ∆psq 2 fpxsq, for every s P G where ∆ : G Ñ R is the modular function of G.

Let π1 and π2 be unitary representations of G. An intertwining operator for π1 and π2 is a bounded linear map T : Hπ1 Ñ Hπ2 such that for all x P G, we have

T π1pxq “ π2pxqT.

The unitary representations π1 and π2 are (unitarily) equivalent if there is a unitary intertwining operator. We write π1 » π2. It is true that λ » ρ. We therefore speak sometimes of the regular representation of G, without specifying left or right.

If the only closed subspaces of Hπ invariant under πpGq are 0 and Hπ, then π is said to be (topologically) irreducible. We write the set of equivalence classes of unitary irreducible representations of G by G and the equivalence class of a unitary irreducible representation π by rπs. Every p locally compact group G has enough unitary irreducible representations to separate points of G:

1.1 Theorem (The Gelfand-Raikov Theorem). If x and y are two distinct points in a locally compact group G, there is a unitary irreducible representation π of G such that πpxq ‰ πpyq.

One of the basic questions in representation theory is to describe all the unitary irreducible reprentations of G, up to equivalence. Many mathematicans have been working on the determination of G for various types of groups G. The answer depends strongly on the structure of G. For con- p

6 nected, simply connected nilpotent Lie groups, a beautiful description of the dual G was found by Kirillov, and will be described next.

Kirillovp Theory

Let G be a nilpotent Lie group with Lie algebra g, and denote the vector space dual of g by g˚. There is an action of G on g˚ called the contragradient of the adjoint action, or the coadjoint action Ad˚. It is deﬁned as follows:

ppAd˚ xqlqpY q “ lppAd x´1qY q,Y P g, l P g˚, and x P G.

A coadjoint orbit, denoted by g˚{ Ad˚pGq, is an orbit of the Lie group G in the space g˚. The stabilizer subgroup of G associated with l P g˚ is deﬁned by

˚ Rl “ tx P G | pAd xql “ lu.

˚ If G is a nilpotent Lie group and l P g , then the stabilizer Rl is connected and ˚ rl “ tX P g | pad Xql “ 0u is its Lie algebra and Rl “ exp rl. See Lemma 1.3.1 in [5] for the proof.

There is also another way to describe this Lie subalgebra rl of Lie algebra ˚ g. Each l P g , deﬁnes a natural bilinear form Bl : g ˆ g Ñ R,

BlpX,Y q “ lprX,Y sq,X,Y P g.

Then Bl is skew-symmetric, i.e., BlpX,Y q “ ´BlpY,Xq. The radical of Bl is, by deﬁnition, tY P g : BlpX,Y q “ 0 for all X P gu which coincides with ˚ rl. If g is a Lie algebra and l P g , its radical rl has even codimension in g. Hence coadjoint orbis are of even dimension. See Lemma 1.3.2 in [5] for proof.

7 Now let V be a real vector space with a skew-symmetric symplectic bilinear form B. Its isotropic subspaces W are those such that Bpw, w1q “ 0, for all w, w1 P W . It is known that maximal isotropic subspaces exist and have the same dimension:

1 1 dimpV { rad Bq ` dimprad Bq “ pdim V ` dim rad Bq, 2 2 where rad B :“ tx P V : Bpx, yq “ 0 for all y P V u. In other words, they 1 have codimension k “ 2 pdim V { rad Bq and lie halfway between the radical rad B and V . Note that rad B is contained in them. ˚ In particular, if V “ g, l P g , and B “ Bl then we call subalgebras 1 m Ď g that are isotropic for Bl and have codimension k “ 2 pdim V { rad Bq as polarizing subalgebras or maximal subordinate subalgebras for l. ˚ Given l P g , the radical rl is uniquely determined, but there can be many polarizing subalgebras m as we shall see in examples below. There is no systematic way to construct all of m and this is one of the complications of the theory. See §1.3 in [5] for more details. Let G be a connected, simply connected nilpotent Lie group. We choose a polarizing subalgebra m for l and let M “ exp m. This is a Lie subgroup of G. Then l|m is an algebra homomorphism from m to R. Deﬁne a map from M to S1 by 2πilpY q χl,M pexp Y q “ e , for Y P m. The isotropy condition insures that χl,M deﬁnes a 1-dimensional representation of the subgroup M “ exp m. Maximal isotropy insures that

χl,M induces to an irreducible unitary representation of G. Then we write G πl,M “ IndM χl,M as the induced representation from M to G. It can be shown that πl,M is independent of the choice of maximal subordinate subalgebra up to equivalence. So we may write πl for πl,M . See §2.2 and

8 §2.4 in [5] for more details.

1.2 Theorem. Let G be a connected, simply connected, nilpotent Lie group. ˚ Given l P g . The is equivalence class of πl depends only on the orbit of l under the coadjoint action and the map

˚ ˚ Ol ÞÑ rπls, g { Ad pGq Ñ G is a bijection from the set of coadjoint orbits to thep dual G of G. It is a homeomorphism with respect to the natural quotient topology on the set of p orbits and the Fell topology on G.

This theorem is proved by Kirillovp in [16], except for the fact that the map rπls ÞÑ Ol is a continuous map, which was proved by Ian Brown 11 years later in [4] in 1973. One of the main ingredients for the proof of the above theorem is Kirillov’s Lemma. In many theorems concerning Kirillov theory, proofs are given by induction on the dimension of the group G (or equivalently on the dimension of g.) If the center zpgq has dimension greater than 1, we usually factor out a proper ideal h Ď zpgq and solve the problem on a quotient algebra of lower dimension. If dim zpgq “ 1, we use Kirillov’s Lemma. This lemma shows that, for any noncommutative nilpotent Lie algebra with one-dimensional center, it contains a specially placed Heisenberg Lie subalgebra.

1.3 Lemma (Kirillov’s Lemma). Let g be a noncommutative nilpotent Lie algebra whose center zpgq is one-dimensional. Then g can be decomposed as

g “ RZ ‘ RY ‘ RX ‘ w “ RX ‘ g0 a vector space direct sum, where

1. RZ “ zpgq;

9 2. rX,Y s “ Z;

3. g0 “ RY ‘ RZ ‘ w is the centralizer of Y , and an ideal in g.

Let’s consider the Kirillov Theory in the case of Heisenberg Lie group ˚ Hn. It can be shown that the coadjoint orbits in hn are

1. The singleton sets in ZK where ZK “ tl | lpZq “ 0u;

2. The hyperplanes λZ˚ ` ZK for λ ‰ 0, where

Z˚pZq “ 1,Z˚pXq “ 0,Z˚pY q “ 0.

The complete classiﬁcation of the irreducible unitary representations of

Heisenberg Lie group Hn is the consequence of the famous Stone-von Neu- mann theorem (See Rosenberg in [26] or Prasad in [25] for more details on Stone-von Neumann theorem). The proof of the classiﬁcation can be found in Corwin and Greenleaf [5] or in Folland [9], [10].

1.4 Proposition. Every irreducible unitary representation of Hn is unitarily equivalent to one and only one of the following representations:

1. For l P ZK, the one-dimensional representation is

2πilpW q πpexppW qq “ e , for W P hn;

2 n 2. For λ P Rzt0u, the corresponding representation on L pR q is deﬁned by

rπpexppxXqqfsptq “ fpt ` λxq

rπpexppyY qqfsptq “ e2πiy¨tfptq

rπpexppzZqqfsptq “ e2πiλzfptq,

where tX,Y,Zu is a basis of Heisenberg Lie algebra hn.

10 Hilbert-Schmidt Operators

Let H be a Hilbert space and T P BpHq where BpHq is the set of all bounded linear operators on H. T is called positive if xT u, uy ě 0 for all u P H. Note that T ˚T is always a positive operator for any bounded operator T on H, so we can deﬁne ? |T | “ T ˚T.

Now let H be a separable Hilbert space. Suppose T is a positive operator on H. We say that T is trace-class if T has an orthonormal eigenbasis tenu with eigenvalues tλnu (where λn ě 0), and λn ă 8. An operator T P BpHq is called trace-classř if the positive operator |T | is trace class. If T is trace-class, we set

trpT q “ xT xn, xny, ÿ where txnu is any orthonormal basis for H. The set of trace-class operators is a two-sided ˚-ideal in BpHq (see p. 258 in [10] for proof). An operator T P BpHq is called Hilbert-Schmidt if T ˚T is trace-class. The inner product on the the space of all Hilbert-Schmidt operators can be deﬁned as ˚ xT,Sy “ trpS T q “ xT xn, Sxny, ÿ for any orthonormal basis txnu. It can be shown that T is Hilbert-Schmidt 2 if and only if }T }“ }T xn} ă 8 for any orthonormal basis txnu. The set of Hilbert-Schmidtř operators form a two-sided ˚-ideal in BpHq. It is indeed a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces H b H˚, where H˚ is the dual vector space of H. See §7.3 and appendix 2 in [10] for more details.

11 Direct Integral Decompositions

Let pA, Mq be a measurable space equipped with a σ-algebra M. A family tHσuσPA of nonzero separable Hilbert spaces indexed by A is called a ﬁeld of Hilbert spaces over A, and an element of αPA Hα - that is, a map f on

A such that fpαq P Hα for each α - is calledś a vector field on A. We write the inner product and norm on Hα by x¨, ÿα and }¨}α.A measurable field of Hilbert spaces over A is a field of Hilbert spaces tHαu together with a 8 countable set teju1 of vector fields with the following properties:

1. the functions α ÞÑ xejpαq, ekpαqyα are measurable for all j, k,

8 2. the linear span of tejpαqu1 is dense in Hα for each α.

Given a measurable ﬁeld of Hilbert spaces tHαu, teju on A, a vector ﬁeld f on A will be called measurable if xfpαq, ejpαqyα is a measurable function on A for each j.

Now let tHαu, teju be a measurable ﬁeld of Hilbert spaces over A, and suppose µ is a measure on A. The direct integral of the spaces Hα with respect to µ, denoted by ‘ Hα dµpαq, ż is the space of measurable vector ﬁelds f on A such that

2 2 }f} “ }fpαq}α dµpαq ă 8. ż ‘ Hα dµpαq is a Hilbert space with inner product ş xf, gy “ xfpαq, gpαqyα dµpαq. ż See §7.4 in [10] for more details. Let’s consider the following two examples:

12 1.5 Example. Consider the quotient space g˚{ Ad˚pGq with the quotient topology and measurable space structure. According to Kirillov (see §2.7 in [19]), under the identiﬁcation

G – g˚{ Ad˚pGq, the Plancherel measure (seep Theorem 2.5 below) gives a measure σ on g˚{ Ad˚pGq such that

fplq dl “ fplq dµplq dσpOq, ˚ ˚ ˚ żg żg { Ad pGq ˆżO ˙ for all nonnegative measurable functions f on g˚. We obtain a measurable ˚ 2 field of Hilbert spaces on g {O by setting HO “ L pOq and by construct- ing measurable sections from all measurable functions f : g˚ Ñ C with 2 g˚ |fplq| dl ă 8 by defining fO “ f|O. ş 1.6 Example. The unitary dual G has the structure of a measurable space. The spaces of Hilbert-Schmidt operators on H pπ P Gq have a structure of a p π measurable field and there is a direct integral decomposition (as in Theorem p 2.5 below).

The Abstract Plancherel Theorem

The Plancherel Theorem is, roughly speaking, the decomposition of the bi- regular representation of locally compact group G as a direct integral of irreducible representations. Now let G be a unimodular locally compact group. Recall that the right ρ and the left λ regular representations of G on L2pGq are deﬁned by

ρpxqfpyq “ fpyxq, λpxqfpyq “ fpx´1yq.

13 We can combine both representations to obtain a new representation β of G ˆ G on L2pGq. It is deﬁned by

βpx, yqfpzq “ fpx´1zyq. and we call β the bi-regular representation of G (although it is actually a representation of G ˆ G). For a second countable, unimodular, postliminal group G, there is a measurable field of irreducible representations over G such that the representation at the point p P G belongs to equivalence class p (see §7.5 in p [10]). Hence we identify the points of G with the representations in this p field. Therefore, if f P L1pGq, we define the Fourier transform of f to be p the measurable field of operators over G given by

fpπq “ fpxqpπpx´1q dx. ż p We want to think fpπq as an element of Hπ b Hπ. However, Hπ b Hπ can be identiﬁed with the space of Hilbert-Schmidt operators (see Appendix 2 p and §7.3 in [10]). It turns out that fpπq is Hilbert-Schmidt for a suitably large class of f’s and π’s. The proof of the theorem below may be found in p §18.8 in [8] or §4.3 in [5].

1.7 Theorem (The Abstract Plancherel Theorem). Suppose G is a second countable, unimodular, postliminal group. There is a measure µ, called Plancherel measure, on G, uniquely determined once the Haar measure on G is ﬁxed, with the following properties. The Fourier transform f ÞÑ f p maps L1pGq X L2pGq into ‘ H b H dµpπq, and it extends to a unitary π π p 2 ‘ map from L pGq onto Hπş b Hπ dµpπq that intertwines the bi-regular rep- ‘ resentation β with πş b π dµpπq. ş

14 The C˚-algebras ApGq and Apg˚q

The ﬁrst C*-algebra, denoted ApGq, acts on the Hilbert space L2pGq and is related to the decomposition of L2pGq into irreducible representations. Recall that the group G acts on L2pGq by both the left and right regular representations, and we shall use both in the construction of ApGq. Namely we shall deﬁne ApGq to be the image of the group C˚-algebra of GˆG under the “bi-regular” representation of G ˆ G on L2pGq. According to the abstract Plancherel theorem, the Hilbert space L2pGq decomposes as an integral of the Hilbert spaces Hπ b Hπ, as π ranges over the irreducible unitary representations of G. Here Hπ is the complex conjugate of the Hilbert space Hπ associated to the representation π. We see in this way that the C˚-algebra ApGq is very closely related to the unitary representation theory of G. The second C˚-algebra is denoted Apg˚q and it acts on the Hilbert space L2pg˚q. It is the image under the natural coadjoint/multiplication action of ˚ ˚ ˚ the crossed product C -algebra C pG, C0pg qq. Another way of describing Apg˚q is to use the Fourier transform isomorphism

L2pgq – L2pg˚q, under which Apg˚q becomes conjugate to the natural coadjoint/translation representation of the group G ˙ g on L2pgq. As we saw from the example above, the Hilbert space L2pg˚q decomposes as an integral of the Hilbert spaces L2pOq associated to the coadjoint orbits O of G. This decomposition is also a decomposition of L2pg˚q into irreducible representations of Apg˚q. We see in this way that the C˚-algebra Apg˚q is very closely related to the space of coadjoint orbits in g˚. The above remarks motivate the conjecture that ApGq and Apg˚q are

15 isomorphic.

Outline and Main Results

In Chapter 2, we breifly review the the essential definitions, theorems and examples related to harmonic analysis, C˚-algebras and Kirillov theory for nilpotent Lie group. In Chapter 3, we explicitly define the two C˚-algebras ApGq and Apg˚q.

Then we will prove the conjecture for the Heisenberg group H1 using the exponential map, which gives a unitary isomorphism from L2pGq to L2pgq. We will show that the unitary isomorphism conjugates ApGq onto Apg˚q. Unfortunately this is not true for all nilpotent groups, as we shall show at the end of this chapter. In Chapter 4, we investigate the structure of the C˚-algebra ApGq in the case of the Heisenberg group H1. We shall determine the spectrum of the C˚-algebra, and use the information we ﬁnd to decompose ApGq as a C˚-algebra extension. We have already proved the main conjecture for the Heisenberg group in previous chapter, but since the method of proof does not carry over to more general nilpotent groups, we hope that a better understanding of the structure of ApGq may eventually lead to proofs of the conjecture in more cases. In Chapter 5, we shall do the same as in Chapter 4 for the C˚-algebra Apg˚q. The conjecture that ApGq is isomorphic to Apg˚q remains open, in general. A goal of future work will be to determine decompositions of the two C˚-algebras by ideas in cases beyond the Heisenberg group, with the aim of using this structure to prove the isomorphism.

16 Chapter 2

Preliminaries

2.1 Harmonic analysis

Locally Compact Groups

A topological space X is called locally compact if every point of X has a compact neighbourhood. So a compact space is automatically locally compact. A topological group is a group G together with a topology on G under which the group operations are continuous; that is, the maps

px, yq ÞÑ xy and x ÞÑ x´1 are both continous for all x, y P G. Any group is a topological group under the discrete topology and in fact is locally compact. The theory of locally compact groups therefore embraces the theory of ordinary groups. Let H be a subgroup of the topological group G. Let q be the natural quotient map of G onto the left coset space G{H. We impose the quotient topology on G{H; that is, U Ă G{H is open if and only if q´1pUq is open in G. Then q maps open sets in G to open sets in G{H; that is qpUq is open if U is open. It is known that if G is a T1 space then G is Hausdorﬀ. If G is not a T1 space then G{teu is a Hausdorﬀ topological group.

17 It is because of this result, it is often unnecessary (for example in representation theory) to assume that a topological group is Hausdorﬀ. With this in mind, therefore the term locally compact group will from now on mean a topological group whose topology is locally compact and Haudorﬀ. n n Common examples are R , pR{Zq and all closed subgroups of the group, n GLpn, Rq, of invertible linear transformation of R . References: §2.1 in [10], §28 in [21].

The Haar Measure

Let G be a second countable, locally compact group (every connected Lie group is second countable). A left (invariant) Haar measure on G is a nonzero Borel measure µ on G with µpxEq “ µpEq for every Borel subset E of G and every x P G. A similar deﬁnition is made for right (invariant) Haar measure. If µ is a Borel measure on the locally compact group G, and we deﬁneµ ˜pEq “ µpE´1q, then it is easy to show that µ is a left Haar measure if and only ifµ ˜ is a right Haar measure. The question to ask is about the existence and uniqueness of Haar measure. There is a simple argument to prove that every Lie group has a Haar measure. It is also well known but not easy to prove that

1. Every locally compact group G posseses a left Haar measure µ.

2. Left Haar measure is unique up to multiplication by a positive con- stant; that is if λ and µ are left Haar measures on G, there exists c P p0, 8q such that µ “ cλ.

3. µpUq ą 0 for every nonempty Borel open subset U of G.

4. G is compact if and only if µpGq ă 8. Therefore Haar measure is customarily normalized so that µpGq “ 1 ( 1 dµ “ 1) for any compact ş 18 groups G.

Some examples of Haar measure for various groups are listed below.

1. A Haar measure on pR, `q which takes the value 1 on the unit inter- val r0, 1s is equal to the restriction of Lebesgue measure to the Borel subsets of R. dx 2. is a Haar measure on the multiplicative group Rˆ. |x| dx dy 3. is a Haar measure on the multiplicative group Cˆ with coor- x2 ` y2 dinates z “ x ` iy. dT 4. is a left and right Haar measure on GLpn, Rq where dT is | det T |n Lebesgue measure on the vector space of all real n ˆ n matrices.

5. dx dy dz is a Haar measure on the 3-dimensional Heisenberg group

1 x z H1 “ 1 y : x, y, z P R . $¨ ˛ , & 1 . ˝ ‚ In general, left invariant Haar% measure need not be- also right invariant. Now let G be a locally compact group with left Haar measure λ. Fix x P G , we deﬁne λxpEq “ λpExq for any Borel subset E of G. Then λx is again a left Haar measure. Therefore by the uniqueness of left Haar measure, there is a positive number ∆pxq such that λx “ ∆pxqλ. The function ∆ : G Ñ p0, 8q is called the modular function of G. It is the fact that a modular function is a continuous homomorphism from G to the multiplicative group Rzt0u; that is

∆peq “ 1, ∆pxyq “ ∆pxq∆pyq, ∆px´1q “ ∆pxq´1, for all x, y P G.

19 If we write dλpxq by dx, we have

dpyxq “ dx, dpxyq “ ∆pyq dx, dx´1 “ ∆pxq´1 dx, for all y P G.

In other words,

fpyxq dx “ fpxq dx żG żG fpxq fpx´1q dx “ dx ∆pxq żG żG ∆pyq fpxq dx “ fpxq dpxyq “ fpxy´1q dx, żG żG żG for all y P G and f P CcpGq, where CcpGq is the set of all continous functions on G with a compact support. G is said to be unimodular, if ∆ ” 1, that is, if left Haar measure is also right Haar measure. Abelian groups, discrete groups, compact groups and connected nilpotent Lie groups are all unimodular. References: §2.2 in [10], §28- §30 in [21].

Group Algebras

From now on we assume that each locally compact group G is equipped with a ﬁxed left Haar measure λ and we write dx for dλpxq when there is no ambiguity. Let L1pGq be the space of absolutely integrable functions with respect to left Haar measure. If f, g P L1pGq, their convolution product is deﬁned by

pf ˚ gqpxq “ fpyqgpy´1xq dy ż “ fpxyqgpy´1q dy ż “ fpy´1qgpyxq∆py´1q dy ż “ fpxy´1qgpyq∆py´1q dy. ż 20 Using Fubini’s theorem, one can show that this integral is absolutely con- vergent for almost every x and that

}f ˚ g}1 ď }f}1}g}1.

Note that when G is unimodular, the term ∆py´1q disappears. Convolution can be extended from L1 to other Lp and we have the following result: Suppose 1 ď p ď 8, f P L1 and g P Lp, we have f ˚ g P Lp and

}f ˚ g}p ď }f}1}g}p.

If G is unimodular, the same conclusions hold with f ˚ g replaced by g ˚ f. We shall use this when p “ 2. The involution is deﬁned by the relation

f ˚pxq “ ∆pxq´1fpx´1q.

˚ 1 1 ˚ It is the fact that f P L if and only if f P L and }f }1 “}f}1. With the convolution and the involution deﬁned above, L1pGq becomes a Banach ˚-algebra, called the L1 group algebra of G. Moreover, it is well-known that

1. G is Abelian if and only if L1pGq is Abelian.

2. If G is separable then so is L1pGq.

3. L1pGq has an approximate identity (See the deﬁnition in “C˚-algebras” section) Speciﬁcally, let U be any neighborhood of e in G, and let

ψU be a function such that supp ψU is compact and contained in U, ´1 ψU ě 0, ψU px q “ ψU pxq, and ψU “ 1. Then tψU | Uu is an approximate identify for L1pGq, i.e.,ş

}ψU ˚ f ´ f}1 Ñ 0, }f ˚ ψU ´ f}1 Ñ 0

for all f P L1pGq.

21 4. L1pGq has an identity if and only if G is discrete. If e is an identity

of G then the function δepxq which is 1 at x “ e and zero else where is an identify of L1pGq. In this case, we write l1pGq. It is well-known

that the group algebra CG consisting of all ﬁnite sum sPG αxδs form a dense subalgebra of l1pGq. ř

References: §13.2 in [8], §2.5 in [10], §31 in [21].

Representation Theory

Let G be a locally compact group. A unitary representation of G is a homomorphism π from G into the group UpHπq of unitary operators on some nonzero Hilbert space Hπ that is continous in the strong operator topology.

In other words, a unitary representation is a map π : G Ñ UpHπq satisfying

πpxyq “ πpxqπpyq, πpx´1q “ πpxq´1 “ πpxq˚, and x ÞÑ πpxqu is continous from G to Hπ for all u P Hπ. The Hilbert space

Hπ is called the representation space of π and its dimension of Hπ is called the dimension or degree of π. It is worth nothing that the strong and weak topologies coincide on UpHπq. In the above, we can therefore replace the strong topology by the weak topology throughout. If f is a function on the topological group G and y P G, then we deﬁne the left and right translates of f through y by

´1 Lypfpxqq “ fpy xq,Rypfpxqq “ fpxyq.

´1 Note that we are using y in Ly and y in Ry to make both y ÞÑ Ly and y ÞÑ Ry group homomorhisms from G to the group of linear operators on the space of functions on G; that is Lxy “ LxLy and Rxy “ RxRy.

22 Now ﬁx a Haar measure on G, it is easy to check that convolutions have the following behavior under translations:

Lzpf ˚ gq “ pLzfq ˚ g, Rzpf ˚ gq “ f ˚ pRzgq.

Basic examples of unitary representation are the left and right regular representations arising from the action of G on itself by left or right translation. Left translations give the left regular representation λ of G on L2pGq, which is deﬁned by

´1 pλpxqfqpyq “ Lxfpyq “ fpx yq,

2 where f P L pGq, x, y P G. Likewise, the right translation operators Rx de- 2 ﬁne a unitary representation ρ on L pG, µrq where µr is right Haar measure on G. They can also be built into a unitary representation ρ on L2pGq with r left Haar measure µL.

pρpxqfqpyq “ Rxfpyq “ fpyxq

1 1 pρpxqfqpyq “ ∆pxq 2 R fpyq “ ∆pxq 2 fpyxq, r x where ∆ : G Ñ R is the modular function of G. Both are called the right regular representation of G. If G is unimodular, both are the same.

If π1 and π2 are unitary representations of G, an intertwining operator from π1 to π2 is a bounded linear map T : Hπ1 Ñ Hπ2 such that for all x P G, we have

T π1pxq “ π2pxqT.

The set of all intertwining operators is denoted by Cpπ1, π2q. If Cpπ1, π2q contains a unitary operator U then π1 and π2 are (unitarily) equivalent and we write π1 » π2. We write Cpπq for Cpπ, πq. This space is called communtant or centralizer of π. It consists of all bounded operators on

23 Hπ that commute with πpxq for all x P G. It is straightforward to check that

1 1. ρ » ρ and f ÞÑ ∆ 2 f is an intertwining operator.

2. ρr » λ and Ufpxq “ fpx´1q is an intertwining operator.

´ 1 1 3. λr » ρ and f 1pxq “ ∆pxq 2 fpx´ q is an intertwining operator.

We therefore use sometimes of the regular representation of G without specifying left or right.

A closed subspace M of Hπ is called an invariant subspace for π if πpxqM Ă M for all x P G. If M ‰ t0u is invariant, the restriction of π to M, M π pxq “ πpxq|M, deﬁnes a representation of G on M. We call it a subrepresentation of π.

Let π be a unitary representation of G and u P Hπ, the closed linear span Mu of tπpxqu | x P Gu in Hπ is called the cyclic subspace generated by u. Mu is invariant under π. If Mu “ Hπ, u is called a cyclic vector for π. If π has a cyclic vector, π is called a cyclic representation. If tπiuiPI is a family of unitary representations, their direct sum ‘πi is the repesentation π on H “‘Hπ, deﬁned by

πpxqp viq “ πipxqvi, pvi P Hπi q. ÿ ÿ It is well-known that every unitary representation is a direct sum of cyclic representations. Also the following conditions are equivalent:

(i) the only closed subspaces of Hπ invariant under πpGq are 0 and Hπ;

(ii) every non-zero vector of Hπ is a cyclic vector for π.

24 If these conditions are satisﬁed, and, in addition, Hπ ‰ 0, then π is said to be topologically irreducible or simply irreducible. If G is Abelian, then every irreducible representation of G is one-dimensional. We denote the set of equivalence classes of unitary irreducible representations of G by G and the equivalence class of a unitary irreducible representation π by rπs. We p also have

2.1 Theorem (Schur’s Lemma). Suppose π1 and π2 are unitary irreducible representations of G. If π1 and π2 are equivalent then Cpπ1, π2q is one- dimensional; otherwise, Cpπ1, π2q “ t0u.

For a locally compact group G, it has enough unitary irreducible representations to separate points according to the Gelfand-Raikov theorem.

2.2 Theorem (The Gelfand-Raikov Theorem). If x and y are two distinct points in a locally compact group G, there is a unitary irreducible representation π of G such that πpxq ‰ πpyq.

The unitary irreducible representations of a locally compact group G are the building blocks of the harmonic analysis associated to G. One of the basic questions of harmonic analysis on G is to describe all the unitary irreducible representations of G, up to equivalence. Many mathematicans have been working on the determination of G for various types of groups G. The answer depends strongly on the structure of G. For connected, simply p connected nilpotent Lie groups, a beautiful description of the dual G was found by Kirillov. We shall discuss it later in this chapter. p References: §13.1 in [8], §3.1 in [10].

25 Representations of a Group and its Group Algebra

When π is a unitary representation of G, it induces a representation π of L1pGq by integration: If f P L1pGq, we deﬁne the bounded operator πpfq r on Hπ by r πpfq “ fpxqπpxq dx. ż This operator-valued integralr is intepreted as follows: For any u P Hπ, we deﬁne πpfqu by specifying its inner product with an arbitrary v P Hπ, which the vector is given by r xπpfqu, vy “ fpxqxπpxqu, vy dx. (2.1) ż Since xπpxqu, vy is a boundedr continuous function of x P G, the integral on the right is the ordinary integral of a function in L1pGq. We have

|xπpfqu, vy| ď }f}1}u}}v}, so πpfq is a bounded linear operator on Hπ with norm r }πpfq} ď |fpxq| dx “}f}1. ż It can be shown that ther map f ÞÑ πpfq is a nondegenerate ˚-representation 1 1 of L pGq on Hπ (means the closed linear span of πpL pGqqH is H). For r example, if λ is the left regular representation of G, then λpfq is simply a convolution with f on the left: r

λpfqg “ fpyqLypgq dy “ f ˚ g. ż In this case nondegeneracyr follows from the existence of approximate units. This representation is called the left regular representation of L1pGq in L2pGq. On the other hand, suppose π is a nondegenerate ˚-representation of L1pGq on the Hilbert space H. Then π arises from a unique unitary repre- r senation of G on H according to (2.1).

26 There is therefore a bijective correspondence between the collection of all unitary representations of G and the collection of all nondegenerate ˚- representations pπ, Hq of L1pGq. Indeed it can be shown further that there is a bijective correspondence between the unitary irreducible representations of G and the topologically irreducible representations of L1pGq because the topological irreducibility of the nondegenerate representation π1 of L1pGq is equivalent to that of an associated unitary representation π of G. References: Chapter VII in [6], §13.3 in [8], §3.2 in [10].

Induced Representation

Let G be a locally compact group, H a closed subgroup, q : G Ñ G{H the canonical quotient map, σ a unitary representation of H on Hσ, and

CpG, Hσq the space of continuous functions from G to Hσ. Let’s consider the following space of vector-valued functions:

H0 “ tf P CpG, Hσq | qpsupp fq is compact and

fpxξq “ σpξ´1qfpxq for x P G, ξ P Hu.

Next proposition tells us that this set is nonempty and how its elements actually look like.

2.3 Proposition. If α : G Ñ Hσ is continuous with compact support, the the function

fαpxq “ σpηqαpxηq dη żH belongs to H0 and is uniformly continuous on G. Moreover, every element of H0 if of the form fα for some α P CcpG, Hσq.

If G{H admits invariant measure µ (This is true for any unimodular group. In general, see Theorem 2.49 p. 57 in [10].), then for f, g P H0,

27 xfpxq, gpxqyσ depends only on the coset qpxq of x and it deﬁnes a function in CcpG{Hq which can be integrated with respect to µ, and so we set

xf, gy “ xfpxq, gpxqyσ dµpxHq. żG{H This is an inner product on H0 and it is preserved by left translations. Let

H be the Hilbert completion of H0. Then the left translations of G on H0

(f ÞÑ Lxf) extend to the unitary operators on H. It is indeed a unitary representation of G. We call it the representation induced by σ and G denote it by IndH σ. References: Chapter 6 in [10], Chapter 2 in [5], or Chapter 13 in [17].

Mackey Machine

Let G be a locally compact group and N a nontrivial closed Abelian normal subgroup of G. Let G act on N by conjugation. This induces an action of G on the dual group N, px, νq Ñ xν, deﬁned by

xn, xνy “p xx´1nx, νy where x P G, ν P N, n P N.

For each ν P N, we denote by Gν the stabilizer of ν,p

p Gν “ tx P G | xν “ νu, which is a closed subgroup of G, and we denote by Oν the orbit of ν:

Oν “ txν | x P Gu.

We shall say that G acts regularly on N if the following two conditions are satisﬁed. p 1. The orbit space is countably separated, that is, there is a countable

family tEju of G-invariant Borel sets in N such that each orbit in N is the intersection of all the E ’s that contain it. j p p 28 2. For each ν P N, the natural map xGν Ñ xν from G{Gν to Oν is a homeomorphism. p When G is second countable, these two conditions are actually equivalent. Let G be a topological group. The semidirect product of two closed subgroups N and H if N is normal in G and the map pn, hq Ñ nh from N ˆ H to G is a homeomorphism; in this case we write G “ N ˙ H. Every element of G can be written uniquely as nh with n P N and h P H, and the group law is ´1 pn1, h1qpn2, h2q “ pn1rh1n2h1 s, h1h2q.

If N is Abelian, for ν P N we deﬁne the little group Hν associated to ν to be p Hν “ Gν X H.

Since Gν Ą N, we then have Gν “ N ˙ Hν and Hν – Gν{N. The character

ν always extends to a homomorphismν ˜ : Gν Ñ T by the formula

ν˜ppn, hqq “ νpnq “ xn, νy.

To see this, consider

´1 ´1 ν˜ppn1, h1q, pn2, h2qq “ ν˜ppn1rh1n2h1 s, h1h2qq “ νppn1rh1n2h1 sq, and since h1 P Hν,

ν˜ppn1, h1q, pn2, h2qq “ νpn1n2q “ νpn1qνpn2q “ ν˜ppn1, h1qqν˜ppn2, h2qq.

If ν P N and ρ is an irreducible representation of Hν, we obtain an irreducible representation of G , which we denote by ν b ρ, by setting p ν

pν b ρqppn, hqq “ νpnqρphq,

29 and every irreducible representation σ of Gν such that σpnq “ xn, νyI for 1 n P N is of this form. Moreover pνbρq|Hν “ ρ, and νbρ is equivalent to νbρ if and only if ρ is equivalent to ρ1. The irreducible representation of G “ N ˙ H can be completely classiﬁed in terms of irreducible representations of N (i.e., the characters ν P N) and the irreducible representations of their little groups H by the following theorem: ν p 2.4 Theorem. Suppose G “ N˙H, where N is Abelian and G acts regularly G on N. If ν P N and ρ is an irreducible representation of Hν, then IndGν pν b ρq is an irreducible representation of G, and every irreducible representation p p of G is equivalent to one of this form. Moreover IndG pνbρq and IndG pν1b Gν Gν1 ρ1q are equivalent if and only if ν and ν1 belong to the same orbit, say ν1 “ xν, ´1 and h Ñ ρphq and h Ñ ρpx hxq are equivalent representations of Hν.

References: §6.6 in [10].

Hilbert-Schmidt Operators

trpT q “ xT xn, xny, ÿ 30 where txnu is any orthonormal basis for H. The set of trace-class operators is a two-sided ˚-ideal in BpHq An operator T P BpHq is called Hilbert-Schmidt if T ˚T is trace-class. The inner product on the the space of all Hilbert-Schmidt operators can be deﬁned as ˚ xT,Sy “ trpS T q “ xT xn, Sxny, ÿ for any orthonormal basis txnu. It can be shown that T is Hilbert-Schmidt 2 if and only if }T }“ }T xn} ă 8 for any orthonormal basis txnu. The set of Hilbert-Schmidtř operators form a two-sided ˚-ideal in BpHq. It is indeed a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces H b H˚, where H˚ is the dual vector space of H. References: §7.3 and Appendix 2 in [10].

Direct Integral Decompositions

1. the functions α ÞÑ xejpαq, ekpαqyα are measurable for all j, k,

8 2. the linear span of tejpαqu1 is dense in Hα for each α.

Given a measurable ﬁeld of Hilbert spaces tHαu, teju on A, a vector ﬁeld f on A will be called measurable if xfpαq, ejpαqyα is a measurable function

31 on A for each j.

2 2 }f} “ }fpαq}α dµpαq ă 8. ż ‘ Hα dµpαq is a Hilbert space with inner product ş xf, gy “ xfpαq, gpαqyα dµpαq. ż References: §7.4 in [10].

The Abstract Plancherel Theorem

ρpxqfpyq “ fpyxq, λpxqfpyq “ fpx´1yq.

We can combine both representations to obtain a new representation β of G ˆ G on L2pGq. It is deﬁned by

βpx, yqfpzq “ fpx´1zyq. and we call β the bi-regular representation of G (although it is actually a representation of G ˆ G).

32 For a second countable, unimodular, postliminal group G, there is a measurable field of irreducible representations over G such that the representation at the point p P G belongs to equivalence class p. Hence we p identify the points of G with the representations in this field. Therefore, if p f P L1pGq, we define the Fourier transform of f to be the measurable field p of operators over G given by

p fpπq “ fpxqπpx´1q dx. ż p We want to think fpπq as an element of Hπ b Hπ. However, Hπ b Hπ can be identiﬁed with the space of Hilbert-Schmidt operators. It turns out that p fpπq is Hilbert-Schmidt for a suitably large class of f’s and π’s.

2.5p Theorem (The Abstract Plancherel Theorem). Suppose G is a second countable, unimodular, postliminal group. There is a measure µ, called Plancherel measure, on G, uniquely determined once the Haar measure on G is ﬁxed, with the following properties. The Fourier transform f ÞÑ f p maps L1pGq X L2pGq into ‘ H b H dµpπq, and it extends to a unitary π π p 2 ‘ map from L pGq onto Hπş b Hπ dµpπq that intertwines the bi-regular rep- ‘ resentation β with πş b π dµpπq. ş References: Appendix 2, §7.3, and §7.5 in [10], §18.8 in [8], §4.3 in [5].

2.2 C˚-algebras

Group C˚-algebras and Fell Topology

Let A be a Banach algebra over C. If A admits a map x ÞÑ x˚ with the following properties:

pλx ` µyq˚ “ λx˚ ` µy˚, pxyq˚ “ y˚x˚, px˚q˚ “ x,

33 for all x, y P A, λ, µ P C, then A is called a Banach ˚-algebra. If an involution ˚ of A satisﬁes the additional condition:

}x˚x}“}x}2, for all x P A then A is called a C˚-algebra. An approximate identity ˚ or approximate unit for a C -algebra A is an increasing net puλqλPΛ of positive elements in the closed unit ball of A such that a “ limλ auλ for all a P A. Equivalently, a “ limλ uλa for all a P A. Let A be a Banach ˚-algebra with an approximate identity. For each x P A, we deﬁne

}x}1 “ supt}πpxq} | π is a ˚ -representation of Au.

For each x P A, it is known that

}x}1 “ supt}πpxq} | π is a topologically irreducible ˚ -representation of Au

˚ 1 “ suptρpx xq 2 | ρ is a continuous positive form of norm ď 1 on Au

˚ 1 “ suptρpx xq 2 | ρ is a pure state on Au.

The map x ÞÑ }x}1 is a seminorm on A satisfying

2 }x}1 ď }x}, }xy}1 ď }x}1}y}1, }x˚}1 “}x}1, }x˚x}1 “}x}1 , for any x, y P A. Let N be the set of x P A such that }x}1 “ 0. This is a closed, self-adjoint, two-sided ideal of A , and } ¨ }1 becomes a C˚-norm on the quotient A{N. Endowed with this norm, A{N satisﬁes all the C˚- algebra axiom except that A{N is not complete in general. The completion of B “ pA{N, } ¨ }1q is called the enveloping C˚-algebra of A. The canonical map of A into B is a norm-reducing ˚-algebra morphism whose image is dense in B. When A is a C˚-algebra, we have }x}1 “}x}

34 and A may be identiﬁed with its own enveloping C˚-algebra. Also we have the following:

2.6 Proposition. Let A be a Banach ˚-algebra with an approximate identity, B the enveloping C*-algebra of A and τ the canonical map of A into B.

1. If π is a ˚-representation of A, there is a unique ˚-representation ρ of B such that π “ ρ ˝ τ, and ρpBq is the C*-algebra generated by πpAq.

2. The map π ÞÑ ρ is a bijection of the set of ˚-representations of A onto the set of ˚-representations of B.

3. ρ is nondegenerate if and only if π is nondegenerate.

4. ρ is topologically irreducible ˚-representation if and only if π is a topologically irreducible ˚-representation.

5. If f is a continuous positive form on A, there is a unique positive form on g on B such that f “ g ˝ τ. Moreover }g}“}f}.

6. The map f ÞÑ g is a bijection of the set of continuous positive forms on A onto the set of positive forms on B.

Now let G be a locally compact group. Note that the L1pGq norm is not a C˚-algebra norm and so L1pGq is not a C˚-algebra. Since L1pGq is a Banach ˚-algebra with an approximate identity, we can form its enveloping C˚-algebra. We call this the (full) C˚-algebra of the group G and it is denoted C˚pGq. If G is discrete, then C˚pGq admits an identity element. If G is separable, C˚pGq is also separable. For f P L1pGq,

}f}1 “ sup }πpfq}

35 where π run through the set of nondegenerate ˚-representations of L1pGq, or the set of unitary representations of G. Then f ÞÑ }f}1 is a seminorm on L1pGq, and indeed, a norm since L1pGq admints an injective representation (the left regular representation of L1pGq in L2pGq is injective). The C˚- algebra of G is simply the completion of L1pGq for this supremum norm. By the proposition above, any ˚-representation of L1pGq extends uniquely to ˚- representation of C˚pGq. Therefore we establish a bijective correspondence between unitary representation of G and nondegenerate ˚-representations of C˚pGq. In other words, this is a bijection between C˚pGq and G where C˚pGq is the set of all nondegenerate ˚-representations of Cp ˚pGq. If π is p such ap representation, its kernel

kerpπq “ tf P C˚pGq | πpfq “ 0u is a closed two-sided ideal of C˚pGq. If π is irreducible, then kerpπq is called a primitive ideal of C˚pGq. The space of all primitive ideals of C˚pGq is denoted by PrimpGq; that is

PrimpGq “ tkerpπq | π P Gu.

For any nonempty subset S of PrimpGq, we deﬁnep S Ă PrimpGq by

S “ tI P PrimpGq | I Ą J u. J PS č Then for any S, T Ă PrimpGq, it can be shown that

H“H,S Ă S, S “ S, S Y T “ S Y T.

It follows from a theorem of Kuratowski (See p.119 in [10]) that there is a unique topology on PrimpGq such that S is the closure of S for all S Ă PrimpGq. This topology is called hull-kernel topology or Jacobson topology. It can be shown that

36 1. This topology is T0.

2. S is a closed subset of PrimpGq if and only if S is exactly the set of primitive ideals containing some (ﬁxed) subset of C˚pGq.

3. Let I P PrimpGq. Then tIu is closed in PrimpGq if and only if I is maximal among primitive ideals.

Let G be a locally compact group and π an irreducible representation of G. Then the kernel kerpπq P PrimpGq depends only on the equivalence class of π, and the map rπs ÞÑ kerpπq is a surjection from G onto PrimpGq. We can therefore pull back the hull-kernel topology on PrimpGq to G; that is, open p sets on G is of the form trπs | kerpπq P Uu where U is open in PrimpGq. p This topology is called the Fell topology on G. p References: §2.7, §3.1, §13.9 in [8], §7.1 in [10], §5.4 in [22]. p Liminal, Postliminal

A unitary representation π of G is primary if the center of Cpπq is trivial, i.e., consists of scalar multiples of I. By Schur’s lemma, every irreducible representation is primary. The group G is said to be type I if every primary representation of G is a direct sum of copies of some irreducible representation. Connected nilpotent Lie groups are type I. Let KpHq be the set of all compact operators from H to itself. A C˚- algebra A is said to be liminal or CCR if πpAq Ă KpHπq for every irreducible representation π; that is, πpfq is compact operator for every irreducible ˚-representation of A. Equivalently, πpAq “ KpHπq. CCR is an acronym for “completely continuous representations” (“completely continuous operator” was an old terminology for compact operator) and liminal is a French synonym for CCR invented by Dixmier. A C˚-algebra A is said

37 to be postliminal if KpHq Ă πpAq. The postliminal C˚-algebras are also called GCR or Type I C˚-algebras. Every liminal C˚-algebra is postliminal. If A is a liminal C˚-algebra, then its C˚-subalgebra and its quoteint C˚-algebras are liminal also. The converse is false in general. On the other hand, A is a postliminal C˚-algebra if and only if I and A{I are postliminal where I is a closed ideal in A. The group G is called liminal (resp. postliminal) if C˚pGq is liminal (resp. postliminal). In this case, G is liminal if and only if πpfq is compact whenever π is irreducible and f P L1pGq, since L1pGq is dense in C˚pGq. Abelian groups, compact groups and connected nilpotent Lie group are liminal. If G is a second countable locally compact group, the following are equivalent:

(i) G is type I.

(ii) The Fell topology on G is T0.

(iii) The map rπs ÞÑ kerpπqpfrom G to PrimpGq is bijective.

(iv) G is postliminal. p

It is also known that a locally compact group G is liminal if and only if the

Fell topology on G is T1. Compare item 3 from previous section. References: §7.2 in [10], [12], §5.6 in [22]. p Weak Containment, Amenability and Reduced Group C˚-algebras

If ϕ : A Ñ B is a linear map between C˚-algebras A and B, then ϕ is said to be positive if ϕpA`q Ă B`, where A` is the set of positive elements of A. A state on a C˚-algebra A is a positive linear functional on A of norm one. Let pπ, Hq be a representation of A. A state (or positive linear functional)

38 ρ on A is said to be associated with π, if there exists ξ P H such that

ρpaq “ xπpaqξ, ξy for all a P A. Now let π1, π2 be two representations of A. We say that π1 is weakly contained in π2 (or π2 weakly contains π1), if ker π2 Ă ker π1. The support of π is the set of σ P G which are weakly contained in π. It can be shown that the following statements are equivalent: p

(i) π1 is weakly contained in π2;

˚ (ii) Each positive functional on A associated with π1 is a weak -limit of

sums of positive functionals associated with π2;

˚ (iii) Each state on A associated with π1 is a weak -limit of states which

are sums of positive functionals associated with π2.

In terms of group, π1 is weakly contained in π2 if every positive definite matrix coefficient xπ1pxqξ, ξy can be approximated uniformly on compact subsets of G by finite sums of positive definite matrix coefficients xπ2pxqfi, fiy 2 where fi P L pGq. Let G be a locally compact group. A group G is called amenable if there exist a left translation invariant mean m for G, that is, a state on L8pGq (view L8pGq as a C˚-algebra) such that

mpLsfq “ mpfq,

8 8 for all s P G, f P L pGq where Ls is left translation action of G on L pGq. Discrete groups and Abelian groups are amenable. So are compact groups. Indeed, if µ is an invariant Haar measure on G with µpGq “ 1, then

mpfq “ fpsq dµpsq, @f P L8pGq, ż 39 is an invariant mean on L8pGq. Let λ be the left regular representation of L1pGq on L2pGq. Then 1 ˚ 1 }f}r “}λpfq} (where f P L pGq) is a C -norm on L pGq. The comple- 1 ˚ tion of pL pGq, } ¨ }rq is called the reduced C -algebra of the group G, ˚ and it is denoted by Cr pGq. In other words,

˚ 1 Cr pGq “ λpL pGqq.

˚ ˚ If G is a discrete amenable group, then C pGq and Cr pGq are isomorphic. For any locally compact group G, the following statements are also equivalent:

(i) G is amenable;

(ii) Any ˚-representation of C˚pGq is weakly contained in its left regular representation, where the left regular representation of C˚pGq is the unique extension of the left regular representation of L1pGq;

(iii) The left regular representation of C˚pGq is faithful;

˚ ˚ (iv) There is an isomorphism between C pGq and Cr pGq.

Reference: Chapter VII in [6], Chapter 16 in [20].

2.3 Kirillov theory

Nilpotent Lie Groups and Nilpotent Lie Algebras

A ﬁnite-dimensional real Lie algebra g is a ﬁnite dimensional vector space over R on which there is a bilinear form named the bracket and denoted by r¨, ¨s with the following properties

rx, ys “ ´ry, xs, rx, ry, zss ` ry, rz, xss ` rz, rx, yss “ 0,

40 for all x, y, z P g. The latter equality is the Jacobi identity. The algebra of all n ˆ n square matrices is an example of Lie algebra with the bracket rA, Bs “ AB ´ BA. A subspace h is said to be an ideal if rx, hs P h for all x P g, and all h P h. A linear map φ : g Ñ h is a Lie algebra homomorphism if φprx, ysq “ rφpxq, φpyqs for all x, y P g. Let g be a Lie algebra. The descending central series of g is deﬁned inductively by

p1q pn`1q pnq pnq g “ g, g “ rg, g s “ R- spantrX,Y s : X P g,Y P g u

It follows that g “ gp1q Ą gp2q Ą ..., as the name suggests, and φpgpnqq Ă hpnq for all n if φ : g Ñ h is a Lie algebra homomorphism. Also rgppq, gpqqs Ă gpp`qq, for all intergers p and q. In particular, gpkq is an ideal in g for all k. We say that g is a nilpotent Lie algebra if its descending central series eventually vanishes, i.e., there is an integer n such that gpn`1q “ 0. If n is the minimum positive integer such that gpn`1q “ 0 but gpnq ‰ 0. Then g is said to be n-step nilpotent. Therefore g is n-step nilpotent if and only if all brackets of at least n ` 1 elements of g are 0 but not all brackets of n elements are. If g is nilpotent, so are all subalgebras and quotient algebras of g. However it is not true that if h is an ideal of g such that h and g{h are nilpotent, then g is necessarily nilpotent. The counter example is g “ R- spantX,Y u with rX,Y s “ X, and h “ R- spantXu. If g is n-step nilpotent, gpnq is central. Therefore g always has a nonempty center. A connected simply-connected Lie group G is said to be nilpotent if its Lie algebra g is nilpotent. A nilpotent Lie group G is one whose Lie algebra g is nilpotent. There is an equivalent deﬁnition of nilpotent Lie group. The descending central series for the group G is deﬁned by

Gp1q “ G, Gpj`1q “ rG, Gpjqs

41 where the bracket rH,Ks is a subgroup generated by all hkh´1k´1, h P H, k P K. G is said to be nilpotent if Gpjq “ teu for some j. It can be shown that Gpjq are Lie subgroups of G and the Lie algebra of Gpjq is indeed gpjq. So the Lie group and Lie algebra deﬁnitions coincide. There is also another way to think about nilpotent Lie algebra using ascending central series. The ascending central series of Lie algebra g is deﬁned inductively by

gp1q “ zpgq, gpiq “ tX P g : rX, gs Ď gpi´1qu, where zpgq is the center of g. Each gpiq is an ideal and zpgq “ gp1q Ă gp2q Ă ..., as the name suggests. The Lie algebra g is n-step nilpotent if and only if g “ gpnq ‰ gpn´1q. Now let’s explore a few examples of nilpotent Lie algebra.

n 1. R with the trivial bracket (rX,Y s “ 0 for all X,Y ) is an Abelian nilpotent Lie algebra. This is (up to isomorphism) the only one-step nilpotent Lie algebra.

2. The p2n ` 1q-dimensional Heisenberg algebra, denoted by hn, is

the Lie algebra with basis tX1,...,Xn,Y1,...,Yn,Zu whose pairwise brackets are equal to zero except for

rXi,Yis “ Z, 1 ď i ď n.

There is a matrix realization for Heisenberg algebra in which zZ ` n i“1pxiXi ` yiYiq corresponds to the pn ` 2q ˆ pn ` 2q matrix

ř 0 x1 . . . xn z . y1 ¨ . ˛ . . . ˚ ‹ ˚ . yn‹ ˚ ‹ ˚0 0 ‹ ˚ ‹ ˝ ‚ Note that the Heisenberg algebra is a two-step nilpotent Lie algebra.

42 3. The pn ` 1q-dimensional Lie algebra kn is the Lie algebra with basis

tX,Y1,...,Ynu and brackets all equal to zero except for

rX,Yis “ Yi`1, 1 ď i ď n ´ 1.

n The matrix realization is obtained by letting xX ` i“1 yiYi correspond to pn ` 1q ˆ pn ` 1q matrix ř

0 x 0 ...... 0 yn 0 x 0 ...... yn´1 ¨ . . . ˛ 0 ...... ˚ ‹ . ˚ 0 x 0 y3 ‹ ˚ ‹ ˚ 0 x y2 ‹ ˚ ‹ ˚ 0 y ‹ ˚ 1 ‹ ˚ 0 ‹ ˚ ‹ ˝ ‚ Note that kn is an n-step nilpotent Lie algebra and that k2 “ h1.

4. Let nn be the Lie algebra of strictly upper triangular n ˆ n matices. This is an pn ´ 1q-step nilpotent algebra of dimension npn ´ 1q{2 and

its center is one-dimensional. Note that n3 “ h1.

Nilpotent Lie groups are usually denoted by capital letters, and the corresponding Lie algebra is denoted by the corresponding lower case gothic letter. For example we use Hn, Kn and Nn for the nilpotent Lie groups corresponding to nilpotent Lie algebras hn, kn and nn. Next are the classical, basic theorems about nilpotent Lie algebras. The ﬁrst one is a special case of Ado’s Theorem (every ﬁnite-dimensional Lie algebra over C is isomorphic to a subalgebra of glpn, Cq).

2.7 Theorem (Birkhoﬀ Embedding Theorem). Let g be a nilpotent Lie algebra over R. Then there is a ﬁnite-dimensional vector space V and an injection i : g Ñ glpV q, such that, for all X P g, ipXq is nilpotent.

43 2.8 Theorem (Engel’s Theorem). Let g be a Lie algebra and let α : g Ñ glpV q be a homomorphism such that αpXq is nilpotent for all X P g. Then there exists a ﬂag (Jordan-Holder series) of subspaces

t0u “ V0 Ď V1 Ď ¨ ¨ ¨ Ď Vn “ V, with dim Vj “ j, such that αpXqVj Ď Vj´1 for all j ě 1 and all X P g. In particular, αpgq is a nilpotent Lie algebra.

So every nilpotent Lie algebra has a faithful embedding in nn for some n. If G is a Lie subgroup of GLpn, Rq, then the exponential map from g to G becomes the ordinary exponential map and the adjoint map Ad x becomes A ÞÑ xAx´1 for A P g, x P G. Connected, simply connected nilpotent Lie groups have several nice properties as we list some of them below.

2.9 Theorem. Let G be a connected, simply connected nilpotent Lie group with Lie algebra g.

1. exp : g Ñ G is an analytic diﬀeomorphism. It carries the Lebesgue measure on g to a left-invariant Haar measure on G. This measure is also right-invariant. In other words, G is unimodular.

2. The Campbell-Baker-Hausdorff formula holds for all X,Y P g. The low order terms in Campell-Baker-Hausdorff formula is 1 1 1 X ˚ Y “ X ` Y ` rX,Y s ` rX, rX,Y ss ´ rY, rX,Y ss 2 12 12 1 1 ´ rY, rX, rX,Y sss ´ rX, rY, rX,Y sss 48 48 ` (commutators in five or more terms).

where X ˚ Y “ logpexp X ¨ exp Y q, where X,Y P g. This series is a ﬁnite sum when G is a nilpotent Lie group.

44 3. Every connected Lie subgroup of G is closed, and also simply connected and nilpotent.

4. G has a faithful embedding as a closed subgroup of Nn for some n.

5. Let g be a nilpotent Lie algebra and let z be the center of g. Then exppzq is the center of G.

From now on, when we talk about a nilpotent Lie group, we will always assume it is connected and simply-connected. Since exp is a diﬀeomorphism of g onto G, we can use it to transfer coordinates from g to G. If we use exp to identify g to G then the group multiplication becomes the Campbell- Baker-Hausdorﬀ product

exppX ˚ Y q “ exp X ¨ exp Y, for all X,Y P g.

If g is equipped with coordinates associated with a linear basis, the corresponding coordinates in G will be called exponential coordinates. Reference: §1.1, §1.2 in [5].

Elements of Kirillov Theory

The next result, Kirillov’s lemma, is a key ingredient for the representation theory for nilpotent Lie groups. This lemma asserts that a noncommutative nilpotent Lie algebra with one-dimensional center is closely related in structure to the Heisenberg Lie algebra.

2.10 Lemma (Kirillov’s Lemma). Let g be a noncommutative nilpotent Lie algebra whose center zpgq is one-dimensional. Then g can be decomposed as

g “ RZ ‘ RY ‘ RX ‘ w “ RX ‘ g0

45 a vector space direct sum, where

RZ “ zpgq, with rX,Y s “ Z; and g0 “ RY ‘ RZ ‘ w is the centralizer of Y , and an ideal.

Let G be a nilpotent Lie group with Lie algebra g, and denote the dual vector space of g by g˚. The group G acts on g˚ by the contragradient of the adjoint map, or the coadjoint map Ad˚; deﬁned by,

ppAd˚ xqlqpY q “ lppAd x´1qY q,Y P g, l P g˚, and x P G.

A coadjoint orbit is an orbit of the Lie group G in the space g˚. The ˚ ˚ diﬀerential dpAd qe, of the coadjoint map at the unit e P g is written ad : g Ñ Endpg˚q, and is given by

ppad˚ xqlqpY q “ lprY,Xsq “ lpadp´XqY q,X,Y P g, l P g˚.

The stabilizer subgroup of G associated with l P g˚ is deﬁned by

˚ Rl “ tx P G | pAd xql “ lu.

˚ If G is a nilpotent Lie group and l P g , then the stabilizer Rl is connected and ˚ rl “ tX P g | pad Xql “ 0u Ă g is its Lie algebra. Thus Rl “ exp rl. See Lemma 1.3.1 in [5] for proof. There is also another way to describe this Lie subalgebra rl of Lie algebra g. For ˚ each l P g , deﬁnes a natural bilinear form Bl : g ˆ g Ñ R,

BlpX,Y q “ lprX,Y sq,X,Y P g.

Then Bl is skew-symmetric, i.e., BlpX,Y q “ ´BlpY,Xq. The radical of Bl is, by deﬁnition, tY P g : BlpX,Y q “ 0 for all X P gu which coincides with rl. The next lemma is helpful for ﬁnding the radical.

46 ˚ 2.11 Lemma. If g is a Lie algebra and l P g , its radical rl has even codimension in g. Hence coadjoint orbis are of even dimension.

See Lemma 1.3.2 in [5] for proof. Now let V be a real vector space with a skew-symmetric symplectic bilinear form B, its isotropic subspaces W are those such that Bpw, w1q “ 0, for all w, w1 P W . It can be shown that maximal isotropic subspaces exist and have the same dimension:

n n 2.12 Example. Consider G “ R (the Abelian case). We have g – R , with trivial bracket. Then for all x P G and X P g, we have pAd xqX “ X. Thus Ad x “ I and so Ad˚ x “ I for all x P G. Therefore coadjoint orbits ˚ ˚ in g are points. For all l P g , we have Rl “ G. So rl “ g. Therefore the unique polarizing subalgebra for l is g, since dim rl “ dim g.

2.13 Example. Let G “ Hn, g “ hn. By using matrix realization, we write w P G, W P g by pn ` 2q ˆ pn ` 2q matrices

47 1 x1 . . . xn z 0 a1 . . . an c 1 y1 ¨ b1 ¨ . . ˛ ¨ . ˛ w “ .. . ,W “ ¨ . . ˚ ‹ ˚ ‹ ˚ 1 yn‹ ˚ ¨ bn‹ ˚ ‹ ˚ ‹ ˚0 1 ‹ ˚0 0 ‹ ˚ ‹ ˚ ‹ ˝ ‚ ˝ ‚ Then

pAd w´1qW “ w´1W w

1 ´x1 ... ´xn ´z ` x ¨ y 1 ´y1 ¨ . . ˛ “ .. . W w ˚ ‹ ˚ 1 ´yn ‹ ˚ ‹ ˚0 1 ‹ ˚ ‹ ˝ ‚ 0 a1 . . . an c ` y ¨ a ´ x ¨ b b1 ¨ . ˛ “ . . ˚ ‹ ˚ bn ‹ ˚ ‹ ˚0 0 ‹ ˚ ‹ ˝ ‚ n where ¨ is the inner product in R . Let tZ,Y1,...,Yn,X1,...,Xnu be a basis ˚ ˚ ˚ ˚ ˚ ˚ for g “ hn. Then tZ ,Y1 ,...,Yn ,X1 ,...,Xnu is a dual basis for g . Let W P g. So n W “ cZ ` paiXi ` biYiq. i“1 ÿ ˚ ˚ n ˚ ˚ Let l P g . Then l “ γZ ` i“1pβjYj ` αjXj q “ lα,β,γ. So ř n lpW q “ cγ ` pαjaj ` βjbjq. j“1 ÿ

48 n Thus if w “ exppzZ ` i“1pyiYi ` xiXiqq, we get ř ˚ ´1 pAd pwqlα,β,γqpW q “ lpAdpw qW q n n “ l pc ` pyjaj ´ xjbjqqZ ` pajXj ` bjYjq j“1 j“1 “ nÿ n ÿ ‰ “ cγ ` pyjaj ´ xjbjqγ ` pajαj ` bjβjq j“1 j“1 ÿn ÿ “ cγ ` pajpαj ` yjγq ` bjpβj ´ xjγqq j“1 ÿ “ lα`yγ,β´γx,γpW q

Case I: For γ ‰ 0, we get

˚ 1 1 n pAd Gqlα,β,γ “ tlα1,β1,γ | α , β P R u which are 2n-dimensional orbits in g˚ of the form γZ˚ ` zK where zK “ tl P ˚ 1 g : lpZq “ 0u. In this case rl “ RZ. Since dim m “ 2 pdim g ` dim rlq “ 1 2 pp2n ` 1q ` 1q “ n ` 1 and rl Ď m, we need to add n more vectors. In this case m is not unique. Few examples of polarizing subalgebras are m “ RZ ` R -spantX1,...,Xnu, m “ RZ ` R -spantY1,...,Ynu and m “

RZ ` R -spantX1,...,Xku ` R -spantYk`1,...,Ynu for 1 ď k ď n ´ 1. Case II: For γ “ 0, we get

˚ pAd Gqlα,β,0 “ tlα,β,0u

K ˚ ˚ ˚ which are the points orbits in z “ RX ` RY Ď g . In this case, rl “ g is the unique polarizing subalgebra for l.

Now we are ready to see the Kirillov theory. Let’s recall the notations here. Let g˚ be the dual vector space of g and G acts on g˚ by the coadjoint ˚ ˚ action Ad pGq. Give l P g , let Bl be the linear form BlpX,Y q “ lprX,Y sq

49 and let rl be its radical. Choose a polarizing subalgebra m for l and let M “ exp m. Deﬁne the map from M to S1 by

χl,M pexp Y q “ exp 2πilpY q,Y P m,

This is a one-dimensional representation of M, since lprm, msq “ 0. Then G we denote the induced representation from M to G by πl,M “ IndM χl,M . We know before that a polarizing subalgebra always exists and indeed we can ﬁnd one that πl,M is irreducible.

2.14 Theorem. Let l P g˚. Then there exists a polarizing subalgebra m for l such that πl,M is irreducible.

The next theorem allows us to write πl instead of πl,M since it is independent of the choices of polarizing subalgebras for l.

2.15 Theorem. Let l P g˚, and let m, m1 be two polarizing subalgebras for l. Then πl,M – πl,M 1 . In particular, πl,M is irreducible whenever m is polarizing subalgebra for l.

Next theorem tells us that all unitary irreducible representation of G must be in the form of πl up to equivalence class. So we have

˚ G “ trπls | l P g u ,

p where rπls is the equivalence class of πl.

2.16 Theorem. Let π be any irreducible unitary representation of G. Then ˚ there is an l P g such that πl – π.

The following theorem gives a nice connection between unitary equivalent irreducible representations and the coadjoint orbits in g˚.

50 1 ˚ 1 2.17 Theorem. Let l, l P g . Then πl – πl1 ô l and l are in the same Ad˚pGq-orbit in g˚. In other words,

1 ˚ rπls “ rπl1 s ô there is an x P G such that, l “ pAd xql.

Combining these four theorems, we receive

2.18 Theorem (Kirillov Theory). There is a bijection between the coadjoint orbits g˚{ Ad˚pGq and the collection of equivalence classes of irreducible unitary representations of G which is denoted by G. Indeed,

g˚{ Ad˚pGq – G p

p via the map l ÞÑ rπl,M s which is independent of the choices of polarizing subalgebras m.

The proofs of the Theorem 2.14-2.17 have some similarities. The proofs are all by induction on dim G and considered in 2 cases. In case 1, we suppose that there is 1-dimensional central subalgebra h on which l “ 0. The general idea is to apply induction hypothesis to g{h and lift it back to g. For second case which we assume that g has a 1-dimensional center on which l is nontrivial, we apply Kirillov’s lemma and induction in stages in all proofs. Roughly speaking, the Heisenberg Lie group is the building block for general nilpotent Lie group. Now let’s consider the exposition of Kirillov n Theory in R and Heisenberg Lie group Hn.

n n 2.19 Example. Consider G “ R . It is well known that G – R , with λ P n corresponding to the 1-dimensional representation χ : G Ñ S1 R pλ deﬁned by 2πiλ¨x χλpxq “ e .

51 n n ˚ Also g “ R , g˚ “ R and Ad x, Ad x are the identity map for all x P G ˚ ˚ n from example 2.12. Thus g { Ad pGq “ R and the map l ÞÑ πl is simply the map l ÞÑ χl.

Let’s consider the Kirillov Theory in the case of Heisenberg Lie group ˚ Hn. We showed in example 2.13 that the adjoint orbits in hn are

1. The singleton sets in ZK where ZK “ tl | lpZq “ 0u;

2. The hyperplanes λZ˚ ` ZK for λ ‰ 0, where

Z˚pZq “ 1,Z˚pXq “ 0,Z˚pY q “ 0.

The complete classiﬁcation of the irreducible unitary representations of

Heisenberg Lie group Hn is the consequence of the famous Stone-von Neu- mann theorem (see Rosenberg in [26] or Prasad in [25] for more details on Stone-von Neumann theorem). The proof of the classiﬁcation can be found in Corwin and Greenleaf [5] or in Folland [9], [10].

2.20 Proposition. Every irreducible unitary representation of Hn is unitarily equivalent to one and only one of the folowing representations:

1. For l P ZK, the one-dimensional representation is

2πilpW q πpexppW qq “ e , for W P hn;

2 n 2. For λ P Rzt0u, the corresponding representation on L pR q is deﬁned by rπpexppxX ` yY ` zZqqfsptq “ e2πipt¨y`λzqfpt ` xq

where X,Y,Z are basis of Heisenberg Lie algebra hn.

Reference: Chapter 2 in [5].

52 Chapter 3

A conjectural Kirillov isomorphism in C˚-algebra theory

In this chapter we shall associate two C*-algebras to a simply connected nilpotent group G, and, inspired by Kirillov theory, we shall conjecture that they are isomorphic. We shall check the conjecture for the Heisenberg group. The ﬁrst C*-algebra, denoted ApGq, acts on the Hilbert space L2pGq and is related to the decomposition of L2pGq into irreducible representations. Recall that the group G acts on L2pGq by both the left and right regular representations, and we shall use both in the construction of ApGq. Namely we shall deﬁne ApGq to be the image of the group C˚-algebra of GˆG under the “bi-regular” representation of G ˆ G on L2pGq. According to the abstract Plancherel theorem, the Hilbert space L2pGq decomposes as an integral of the Hilbert spaces Hπ b Hπ, as π ranges over the irreducible unitary representations of G. Here Hπ is the complex conjugate of the Hilbert space Hπ associated to the representation π. We see in this way that the C˚-algebra ApGq is very closely related to the unitary representation theory of G.

53 The second C˚-algebra is denoted Apg˚q and it acts on the Hilbert space L2pg˚q. It is the image under the natural coadjoint/multiplication action of ˚ ˚ ˚ the crossed product C -algebra C pG, C0pg qq. Another way of describing Apg˚q is to use the Fourier transform isomorphism

L2pgq – L2pg˚q, under which Apg˚q becomes conjugate to the natural coadjoint/translation representation of the group G ˙ g on L2pgq. As we noted in the introduction, the Hilbert space L2pg˚q decomposes as an integral of the Hilbert spaces L2pOq associated to the coadjoint orbits of G. This decomposition is also a decomposition of L2pg˚q into irreducible representations of Apg˚q. We see in this way that the C˚-algebra Apg˚q is very closely related to the space of coadjoint orbits in g˚. The above remarks motivate the conjecture. We will prove the conjecture for the Heisenberg group at the end of the chapter using the exponential map, which gives a unitary isomorphism from L2pGq to L2pgq. We will show that the unitary isomorphism conjugates ApGq onto Apg˚q. Unfortunately this is almost certainly not true for all nilpotent groups in view of Remark 3.14.

3.1 Deﬁnition of ApGq

For each g1, g2 P G, we deﬁne a map

2 2 g1 g2 Tg1,g2 : L pGq Ñ L pGq, by f ÞÑ f ,

g1 g2 ´1 where f pgq “ fpg1 gg2q for every g P G. Therefore

´1 pTg1,g2 fqpgq “ fpg1 gg2q.

54 3.1 Remark. T is a unitary representation of G ˆ G on L2pGq by sending pg1, g2q ÞÑ Tg1,g2 . To see that Tg1,g2 is a homomorphism, consider

´1 rTg1,g3 pTg2,g4 fqs pgq “ pTg2,g4 fqpg1 gg3q ´1 ´1 “ fpg2 g1 gg3g4q ´1 “ f pg1g2q gpg3g4q ` ˘ “ pTpg1g2,g3g4qfqpgq

“ pTpg1,g3qpg2,g4qfqpgq.

In other words, Tg1,g3 ˝ Tg2,g4 “ Tpg1,g3qpg2,g4q. Also Tg1,g2 is a unitary operator, since

˚ xTg1,g2 f, hy “ xf, Tg1,g2 hy

“ fpgqTg1,g2 hpgq dg żG ´1 “ fpgqhpg1 gg2q. żG ´1 ´1 Changing variables by setting k “ g1 gg2. Then g “ g1kg2 and dk “ dg. Therefore

˚ ´1 xTg1,g2 f, hy “ fpg1kg2 qhk dk żG “ xT ´1 ´1 f, hy g1 ,g2 ´1 “ xT ´1 f, hy “ xT f, hy pg1,g2q pg1,g2q

˚ ´1 In other words, Tg1,g2 “ Tg1,g2 .

By the remark above, we therefore obtain a representation

C˚pG ˆ Gq Ñ BpL2pGqq.

We deﬁne ApGq as the image under this map. This is automatically closed, so a C˚-algebra. In other words, ApGq Ă BpL2pGqq is the C˚-algebra closure

55 of all the operators

T “ ϕpg1, g2qTg1,g2 dg1 dg2, żG żG 8 where ϕ P Cc pG ˆ Gq. This means that

xT f1, f2y “ ϕpg1, g2qxTg1g2 f1, f2yL2pGq dg1 dg2, żG żG 2 for all f1, f2 P L pGq.

3.2 Deﬁnition of Apg˚q

For each g P G, X P g, we deﬁne a map

2 2 g X Sg,X : L pgq Ñ L pgq, f ÞÑ f , where gf X pY q “ fpg´1Y g ´ Xq for all Y P g. Hence

´1 pSg,X pfqqpY q “ fpAdg´1 pY q ´ Xq “ fpg Y g ´ Xq.

3.2 Remark. S is a unitary representation of G ˙ g on L2pgq by pg, Xq ÞÑ

Sg,X . To see this, let pg1,X1q, pg2,X2q P G ˙ g. We consider

´1 pSg1,X1 pSg2,X2 fqq pY q “ pSg2,X2 fqpg1 Y g1 ´ X1q ´1 ´1 “ fpg2 pg1 Y g1 ´ X1qg2 ´ X2q ´1 ´1 “ fppg1g2q Y pg1g2q ´ pg2 X1g2 ` X2qq

“ pS ´1 fqpY q g1g2,g2 X1g2`X2

“ pSpg1,X1qpg2,X2qfqpY q,

2 therefore Spg1,X1qSpg2,X2q “ Spg1,X1qpg2,X2q. Moreover for any f, h P L pgq,

˚ xSg,X f, hy “ xf, Sg,X hy

“ fpZqSg,X hpZq dZ żg “ fpZqhpg´1Zg ´ Xq dZ. żg

56 Let Y “ g´1Zg ´ X. Then Z “ gpY ` Xqg´1, dZ “ dY and so

˚ ´1 xSg,X f, hy “ fpgpY ` Xqg qhpZq dZ żg “ fpgY g´1 ` gXg´1qhpZq dZ żg

“ pSg´1,´gXg´1 fqpZqhpZq dz żg ´1 “ xSpg,Xq´1 f, hy “ xSpg,Xqf, hy.

˚ ´1 Therefore Sg,X “ Sg,X . In other words, Sg,X is a unitary homomorphism from L2pgq to L2pgq.

From the remark above, therefore we get a representation

C˚pG ˙ gq Ñ BpL2pgqq.

We deﬁne Apg˚q as the image under this map. So Apg˚q Ă BpL2pgqq is the C˚-algebra closure of all the operators

S “ ψpg, XqSg,X dg dX, żg żG 8 where ψ P Cc pG ˙ gq. This means that

xSf1, f2y “ ψpg, XqxSg,X f1, f2yL2pgq dg dX, żg żG 2 for all f1, f2 P L pgq.

3.3 Remark. Alternatively, we can deﬁne Apg˚q by using L2pg˚q since Fourier transform is a unitary isomorphism from L2pgq to L2pg˚q by f ÞÑ f, where p fplq “ fpXqeilpXq dX żg for a suitable choice of translationp invariant measure on g˚. Here we can choose dX so that X ÞÑ exppXq from g to G is measure preserving. See Remark 5.1 in Chapter 5.

57 3.3 The conjecture

We conjecture that there is an isomorphism between ApGq and Apg˚q at least in some cases. We will prove the conjecture for the Heisenberg group

H1 in the next section using the exponential map, which gives a unitary isomorphism from L2pGq to L2pgq. Unfortunately this method almost certainly fails for more general nilpotent groups as we shall also see at the end of this chapter. Recall that for any nilpotent Lie group G and its Lie algebra g,

exp : g Ñ G is a measure-preserving diﬀeomorphism. This gives a unitary isomorphism map U : L2pGq Ñ L2pgq, f ÞÑ f ˝ exp, for every f P L2pGq. Its adjoint U ˚ “ U ´1 is deﬁned by f ÞÑ f ˝ log. Consequently, we obtain a map

´1 2 2 UTg1,g2 U : L pgq Ñ L pgq,

2 for each pair of elments g1, g2 P G. For any f P L pgq, we have

´1 g1 g2 g1 g2 pUTg1,g2 U qpfq “ UTg1,g2 pf ˝ logq “ Up pf ˝ logq q “ pf ˝ logq ˝ exp .

Therefore ´1 pUTg1,g2 U pfqqpY q “ fplogpg1pexp Y qg2qq, for any Y P g.

´1 3.4 Remark. Note that if x P G ˆ G and x “ pg1 , g2q then

pTxfqpgq “ fpg1gg2q.

58 Similarly, if z P G ˙ g and z “ pg´1, ´Xq then

´1 pSzfqpY q “ fpgY g ` Xq.

To simplify our calculation, we shall use the following notations from now on ´1 ´1 pTg1,g2 fqpgq “ fpg1 gg2q, pSg,X pfqqpY q “ fpgY g ` Xq.

Next lemma is straightforward to prove. And we shall use in the next section.

3.5 Lemma. For g1, g2, g3, g4 P G and X1,X2 P g, we have

´1 ´1 ´1 UTg3,g4 U UTg1,g2 U “ UTg1g3,g4g2 U ` ˘ ` ˘ Sg2,X2 Sg1,X1 “ S ´1 . g1g2,g1X2g1 `X1

2 Proof. Let g1, g2, g3, g4 P G, f P L pgq and X P g. We have

´1 ´1 ´1 UTg3,g4 U UTg1,g2 U pfq pXq “ UTg1,g2 U pfq plogpg3pexp Xqg4qq

`` ˘ ` ˘ ˘ “ f``plogpg1 expplog˘ pg3˘pexp Xqg4qqg2qq

“ fplogpg1g3pexp Xqg4g2qq

´1 “ UTg1g3,g4g2 U pfq pXq. `` ˘ ˘ Also, for any Y P g, we have

pSg2,X2 Sg1,X1 pfqq pY q “ Sg2,X2 pSg1,X1 pfqq pY q

´1 “ pSg1,X1 pfqq pg2Y g2 ` X2q

´1 ´1 “ f g1pg2Y g2 ` X2qg1 ` X1

´1 ´1 ´1 “ fp`g1g2pY qg2 g1 ` g1X2g1 ˘` X1q

´1 “ Sg1g2,g1X2g1 `X1 pfq pY q. ` ˘ 59 3.4 The case of the Heisenberg group

Now we focus only on the Heisenberg group H1 and its Lie algebra h1. For simplicity, we write

1 x z px, y, zq for 1 y P H1, and ¨ 1˛ ˝0 a c‚ ra, b, cs for 0 b P h1. ¨ 0˛ ˝ ‚ 3.6 Lemma. For all a, b, c, x, y, z P R, we have

ab 1. expra, b, cs “ pa, b, c ` 2 q

xy 2. logpx, y, zq “ rx, y, z ´ 2 s

3. px, y, zqpa, b, cq “ px ` a, y ` b, z ` c ` xbq

4. px, y, zq´1 “ p´x, ´y, xy ´ zq

5. ra, b, cspx, y, zq “ ra, b, ay ` cs

6. px, y, zqra, b, cs “ ra, b, xb ` cs

7. rx, y, zs ` ra, b, cs “ rx ` a, y ` b, z ` cs

Proof. We shall show only ﬁrst two identities since the rest is straightforward. Let a, b, c, x, y, z be real numbers.

8 ra, b, csn ra, b, cs2 expra, b, cs “ “ I ` ra, b, cs ` ` 0 n! 2 n“0 ÿ 1 ab “ I ` ra, b, cs ` r0, 0, abs “ pa, b, c ` q. 2 2

60 Also, we have

8 pI ´ px, y, zqqn logpx, y, zq “ logrI ´ pI ´ px, y, zqqs “ ´ n n“1 ÿ pI ´ px, y, zqq2 “ ´pI ´ px, y, zqq ´ ´ 0 2 1 “ rx, y, zs ´ r´x, ´y, ´zs2 2 1 1 “ rx, y, zs ´ r0, 0, xys “ rx, y, z ´ xys. 2 2

1 1 1 3.7 Lemma. Let pu, v, wq, px, y, zq, px , y , z q P H1, and rα, β, γs, ra, b, cs P 2 h1. For any f P L pgq, we have

´1 pUTpx,y,zqpx1,y1,z1qU pfqqpra, b, csq ab “ fprx ` a ` x1, y ` b ` y1, z ` c ` z1 ` ` xb ` xy1 ` ay1 2 px ` a ` x1qpy ` b ` y1q ´ sq 2 1 “ fprx ` a ` x1, y ` b ` y1, z ` c ` z1 ` pxy1 ´ xy ´ x1y ´ x1y1q 2 x ´ x1 y ´ y1 ` b ´ asq 2 2

pSpu,v,wq,rα,β,γspfqqpra, b, csq “ fpra ` α, b ` β, c ` γ ` ub ´ avsq.

61 Proof. By using identities in Lemma 3.6, we have

´1 pUTpx,y,zqpx1,y1,z1qU pfqqpra, b, csq “ fplogppx, y, zqpexpra, b, csqpx1, y1, z1qqq ab “ fplogppx, y, zqpa, b, c ` qpx1, y1, z1qqq 2 ab “ fplogppx ` a, y ` b, z ` c ` ` xbqpx1, y1, z1qqq 2 ab “ fplogppx ` a ` x1, y ` b ` y1, z ` c ` z1 ` ` xb ` xy1 ` ay1qq 2 ab “ fprx ` a ` x1, y ` b ` y1, z ` c ` z1 ` ` xb ` xy1 ` ay1 2 px ` a ` x1qpy ` b ` y1q ´ sq 2 1 “ fprx ` a ` x1, y ` b ` y1, z ` c ` z1 ` pxy1 ´ xy ´ x1y ´ x1y1q 2 x ´ x1 y ´ y1 ` b ´ asq. 2 2

Similarly, we have

´1 pSpu,v,wq,rα,β,γspfqqpra, b, csq “ fppu, v, wqra, b, cspu, v, wq ` rα, β, γsq “ fppu, v, wqra, b, csp´u, ´v, uv ´ wq ` rα, β, γsq

“ fpra, b, ub ` csp´u, ´v, uv ´ wq ` rα, β, γsq

“ fpra, b, ´av ` ub ` cs ` rα, β, γsq

“ fpra ` α, b ` β, c ` γ ` ub ´ avsq.

1 1 1 1 1 1 3.8 Lemma. For any pu, v, wq, pα, β, γq, pu , v , w q, pα , β , γ q P H1, and 1 1 1 rα, β, γs, rα , β , γ s P h1, we have

´1 ´1 UTpu1,v1,w1qpα1,β1,γ1qU UTpu,v,wqpα,β,γqU

´1 ` ˘ `“ UTpu`u1,v`v1,w`w1˘`uv1qpα1`α,β1`β,γ1`γ`α1βqU ,

Spu1,v1,w1qrα1,β1,γ1sSpu,v,wqrα,β,γs7 “ Spuù1,v`v1,w`w1ùv1qrα`α1,β`β1,γ`γ1ùβ1´vα1s.

62 Proof. We use the relationships we found in Lemma 3.5, then apply identities in Lemma 3.6 to obtain

´1 ´1 UTpu1,v1,w1q,pα1,β1,γ1qU UTpu,v,wqpα,β,γqU

´1 ` ˘ `“ UTpu,v,wqpu1,v1,w1q,p˘α1,β1,γ1qpα,β,γqU

´1 “ UTpu`u1,v`v1,w`w1`uv1q,pα1`α,β1`β,γ1`γ`α1βqU .

Also, we have

Spu1,v1,w1qrα1,β1,γ1sSpu,v,wqrα,β,γs

“ Spu,v,wqpu1,v1,w1q,pu,v,wqrα1,β1,γ1spu,v,wq´1`rα,β,γs

“ Spuù1,v`v1,w`w1ùv1q,rα1,β1,uβ1`γ1spú,´v,uv´wq`rα,β,γs

“ Spuù1,v`v1,w`w1ùv1q,rα1,β1,´α1vùβ1`γ1s`rα,β,γs

“ Spuù1,v`v1,w`w1ùv1qrα`α1,β`β1,γ`γ1ùβ1´vα1s.

3.9 Proposition. For the Heisenberg group H1 and its Lie algebra h1, every UTU ´1 is equal to some S and every U ´1SU is equal to some T . More explicitly, we have

´1 1 UTpx,y,zqpx1,y1,z1qU “ S x´x1 yý 1 1 1 1 1 1 1 1 1 (3.1) p 2 , 2 ,z qrx`x ,y`y ,z`z ` 2 pxy ´xy´x y´x y qs ´1 U Spu,v,wq,rα,β,γsU “ T α β αβ uβ αv α β . (3.2) pu` 2 ,v` 2 ,γùv` 4 ´ 2 ` 2 ´wq,pú` 2 ,´v` 2 ,wq

In other words, there are polynomial maps P : H1 ˆ H1 Ñ H1 ˆ h1 and

Q : H1 ˆ h1 Ñ H1 ˆ H1, which are deﬁned by

P px, y, z, x1, y1, z1q x ´ x1 y ´ y1 1 “ , , z1, x ` x1, y ` y1, z ` z1 ` pxy1 ´ xy ´ x1y ´ x1y1q 2 2 2 ˆ ˙ Qpu, v, w, α, β, γq α β αβ uβ αv α β “ u ` , v ` , γ ` uv ` ´ ` ´ w, ´u ` , ´v ` , w 2 2 4 2 2 2 2 ˆ ˙ satisfying

63 1. P pQpg, Xqq “ pg, Xq and QpP pg1, g2qq “ pg1, g2q

´1 2. UTg1,g2 U “ SP pg1,g2q

´1 3. U Sg,X U “ TQpg,Xq, for all g, g1, g2 P H1 and X P h1. Moreover, the maps P and Q are diﬀeo- morphisms.

Proof. The identity (3.1) is easily followed by Lemma 3.7:

1 S x´x1 y´y 1 1 1 1 1 1 1 1 1 pfq pra, b, csq p 2 , 2 ,z qrx`x ,y`y ,z`z ` 2 pxy ´xy´x y´x y qs ´ ¯ 1 “ fprx ` a ` x1, y ` b ` y1, z ` c ` z1 ` pxy1 ´ xy ´ x1y ´ x1y1q 2 x ´ x1 y ´ y1 ` b ´ asq 2 2 ´1 “ pUTpx,y,zqpx1,y1,z1qU pfqqpra, b, csq,

2 for all f P L ph1q and ra, b, cs P h1. Similarly, by applying Lemma 3.7, we have

´1 pUT α β αβ uβ αv α β U pfqqpra, b, csq pu` 2 ,v` 2 ,γ`uv` 4 ´ 2 ` 2 ´wq,p´u` 2 ,´v` 2 ,wq αβ uβ αv “ fpra ` α, b ` β, γ ` uv ` ´ ` ` c ` : ` ub ´ vasq, (3.3) 4 2 2 where

1 α β α β :“ u ` ´v ` ´ u ` v ` 2 2 2 2 2 ˆ ˙ ˆ ˙ ” ´ ¯ ´α ¯ β α β ´ ú ` v ` ´ ú ` ´v ` , 2 2 2 2 ´ ¯ ˆ ˙ ´ ¯ ˆ ˙ ı which can be simplified to

1 αβ :“ ´2uv ´ αv ` uβ ´ . (3.4) 2 2 ˆ ˙

64 Therefore

αβ uβ αv γ ` uv ` ´ ` c ` : ` ub ´ va 4 2 2 αβ uβ αv 1 αβ “ γ ` uv ` ´ ` ` c ` ´2uv ´ αv ` uβ ´ ` ub ´ va 4 2 2 2 2 ˆ ˙ “ c ` γ ` ub ´ av. (3.5)

Substitute (3.5) into (3.3),

´1 pUT α β αβ uβ αv α β U pfqqpra, b, csq pu` 2 ,v` 2 ,γ`uv` 4 ´ 2 ` 2 ´wq,p´u` 2 ,´v` 2 ,wq “ fpra ` α, b ` β, c ` γ ` ub ´ avsq

“ pSpu,v,wq,rα,β,γspfqqpra, b, csq,

2 for all f P L ph1q and ra, b, cs P h1. Therefore (3.2) is followed. Now let’s check the conditions P ˝ Q “ Id and Q ˝ P “ Id. We shall consider only at the 6th cooordinate of P ˝ Q and the third coordinate of Q ˝ P since the rest is easy to see. Using (3.4),

αβ uβ αv P pQpγqq “ γ ` uv ` ´ ` ` : 4 2 2 ˆ ˙ αβ uβ αv 1 αβ “ γ ` uv ` ´ ` ` ´2uv ´ αv ` uβ ´ 4 2 2 2 2 ˆ ˙ ˆ ˙ “ γ.

Also, we have

1 QpP pzqq “ z ` xy1 ´ xy ´ x1y ´ x1y1 ` ˚, (3.6) 2 ` ˘ where

1 x ´ x1 y ´ y1 px ` x1qpy ` y1q ˚ “ ` 2 2 2 4 ˆ ˙ ˆ ˙ ” x ´ x1 y ` y1 x ` x1 y ´ y1 ´ ´ 2 2 2 2 ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ ı 65 which can be simpliﬁed to

1 ˚ “ ´xy1 ` xy ` x1y ` x1y1 . (3.7) 2 ` ˘ Substitute (3.7) into (3.6),

1 1 QpP pzqq “ z ` xy1 ´ xy ´ x1y ´ x1y1 ` ´xy1 ` xy ` x1y ` x1y1 “ z. 2 2 ` ˘ ` ˘ Now we shall consider the determinant of Jacobian matrices JP and JQ as follow: 1 1 2 0 0 ´ 2 0 0 1 1 0 2 0 0 ´ 2 0

0 0 0 0 0 1 JP “ 1 0 0 1 0 0

0 1 0 0 1 0 y1ý ´x´x1 ýý1 x´x1 2 2 1 2 2 1

1 0 0 0 0 0

0 1 0 0 0 0 1 R R R 2 4` 1Ñ 1 0 0 0 0 0 1 “ 1 R `R ÑR 1 0 0 1 0 0 2 5 2 2 0 1 0 0 1 0 y1ý ´x´x1 ýý1 x´x1 2 2 1 2 2 1

1 0 0 0 0 0

0 1 0 0 0 0

C3ØC6 0 0 1 0 0 0 “ p´1q 1 0 0 1 0 0

0 1 0 0 1 0 y1ý ´x´x1 ýý1 x´x1 2 2 1 2 2 1 “ ´1.

66 Similarly, we have

1 1 0 0 2 0 0 1 0 1 0 0 2 0 β α β v α u v ´ 2 u ` 2 ´1 4 ` 2 4 ´ 2 1 JQ “ ´1 0 0 1 0 0 2 0 ´1 0 0 1 0 2 0 0 1 0 0 0

2 0 0 0 0 0

0 2 0 0 0 0 β α β v α u ´R4`R1ÑR1 v ´ 2 u ` 2 ´1 4 ` 2 4 ´ 2 1 “ 1 ´R5`R2ÑR2 ´1 0 0 0 0 2 0 ´1 0 0 1 0 2 0 0 1 0 0 0

2 0 0 0 0 0

0 2 0 0 0 0

R3ØR6 0 0 1 0 0 0 “ p´1q 1 ´1 0 0 2 0 0 1 0 ´1 0 0 2 0 β α β v α u v ´ 2 u ` 2 ´1 4 ` 2 4 ´ 2 1 “ ´1.

By proposition above, it follows easily that two C˚-algebras ApGq and ˚ Apg q are isomorphic in the case of Heisenberg group H1.

3.10 Corollary. For Heisenberg group G “ H1 and its Lie algebra g “ h1,

UApGqU ´1 “ Apgq.

Consequently, U is an isomorphism from ApGq onto Apg˚q.

67 Proof. By using the properties from previous proposition, we have

´1 ´1 U ϕpg1, g2qTg1g2 dg1 dg2 U “ ϕpg1, g2qUTg1g2 U dg1 dg2 ˆżG żG ˙ żG żG

“ ϕpg1, g2qSP pg1,g2q dg1 dg2 żG żG “ ϕpQpg, XqqSg,X | det JP | dg dX żG żg “ ϕpQpg, XqqSg,X dg dX, żG żg 8 for any ϕ P Cc pG ˆ Gq.

3.5 Other nilpotent groups

In this section, we shall see that the same proof fails for Lie group K3. For

Lie group K3 and its Lie algebra k3, we write

w2 1 w 2 z 1 w y pw, x, y, zq for ¨ ˛ P K3, 1 x ˚ 1‹ ˚ ‹ ˝0 a 0 d ‚ 0 a c ra, b, c, ds for P k , and ¨ 0 b˛ 3 ˚ 0‹ ˚ ‹ ˝0 a e d‚ 0 a c ra, b, c, ds for . e ¨ 0 b˛ ˚ 0‹ ˚ ‹ ˝ ‚ Similar to what we did for Heisenberg group H1, we start by couple ´1 lemmas involving calculation to obtain formulas for UTg1,g2 U and Sg,X .

68 1 1 1 1 3.11 Lemma. Let pw, x, y, zq, pw , x , y , z q P K3, and ra, b, c, ds P k3. Then

ab ac a2b expra, b, c, ds “ pa, b, c ` , d ` ` q 2 2 6 wx wy w2x logpw, x, y, zq “ rw, x, y ´ , z ´ ` s 2 2 12 w2x1 pw, x, y, zqpw1, x1, y1, z1q “ pw ` w1, x ` x1, y ` y1 ` wx1, z ` z1 ` wy1 ` q 2 w2x pw, x, y, zq´1 “ p´w, ´x, wx ´ y, ´ ` wy ´ zq 2

ra, b, c, dspw, x, y, zq “ ra, b, c ` ax, d ` ayswa w2b pw, x, y, zqra, b, c, ds “ ra, b, c ` wb, d ` wc ` s . 2 wa

Proof. Again, we shll show only the ﬁrst two identities.

8 ra, b, c, dsn expra, b, c, ds “ n! n“0 ÿ ra, b, c, ds2 ra, b, c, ds3 “ I ` ra, b, c, ds ` ` ` 0 2! 3! 0 a 0 d 0 0 a2 ac 0 0 0 a2b 0 a c 1 0 0 ab 1 0 0 0 “ I ` ¨ ˛ ` ¨ ˛ ` ¨ ˛ 0 b 2 0 0 6 0 0 ˚ 0‹ ˚ 0 ‹ ˚ 0 ‹ ˚ ‹ ˚ ‹ ˚ ‹ ˝ a2 ‚ac a2˝b ‚ ˝ ‚ 1 a 2 d ` 2 ` 6 1 a c ` ab “ ¨ 2 ˛ 1 b ˚ 1 ‹ ˚ ‹ ˝ ab ac a2b‚ “ pa, b, c ` , d ` ` q. 2 2 6

69 Also, we have logpw, x, y, zq “ log I ´ pI ´ pw, x, y, zqq 8 pI ´ pw, x, y, zqqn “ ´ n n“1 ÿ pI ´ pw, x, y, zqq2 pI ´ pw, x, y, zqq3 “ ´pI ´ pw, x, y, zqq ´ ´ ´ 0 2 3 w2 2 w2x 0 ´w ´ 2 ´z 0 0 w wy ` 2 0 ´w ´y 1 0 0 wx “ ´ ¨ ˛ ´ ¨ ˛ 0 ´x 2 0 0 ˚ 0 ‹ ˚ 0 ‹ ˚ ‹ ˚ ‹ ˝ 0 0 0 ´w2x ‚ ˝ ‚ 1 0 0 0 ´ ¨ ˛ 3 0 0 ˚ 0 ‹ ˚ ‹ ˝ wy ‚w2x 0 w 0 z ´ 2 ` 12 0 w y ´ wx “ ¨ 2 ˛ 0 x ˚ 0 ‹ ˚ ‹ ˝ wx wy w‚2x “ rw, x, y ´ , z ´ ` s. 2 2 12

3.12 Lemma. For Lie group K3 and its Lie algebra k3, we have

´1 pUTpw1,x1,y1,zqpw1,x1,y1,z1qU pfqqra, b, c, ds 1 “ fpra ` w ` w1, b ` x ` x1, c ` y ` y1 ` pwb ` wx1 ` ax1q 2 1 ´ pax ` wx ` w1b ` w1x ` w1x1q, ˚sq, 2 where

1 1 ˚ “ d ` z ` z1 ` pwc ` wy1 ` ay1q ´ pay ` wy ` w1c ` w1y1 ` w1yq 2 2 1 1 ´ pw1wb ` w1wx1 ` w1ax1q ` pwax1 ` wax ` w1ax ` w1wxq 3 6 1 1 ` pw2x1 `a2x1 `w2b`w12b`a2x`w2x`w12x`w12x1q´ pwab`w1abq 12 12 (3.8)

70 and

pSpw,x,y,zq,rαβγδspfqqpra, b, c, dsq w2b “ fpra ` α, b ` β, c ` γ ` wb ´ xa, d ` δ ` wc ´ ya ` sq, 2

1 1 1 1 for all pw, x, y, zq, pw , x , y , z q P K3, and rα, β, γ, δs P k3.

Proof. Using identities we obtained in Lemma 3.11,

1 1 1 1 pSpw,x,y,zq,rαβγδspfqqpra, b, c, dsq “ fppw, x, y, zqra, b, c, dspw , x , y , z q ` rα, β, γ, δsq w2b w2x “ fpra, b, c ` wb, d ` wc ` s p´w, ´x, wx ´ y, ´ ` wy ´ zq 2 wa 2 ` rα, β, γ, δsq w2b “ fpra, b, c ` wb ´ xa, d ` wc ´ ya ` s ` rα, β, γ, δsq 2 w2b “ fpra ` α, b ` β, c ` γ ` wb ´ xa, d ` δ ` wc ´ ya ` sq 2

And also

´1 pUTpw,x,y,zqpw1,x1,y1,z1qU pfqqpra, b, c, dsq “ fplogppw, x, y, zqpexpra, b, c, dsqpw1, x1, y1, z1qqq ab ac a2b “ fplogppw, x, y, zqpa, b, c ` , d ` ` qpw1, x1, y1, z1qqq 2 2 6 ab ac a2b wab w2b “ fplogppw ` a, x ` b, y ` c ` ` wb, z ` d ` ` ` wc ` ` q 2 2 6 2 2 pw1, x1, y1, z1qqq ab “ fplogppa ` w ` w1, b ` x ` x1, c ` y ` y1 ` ` wb ` pw ` aqx1, :qqq, 2 where

ac a2b wab w2b pw ` aq2x1 :“ d ` z ` z1 ` ` ` wc ` ` ` pw ` aqy1 ` . 2 6 2 2 2

71 Therefore

´1 pUTpw,x,y,zqpw1,x1,y1,z1qU pfqqpra, b, c, dsq ab “ fpra ` w ` w1, b ` x ` x1, c ` y ` y1 ` ` wb ` pw ` aqx1 2 pa ` w ` w1qpb ` x ` x1q ´ s, ∇q 2 1 “ fpra ` w ` w1, b ` x ` x1, c ` y ` y1 ` pwb ` wx1 ` ax1q 2 1 ´ pax ` wx ` w1b ` w1x ` w1x1q, ∇sq, 2 where

pa ` w ` w1qpc ` y1 ` y ` ab ` wb ` pw ` aqx1q ∇ “ : ´ 2 2 pa ` w ` w1q2pb ` x ` x1q ` 12 “ ˚.

The following proposition tells us that the group K3 can not be handled in the same way as the Heisenberg group for the purposes of our conjecture.

3.13 Proposition. For Lie group K3 and its Lie algebra k3, there is some T that U ´1TU ‰ S for any S. On the other hand, there is some S that S ‰ UTU ´1 for any T .

´1 Proof. Suppose that we can write UTp1,1,0,0qp0,0,0,0qU as a certain S, which means we can ﬁnd k, l, m, n, α, β, γ, δ, such that

´1 Spk,l,m,nqrα,β,γ,δs “ U Tp1,1,0,0qp0,0,0,0qU.

72 From previous lemma, we have

pSpk,l,m,nq,rαβγδspfqqpra, b, c, dsq k2b “ fpra ` α, b ` β, c ` γ ` kb ´ la, d ` δ ` kc ´ ma ` sq, 2 ´1 pUTp1,1,0,0qp0,0,0,0qU pfqqra, b, c, ds b a 1 c a b a2 1 ab “ fpra ` 1, b ` 1, c ` ´ ´ , d ` ` ` ` ` ´ sq. 2 2 2 2 6 12 12 12 12 for any a, b, c, d P R. Consequently,

a ` α “ a ` 1

b ` β “ b ` 1 b a 1 c ` γ ` kb ´ la “ c ` ´ ´ 2 2 2 k2b c a b a2 1 ab d ` δ ` kc ´ ma ` “ d ` ` ` ` ` ´ , 2 2 6 12 12 12 12

1 1 for all a, b, c, d P R. First two equations give α “ β “ 1. Also γ “ ´ 2 , k “ 2 1 1 and l “ 2 by third equation. Substitute k “ 2 into the last equation, we then have a b a2 1 ab δ ´ ma “ ´ ` ` ´ . 6 24 12 12 12 This is a contradiction since δ and m are simply constants and they are independent of variable b. ´1 Now suppose that we can write Sp1,1,1,0qr2,2,0,0s as a certain UTU , which means we can ﬁnd w, x, y, z, w1, x1, y1, z1, such that

´1 UTpw,x,y,zqpw1,x1,y1,z1qU “ Sp1,1,1,0qr2,2,0,0s.

From previous lemma, we have

b pS pfqqpra, b, c, dsq “ fpra ` 2, b ` 2, c ` b ´ a, d ` c ´ a ` sq. p1,1,1,0qr2,2,0,0s 2

73 Therefore if ˚ is the term in

a ` w ` w1 “ a ` 2

b ` x ` x1 “ b ` 2 w ´ w1 x ´ x1 wx1 ´ wx ´ w1x ´ w1x1 c ` y ` y1 ` b ´ a ` “ c ` b ´ a 2 2 2 b ˚ “ d ` c ´ a ` , 2 (3.9) for all a, b, c, d P R where ˚ is the one in (3.8). First two equations give w ` w1 “ 2 and x ` x1 “ 2, while the third equation gives w ´ w1 “ 2, x ´ x1 “ 2 and wx1 ´ wx ´ w1x ´ w1x1 y ` y1 ` “ 0. (3.10) 2 Then we have w “ x “ 2 and w1 “ x1 “ 0 which transform (3.10) to y ` y1 “ 2 and (3.8) to

1 1 1 ˚ “ d ` z ` z1 ` p2c ` 2y1 ` ay1q ´ pay ` 2yq ` p4aq 2 2 6 1 1 ` p4b ` 2a2 ` 8q ´ p2abq 12 12 ay1 ay 2a b a2 2 ab “ d ` z ` z1 ` c ` y1 ` ´ ´ y ` ` ` ` ´ 2 2 3 3 6 3 6 a 2a b a2 2 ab “ d ` z ` z1 ` c ` p1 ` qpy1 ´ yq ` ` ` ` ´ . (3.11) 2 3 3 6 3 6

Then we substitute (3.11) into (3.9), use y1 “ 2 ´ y and then simplify it to obtain 5a b a2 2 ab 0 “ z ` z1 ` p2 ` aqp1 ´ yq ` ´ ` ` ´ , 3 6 6 3 6 for a, b, c, d P R. This is again a contradiction since z, z1 and y are simply constants and they are independent of variable b.

3.14 Remark. This proposition does not conclusively show that U fails to conjugate ApGq into Apg˚q, but it oﬀers strong evidence that this is so. In

74 any case, the method of proof that we used for the Heisenberg group H1 certainly fails for K3.

75 Chapter 4

Structure of ApGq for the Heisenberg group

The purpose of this chapter is to investigate the structure of the C˚-algebra ApGq in the case of the Heisenberg group (in the next chapter we shall do the same for the C˚-algebra Apg˚q). We shall determine the spectrum of the C˚-algebra, and use the information we ﬁnd to decompose ApGq as a C˚-algebra extension. We have already proved the main conjecture for the Heisenberg group, but since the method of proof does not carry over to more general nilpotent groups, we hope that a better understanding of the structure of ApGq may eventually lead to proofs of the conjecture in more cases.

4.1 The unitary dual of a product

There is a natural map from G1 ˆ G2 into G1 ˆ G2 deﬁned by

x x { prπ1s, rπ2sq ÞÑ rπ1 b π2s.

It is indeed a well-deﬁned injection map. In what follows we shall use the following more precise theorem of Wulfsohn (see [31] for proof and details).

76 4.1 Theorem (Wulfsohn). If either G1 or G2 is postliminal, then the map- ping

prπ1s, rπ2sq ÞÑ rπ1 b π2s is a homeomorphism from pG1qr ˆ pG2qr onto pG1 ˆ G2qr.

4.2 The diagonal{ in the{ unitary{ dual of a product

Since ApGq is a quotient of the C˚-algebra of G ˆ G, its spectrum can be viewed as a closed subset of the unitary dual of G ˆ G. In this section we shall begin by considering the “diagonal” representations π b π˚. Our ﬁrst goal is to show that they lie in the spectrum of ApGq.

4.2 Theorem. Let G be an amenable liminal Lie group. If π is an irreducible representation of G then π b π˚ is weakly contained in the bi-regular representation β. So it is in the dual of ApGq.

Proof. We’ll prove by contradiction. We suppose that π b π˚ is not weakly contained in β. Then kerpβq Ć kerpπ b π˚q.

Consequently, there is a basic open set in the product topology of G ˆ G (thanks to previous theorem), p p

UI1 ˆ UI2 “ tpπ1, π2q | π1pI1q ‰ 0, and π2pI2q ‰ 0u

˚ containing π b π and not intersecting Supppβq. As indicated, UI1 ˆ UI2 ˚ ˚ ˚ corresponds to an ideal I1 b I2 Ă C pG ˆ Gqp“ C pGq b C pGqq. Since the open set UI1 ˆ UI2 is disjoint from β, we have

βpI1 b I2q “ 0.

77 8 8 If f P Cc pGq, then we deﬁne f P Cc pGq by

fpgq “ fpgq, for all g P G. The map f ÞÑ f extends to a map

C˚pGq Ñ C˚pGq.

So we have that πpfq “ 0 if and only if π˚pfq “ 0. Now we let

J1 “ I1 X I2 and J2 “ I2 X I1.

Then we have J1 Ă I1, J2 Ă I2 and J1 “ J2. Therefore

βpJ1 b J2q “ βpJ1 b J1q “ 0.

By liminality, πrI1s “ KpHπq and πrI2s “ KpHπ˚ q. Choose i1 P I1 so that πpi1q ‰ 0 and similarly choose i2 P I2, so that

˚ π pi2q “ πpi1q.

Then consider the element

˚ i2 i1 P I2 ¨ I1 “ I2 X I1 “ J1.

Here we use the fact that I X J “ I ¨ J for any ideals I,J in C˚-algebra. Therefore

˚ ˚ ˚ ˚ ˚ πpi2 i1q “ πpi2 qπpi1q “ π pi2q πpi1q “ πpi1q πpi1q ‰ 0.

Therefore πpJ1q ‰ 0. Consequently, βpJ1 b J¯1q “ 0. So we have J1 ‰ 0, yet

βpJ1 b J1q “ 0.

8 ˚ 8 Pick any nonzero x P J1. Then for any y P Cc pGq, we have yy P Cc pGq and so yy˚ P L2pGq. In addition, βpx b xqpyy˚q “ xyy˚x˚. So

xyy˚x˚ “ 0

78 8 ˚ ˚ 2 for all y P Cc pGq. The convolution xyy x is a smooth function in L pGq, and ˚ ˚ 2 pxyy x qpeq “ }xy}L2pGq. Here we use the fact that if f P L2pGq is any function then

˚ ˚ ´1 2 2 pf ˙ f qpeq “ fpgqf pg q dg “ fpgqfpgq dg “ |fpgq| dg “}f}L2pGq. ż ż ż 8 8 So xy “ 0 for all y P Cc pGq and hence x “ 0 in Cc pGq. This is a contradiction. Note that when we wrote xyy˚x˚, we meant βpx b xqpyy˚q P L2pGq. We used the fact that this is equal to the convolution of λpxqy P L2pGq with pλpxqyq˚ P L2pGq. That is

βpx b xqpyy˚q “ pλpxqyqpλpxqyq˚.

4.3 Remark. Given a Hilbert space H, its dual space H˚ is the space of all bounded linear functional from H to C. Also its complex conjugate space H is the same set as Hilbert space H with the following rules for addition, scalar multiplication and inner product:

v ` w :“ v ` w, αv :“ αv, xv, wyH :“ xw, vyH.

Its complex conjugate space H is the same as its dual space H˚ via the map v ÞÑ pw ÞÑ xw, vyq. Therefore, if π is a representation of G on H, we can ˚ ˚ identify a contragredient representation π of G on Hπ˚ “ H with a complex conjugate representation π of G on Hπ “ H, where

rπ˚pgqpϕqs pvq “ ϕpπpg´1qvq pϕ P H˚, v P Hq

πpgqv “ πpgqv pv P Hq.

We have now shown that certain representations of G ˆ G belong to the spectrum of ApGq. Our next goal is to show that certain other representations do not belong to the spectrum. We shall do this by studying the center of the group G.

79 ˚ 4.4 Theorem. If π1 b π2 is weakly contained in α then π1|zpGq “ π2|zpGq.

Proof. Let π1 and π2 be unitary representations of G on H1 and H2 respec- ˚ tively. Let v1 P H1 and v2 P H2. Assume that π1 b π2 is weakly contained 2 in bi-regular representation β. Therefore there are unit vectors fn P L pGq such that

xβpg1, g2qfn, fny Ñ xπ1pg1qv1, v1yxπ2pg2qv2, v2y which is

xβpg1, g2qfn, fny Ñ xπ1pg1qv1, v1yxv2, π2pg2qv2y (4.1) for all g1, g2 P G as n Ñ 8. For g1 “ g2 P zpGq, we have

´1 ´1 pβpg1, g2qfnqpγq “ fnpg1 γg2q “ fnpg1 γg1q “ fnpγq.

Also by Schur’s lemma, π1|zpGq “ z1 IdzpGq and π2|zpGq “ z2 IdzpGq, where z1, z2 are unit vectors in C. Consider (4.1) particularly when g1 “ g2 P zpGq, it becomes

xfn, fny Ñ xz1v1, v1yxv2, z2v2y as n Ñ 8 ñ z1z2 “ 1.

Multiplying z2 to both sides of the equation, we obtain z1 “ z2. In other words, π1|zpGq “ π2|zpGq.

4.3 The spectrum of ApGq for the Heisenberg group

In the previuos section we obtained some information about the spectrum ApGq. In this section, we shall use the fact that the spectrum is a closed subset of the dual of G ˆ G to complete the computation of the spectrum.

Let G be a Heisenberg group H1 and its Lie algebra g “ h1. Recall that there are two inequivalent classes of unitary irreducible representations

80 2 πc : G Ñ UpL pRqq and πab : G Ñ UpCq which are deﬁned by

icz ixt pπcpgqfqptq “ e e fpt ´ cyq

ipax`byq πabpgq “ e ,

˚ where a, b, c P R and for g “ px, y, zq P H1. We shall show that tπαβ b πγδ | ˚ α, β, γ, δ P Ru is weakly contained in tπc b πc | c P Rzt0uu. To do that, we need the following lemma.

4.5 Lemma. We have

2 αx2`βx`γ π ´ β `γ e dx “ ¨ e 4α , ´α żR c where α ă 0, β and γ are constants.

Proof. Note that

β 2 β2 αx2 ` βx ` γ “ α x ` ` ´ ` γ . 2α 4α ˆ ˙ ˆ ˙ β β2 We write δ “ 2α and “ ´ 4α ` γ. Then

2 2 2 eαx `βx`γ dx “ eαpx`δq ` dx “ e eαpx`δq dx. żR ż ż ? ? Let u “ ´αpx ` δq. Then du “ ´α dx and u2 “ ´αpx ` δq2. Then

2 2 du e 2 eαx `βx`γ dx “ e e´u ? “ ? e´u du ´α ´α żR ż ż 2 e ? π ´ β `γ “ ? π “ e 4α . ´α ´α ˆc ˙

2 ? The second to last equality uses the fact that e´x dx “ π. żR Before completing our calculation of the spectrum of ApGq, we prove a simpler result concerning the dual of G rather than the dual of G ˆ G.

4.6 Lemma. tπab | a, b P Ru is weakly contained in tπc | c P Rzt0uu.

81 2 2 Proof. Given pa, bq P R , it suﬃces to ﬁnd f P L pRq such that

xπcpgqf, fy Ñ xπabpgq1, 1y, for all g P G as c Ñ 0.

2 pa´tq b 4 2 ´ c ´i c t We claim that fptq “ πc ¨ e does the job. For g “ px, y, zq, we have b

xπcpgqf, fy “ pπcpgqfq ptqfptq dt ż “ eiczeixtfpt ´ cyqfptq dt ż 2 2 icz ixt 4 2 ´ pa´ptćyqq í b ptćyq 4 2 ´ pa´tq ì b t “ e e ¨ e c c ¨ e c c dt πc πc ż ˜c ¸ ˜c ¸ 2 2 2 ibyìcz ixt ´ pa´ptćyqq ´ pa´tq “ ¨ e e e c c dt. (4.2) πc c ż Note

pa ´ pt ´ cyqq2 pa ´ tq2 1 ixt ´ ´ “ ixt ´ pa ´ t ` cyq2 ` pa ´ tq2 c c c 1 “ ixt ´ `2pa ´ tq2 ` 2pa ´ tqcy `˘c2y2 c 1 “ ixt ´ `2a2 ´ 4at ` 2t2 ` 2acy ´ 2tcy˘ ` c2y2 c 2 ` 4a ˘ “ ´ t2 ` ` 2y ` ix t c c ˆ ˙ ˆ ˙ 2a2 ` ´ ´ 2ay ´ cy2 . c ˆ ˙ Now we apply Lemma 4.5,

4a 2 p `2yìxq 2a2 a t cy 2 a t 2 c 2 p ´p ´ qq p ´ q π ´ 2 `p´ c ´2ayćy q ixt ´ c ´ c 4p´ c q e e dt “ 2 ¨ e ż c c 2 2 πc c p 16a `4y2´x2` 16ay `4ixy` 8aix q´ 2a ´2ayćy2 “ ¨ e 8 c2 c c c 2 c 2 2 πc ´ cy ´ x c ` icxy ìax “ ¨ e 2 8 2 . (4.3) 2 c

82 Then we substitute (4.3) into (4.2),

2 2 2 ibyìcz πc ´ cy ´ x c ` icxy ìax xπ pgqf, fy “ ¨ e ¨ e 2 8 2 c πc 2 c c 2ˆ 2 ˙ ibyìax icz´ cy ´ x c ` icxy “ e e 2 8 2 ,

ipax`byq which converges to e “ xπabpgq1, 1y as c Ñ 0.

˚ 4.7 Proposition. tπαβ b πγδ | α, β, γ, δ P Ru is weakly contained in ˚ γδ αβ 2 2 tπc b πc | c P Rzt0uu. In other words, we can ﬁnd hc P L pRq b L pRq such that

˚ γδ αβ γδ αβ ˚ xpπcpg1q b πc pg2qq hc , hc y Ñ παβpg1qπγδpg2q, as c Ñ 0 for all g1, g2 P H1 .

αβ Proof. Let g1 “ px, y, zq and g2 “ pu, v, wq. Deﬁne fc as follows:

2 αβ 4 2 ´ pα´tq ´i β t f ptq “ ¨ e c e c . c πc c From the proof of Lemma 4.6, we have

2 2 αβ αβ iαx`iβy icz´ cy ´ x c ` icxy xπcpg1qfc , fc y “ e e 2 8 2 .

If we can show that

cv2 icuv u2c ˚ γδ γδ íγuíδv ícw´ ´ ´ xπc pg2qfc , fc y “ e e 2 2 8 , then we are done since

˚ αβ γδ αβ γδ pπcpg1q b πc pg2qqpfc b fc q, fc b fc

A αβ ˚ γδE αβ γδ “ πcpg1qfc b πc pg2qfc , fc b fc

A αβ αβ ˚ γδ γδ E “ πcpg1qfc , fc πc pg2qfc , fc

A cyEA2 2 icxy E iαxìβy icz´ ´ x c ` “ e e 2 8 2 ˆ ˙ 2 2 íγuíδv ícw´ cv ´ icuv ´ u c ¨ e e 2 2 8 ˆ ˙

83 iαxìβy íγuíδv ˚ which converges to e e “ παβpg1qπγδpg2q, as c Ñ 0. Now, let’s ˚ γδ find the formula for πc pg2qfc . Consider

˚ γδ γδ γδ γδ xπc pg2qfc , fc y “ xfc , πcpg2qfc y γδ γδ “ fc ptq πcpg2qfc ptq ż ´ ¯ γδ ícw íut γδ “ fc ptqe e fc pt ´ cvq dt ż 2 4 2 ´ pγ´tq í δ t ícw íut “ ¨ e c c e e πc ż ˜c ¸ 2 4 2 ´ pγ´ptćvqq ì δ ptćvq ¨ ¨ e c c dt πc ˜c ¸ 2 2 2 ícw íδv íut ´ pγ´t`cvq ´ pγ´tq “ ¨ e e e e c c dt. πc c ż Note that

pγ ´ t ` cvq2 pγ ´ tq2 íut´ ´ c c 1 “ íut ´ pγ ´ t ` cvq2 ` pγ ´ tq2 c 1 “ íut ´ `2pγ ´ tq2 ` 2pγ ´ tqcv `˘c2v2 c 1 “ íut ´ `2γ2 ´ 4γt ` 2t2 ` 2γcv ´ 2tcv˘ ` c2v2 c 2 ` 4γ 2γ2 ˘ “ ´ t2 ` ` 2v ´ iu t ` ´ ´ 2γv ´ cv2 . c c c ˆ ˙ ˆ ˙ ˆ ˙

84 Again we apply Lemma 4.5,

2 2 ´iut ´ pγ´t`cvq ´ pγ´tq e e c c dt

ż 4γ 2 2 p c `2víuq 2γ 2 π ´ 2 `p´ c ´2γvćv q 4p´ c q “ 2 ¨ e c c 2 2 πc c p 16γ ` 16γv ´ 8γiu `4v2´4viuú2q´ 2γ ´2γvćv2 “ ¨ e 8 c2 c c c 2 c 2 2 2 2 πc 2γ `2γvíγu` cv ´ icuv ´ u c ´ 2γ ´2γvćv2 “ ¨ e c 2 2 8 c 2 c 2 2 πc íγu´ cv ´ icuv ´ u c “ ¨ e 2 2 8 . 2 c Therefore

2 2 γδ γδ 2 ícw íδv πc íγu´ cv ´ icuv ´ u c xπ˚pg qf , f y “ ¨ e e ¨ e 2 2 8 c 2 c c πc 2 c 2ˆc 2 ˙ íγuíδv ícw´ cv ´ icuv ´ u c “ e e 2 2 8 , as we claimed above. So the proof is complete.

˚ ˚ 4.8 Corollary. For a, b, d P R, c P Rzt0u and c ‰ d, πc b πab, πab b πc and ˚ πc b πd are not in ApGq

Proof. The center zzpGq of Heisenberg group is tp0, 0, zq | z P Ru. By the deﬁnition, we have

1 0 z 1 0 z icz πc 0 1 0 “ e Id , πab 0 1 0 “ Id , $¨ ˛, zpGq $¨ ˛, zpGq & 0 0 1 . & 0 0 1 . ˝ ‚ ˝ ‚ for all z P%R. Therefore-πc ‰ πab for all nonzero% real number- c and πc ‰ πd if c ‰ d. Then this corollary immediately follows from Theorem 4.4.

Let’s summary what we have shown so far

˚ ˚ 1. All πc b πc and παβ b παβ are in the spectrum of ApGq.

85 ˚ ˚ ˚ 2. πc b πab, πab b πc and πc b πd are not in the spectrum of ApGq.

˚ 3. tπαβ b πγδ | α, β, γ, δ P Ru is weakly contained in ˚ tπc b πc | c P Rzt0uu.

Combine these results altogether, we have the following

˚ 4.9 Theorem. The spectrum of ApGq is the union of tπc b πc | c P zt0uu ˚ and tπαβ b πγδ | α, β, γ, δ P Ru.

4.10 Theorem. Let G be a Heisenberg group H1, ApGq is the closure of the set tπ b π˚ | c ‰ 0u in pG ˆ Gq. c c z {

86 Chapter 5

Structure of Apg˚q for the Heisenberg group

In this chapter we shall continue the theme of the previous chapter by calcu- lating the spectrum of Apg˚q and using the information obtained to decompose the C˚-algebra Apg˚q as an extension of more elementary C˚-algebras. Of course the results will parallel the work in the previous chapter, since the two C˚-algebras have been proved to be isomorphic in Chapter 3. But our methods here, while they follow those of the previous chapter, will be independent of them.

5.1 Alternative deﬁnition of Apg˚q

In this section, we shall explain how to view Apg˚q as the C˚-algebra generated by a unitary representation of the semidirect product group G ˙ g on L2pg˚q. We deﬁne the map α : G ˙ g Ñ UpL2pg˚qq by

αpg, Xqf “ α|Gpgq pα|gpXqfq ,

87 where g P G and X P g. For f P L2pg˚q and l P g˚,

˚ ilpXq pα|Gpgqfq plq “ fpAdg´1 lq, pα|gpXqfq plq “ e fplq,

˚ where Adg l “ lpAdg´1 q. Therefore, by the deﬁnition, we have

pαpg, Xqfq plq “ rα|Gpgq pα|gpXqfqs plq

˚ “ pα|gpXqfq pAdg´1 lq ipAd˚ lqpXq g´1 ˚ “ e fpAdg´1 lq ilpgXg´1q ˚ “ e fpAdg´1 lq.

5.1 Remark. As we pointed out in Chapter 3, this is an alternative deﬁ- nition of Apg˚q via a unitary isomorphism Fourier transform from L2pgq to L2pg˚q by f ÞÑ f, where

p fplq “ fpXqeilpXq dX. żg p We also write Fpfq for f. In other words, the following diagram commutes:

f P L2pGpq F f P L2pg˚q

Sg,X α p 2 F 2 ˚ Sg,X f P L pGq α ˝ F “ F ˝ Sg,X P L pg q

This is true by considering Fourier transform of Sg,X f. Recall that

´1 Sg,X f : Y ÞÑ fpg Y g ´ Xq.

Then for any l P g˚,

ilpY q ilpY q ´1 FpSg,X fqplq “ e pSg,X fqpY q dY “ e fpg Y g ´ Xq dY. żg żg

88 Let Z “ g´1Y g ´ X. Then dZ “ dY and Y “ gpZ ` Xqg´1 and so

ilpgpZ`Xqg´1q FpSg,X fqplq e fpZq dZ żg ´1 ´1 “ eilpgXg q eilpgZg qfpZq dZ żg ´1 i Ad˚ lpZq “ eilpgXg q e g´1 fpZq dZ żg ilpgXg´1q ˚ “ e fpAdg´1 lq

“ αpFfqplq.p

˚ ˚ ˚ 5.2 Remark. For g1, g2 P G, Ad ´1 Ad ´1 “ Adpg g q´1 To see this, consider g2 g1 1 2

˚ ˚ ˚ ´1 ´1 ´1 ˚ pAd ´1 Ad ´1 lqpXq “ Ad ´1 lpg2Xg2 q “ lpg1g2Xg1 g2 q “ Adpg g q´1 lpXq, g2 g1 g1 1 2 for any X P g.

5.3 Remark. α is a unitary representation. To see that αpg, Xq is homomorphism, we consider

´1 ilpg1X1g1 q ˚ rαpg1,X1qpαpg2,X2qfqs plq “ e pαpg2,X2qfqpAd ´1 lq g1 ˚ ´1 ´1 i Ad ´1 lpg2X2g2 q ilpg1X1g1 q g ˚ ˚ “ e e 1 fpAd ´1 Ad ´1 lq g2 g1 ´1 ´1 ilpg1X1g1 q ilpg1g2X2pg1g2q q ˚ ˚ “ e e fpAd ´1 Ad ´1 lq g2 g1 ´1 ´1 ilpg1X1g1 `g1g2X2pg1g2q q ˚ ˚ “ e fpAd ´1 Ad ´1 lq g2 g1 ´1 ´1 ilpg1g2pg2 X1g2`X2qpg1g2q ˚ e f Ad ´1 l “ p pg1g2q q ´1 “ pαpg1g2, g2 X1g2 ` X2qfqplq

“ pα ppg1,X1qpg2,X2qq fqplq.

Therefore, for any pg1,X1q, pg2,X2q P G ˙ g,

αpg1,X1qαpg2,X2q “ α ppg1,X1qpg2,X2qq .

89 Additionally αpg, Xq is a unitary map, which is αpg, Xq˚ “ αpg, Xq´1. In- deed, for any f, h P L2pg˚q,

xαpg, Xq˚f, hy “ xf, αpg, Xqhy

“ fplqαpg, Xqhplq dl ˚ żg ´ilpgXg´1q ˚ “ fplqe hpAdg´1 lq dl. ˚ żg ˚ ˚ ´1 Changing variables by setting k “ Adg´1 . Then l “ Adg k, and lpgXg q “ kpXq. Therefore

˚ ´ikpXq ˚ xαpg, Xq f, hy “ e fpAdg kqhpkq dk ˚ żg ikpg´1p´gXg´1qgq ˚ “ e fpAdg kqhpkq dk ˚ żg “ αpg´1, ´gXg´1qf pkqhpkq dk ˚ żg “ xαpg“´1, ´gXg´1qf, hy ‰

“ xαppg, Xq´1qf, hy “ xαpg, Xq´1f, hy.

From the remark above, we further obtain a representation

C˚pG ˙ gq Ñ BpL2pg˚qq.

The C˚-algebra Apg˚q is the image under this map.

5.2 Representations of G ˙ g

Let G be the Heisenberg group H1 and g “ h1 its Lie algebra. We shall use Mackey Machine to obtain all irreducible unitary representations of G ˙ g. Firstly, recall that all characters ϕ : g Ñ S1 are in the following form

ipau`bv`cwq ϕabcpru, v, wsq “ e , for ru, v, ws P g.

90 ˚ ˚ 5.4 Remark. Let l P g and tX,Y,Zu a basis of H1. Then labc “ aX ` ˚ ˚ ˚ ˚ ˚ ˚ bY ` cZ where X ,Y ,Z are dual basis of H1 . Also the character ϕabc corresponds to labc by

ilabc ϕabc “ e .

Next two lemmas will be used to ﬁnd the stabilizer subgroup of G with respect to ϕabc for all a, b, c P R.

5.5 Lemma. For g “ px, y, zq P G, we have

˚ Adg ϕabc “ ϕa`cy,b´cx,c

˚ Adg´1 ϕabc “ ϕa´cy,b`cx,c.

Proof. For u, v, w P R, we have

˚ ´1 Adg ϕabcpru, v, wsq “ ϕabc g ru, v, wsg

“ ϕabc `pp´x, ´y, xy ´˘ zqru, v, wspx, y, zqq

“ ϕabcpru, v, uy ` w ´ xvsq

“ eipau`bv`cpuy`w´xvqq

“ eippa`cyqu`pb´cxqv`cwq

“ ϕa`cy,b´cx,cpru, v, wsq.

The second equation can be shown similarly.

Now we introduce the notation » to denote orbit equivalence in g˚. That is, if l1 and l2 are in the same coadjoint orbit then we write l1 » l2. In other ˚ words, l2 “ Adg pl1q for some g P G.

5.6 Lemma. For any nonzero c, ϕabc » ϕa1b1c. In particular, ϕabc » ϕ00c.

91 ˚ Proof. If ϕabc » ϕa1b1c1 then ϕa1b1c1 “ Adg ϕabc. Therefore, for all u, v, w P R, we have ˚ ϕa1b1c1 pru, v, wsq “ Adg ϕabcpru, v, wsq.

Applying previous lemma with g “ px, y, zq P G, we have

a1u ` b1v ` c1w “ pa ` cyqu ` pb ´ cxqy ` cw.

This is possible if and only if c “ c1. If both are zero, we then have a “ a1 and b “ b1 which is not an interesting case. If c “ c1 are nonzero, then we have, a1 ´ a b ´ b1 a ` cy “ a1, b ´ cx “ b1 ñ y “ , x “ . c c

Therefore ϕabc » ϕa1b1c, as desired. The second statement is immediate followed.

Now we are ready to classify all of the stabilizer subgroup of G with respect to ϕabc.

5.7 Lemma. Let a, b, c P R. The stabilizer subgroups of G with respect to

ϕabc are zpGq “ tp0, 0, zq | z P Ru if c ‰ 0 Gϕabc “ #G if c “ 0. Proof. By the deﬁnition of stabilizer subgroup and previous lemma,

˚ Gϕabc “ tg P G | ϕabcpXq “ Adg ϕabcpXq for all X P gu

´1 “ tg P G | ϕabcpXq “ ϕabcpg Xgq for all X P gu

´1 “ tg P G | ϕabcpru, v, wsq “ ϕabcpg ru, v, wsgq for all ru, v, ws P gu

ipau`bv`cwq ippa`cyqu`pb´cxqv`cwq “ px, y, zq P G | e “ e for all u, v, w P R ! ) “ tpx, y, zq P G | a “ a ` cy, b “ b ´ cx for all u, v, w P Ru “ tpx, y, zq P G | cy “ 0, cx “ 0u .

92 If c “ 0, then there is no condition on x, y, z. So Gϕabc “ G. If c ‰ 0, then cy “ cx “ 0 implies y “ x “ 0. Therefore Gϕabc “ zpGq.

Recall that the inequivalent unitary irreducible representations of G are of the form

ipγz`xtq pπγppx, y, zqqfq ptq “ e fpt ´ γyq

ipαx`βyq παβppx, y, zqq “ e .

Now we apply Mackey machine process to obtain all representations on G ˙ g. The results are shown below.

5.8 Theorem. The unitary dual of G ˙ g consists of the following three families of irreducible unitary representations:

IndG˙g π , where λ is any real number and c is any nonzero real • Gϕ˙g λ,c

number, and πλ,c is the one-demensional representation given by

ipλz`cwq πλ,cpp0, 0, zq, ru, v, wsq “ e ,

for any p0, 0, zq P zpGq, ru, v, ws P g. Here ϕ “ ϕ00c.

2 • πγ,ab, where γ, a, b are any real numbers, is the representation on L pRq given by

ipγz`xtq ipau`bvq rπγ,abppx, y, zq, ru, v, wsqfs ptq “ e e fpt ´ γyq,

2 for any px, y, zq P G, ru, v, ws P g, f P L pRq, and t P R.

• παβ,ab, where α, β, a, b are any real numbers, is the one-dimensional representation given by

ipαx`βyq ipau`bvq παβ,abppx, y, zq, ru, v, wsq “ e e ,

for any px, y, zq P G, ru, v, ws P g.

93 In other words, G g IndG˙g π , π , π . ˙ “ t Gϕ˙g λ,c γ,ab αβ,abu

Proof. If c ‰ 0,{ by previous lemma, Gϕabc “ zpGq “ tp0, 0, zq | z P Ru. In λ this case, Gϕabc “ zpGq “ tπ u, where

λ iλz { zπ pp0, 0, zqq “ e , for all z P R.

Since ϕabc and ϕ00c are in the same orbit by Lemma 5.6, it follows that λ λ π b ϕabc is equivalent to π b ϕ00c. We shall use the latter for simplicity. For g “ p0, 0, zq P zpGq and X “ ru, v, ws P g, then

λ λ ipλz`cwq pπ b ϕ00cqpg, Xq “ π pgqϕabcpXq “ e .

λ We write πλ,c for π b ϕ00c from now on. Now let’s consider when c “ 0, we learn from previous lemma that Gϕabc “ G. Then for g “ px, y, zq P G and X “ ru, v, ws P g, we have

ipγz`xtq ipau`bvq rpπγ b ϕab0qpg, Xqfs ptq “ rπγpgqϕab0pXqfsptq “ e fpt ´ γyqe

ipαx`βyq ipau`bvq pπαβ b ϕab0qpg, Xq “ παβpgqϕab0pXq “ e e .

We write πγ,ab for πγ b ϕab0 and παβ,ab for pπαβ b ϕab0q. Then induced representations of πλ,c, πγ,ab, παβ,ab from Gϕ ˙ g to G ˙ g form the complete list of representations on G ˙ g up to equivelence class by Mackey machine process.

5.3 Spectrum of Apg˚q for the Heisenberg group

The next question we consider is “Which irreducible unitary representations of G ˙ g are weakly contained in α?” First we shall exclude some representations. The following calculation is analogous to Theorem 4.4.

5.9 Theorem. If π P G ˙ g is weakly contained in α then

π pp{0, 0, zq, 0q “ Id, for all z P R.

94 Proof. Let π be weakly contained in α. Then there are unit vectors fn and v in L2pg˚q satisfying

xαpg, Xqfn, fny Ñ xπpg, Xqv, vy as n Ñ 8 for all g P G, X P g. Let g “ p0, 0, zq P zpGq and X “ 0. Then we have,

˚ 2 xαpg, Xqfn, fny “ fnpAdg ξqfnpξq dξ “ fnpξqfnpξq dξ “}fn}L2pg˚q “ 1, ˚ ˚ żg żg for all n. On the other hand, π|zpGq is a character. Therefore, by Schur’s lemma, π pg, Xq “ cz Id . Consequently,

xπpg, Xqv, vy “ czxv, vy “ cz.

By the deﬁnition of weak containment above, cz “ 1. Therefore

π pp0, 0, zq, 0q “ Id, as desired.

5.10 Corollary. πγ,ab is not weakly contained in α.

iγz Proof. Since πγ bπab0 pp0, 0, zq, 0q “ e , which is not the identity. Therefore

πγ,ab is not weakly contained in α from theorem above.

Next we consider whether IndG˙g π is weakly contained in α by us- Gϕ˙g λ,c ing the same criteria. In order to to resolve this issue, we need to ﬁnd a formula for the action of G ˙ g in the induced representation. Recall that the multiplication in G ˙ g is deﬁned by

´1 pg1,X1qpg2,X2q “ pg1g2, g2 X1g2 ` X2q, where pg1,X1q, pg2,X2q P G ˙ g.

95 Now let G be the Heisenberg group H1 and g “ h1 its Lie algebra.

Let g1 “ px1, y1, z1q, g2 “ px2, y2, z2q P G and X1 “ ru1, v1, w1s,X2 “ ru2, v2, w2s P g. Then

´1 pg1,X1qpg2,X2q “ pg1g2, g2 X1g2 ` X2q ´1 “ ppx1, y1, z1qpx2, y2, z2q, px2, y2, z2q ru1, v1, w1spx2, y2, z2q ` ru2, v2, w2sq

“ ppx1 ` x2, y1 ` y2, z1 ` z2 ` x1y2q, p´x2, ´y2, x2y2 ´ z2qru1, v1, w1spx2, y2, z2q

` ru2, v2, w2sq

“ ppx1 ` x2, y1 ` y2, z1 ` z2 ` x1y2q, ru1, v1, ´x2v1 ` w1spx2, y2, z2q ` ru2, v2, w2sq

“ ppx1 ` x2, y1 ` y2, z1 ` z2 ` x1y2q, ru1, v1, u1y2 ` w1 ´ x2v1s ` ru2, v2, w2sq

“ ppx1 ` x2, y1 ` y2, z1 ` z2 ` x1y2q, ru1 ` u2, v1 ` v2, w1 ` w2 ` u1y2 ´ x2v1sq.

With x1 “ x, y1 “ y, z2 “ z, u2 “ u, v2 “ v, w2 “ w and setting the rest equal to zero, we obtain the following identity:

ppx, y, zq, ru, v, wsq “ ppx, y, 0q, r0, 0, 0sqpp0, 0, zq, ru, v, wsq, (5.1) where ppx, y, 0q, r0, 0, 0sq P G ˙ g and pp0, 0, zq, ru, v, wsq P Gϕ ˙ g. It is easy to check that pg, Xq´1 “ pg´1, ´gXg´1q. Therefore

pp0, 0, zq, ru, v, wsq´1 “ pp0, 0, zq´1, ´p0, 0, zqru, v, wsp0, 0, zq´1q

“ pp0, 0, ´zq, ´p0, 0, zqru, v, wsp0, 0, ´zqq

“ pp0, 0, ´zq, ´ru, v, wsp0, 0, ´zqq

“ pp0, 0, ´zq, ´ru, v, wsq

6 pp0, 0, zq, ru, v, wsq´1 “ pp0, 0, ´zq, r´u, ´v, ´wsq. (5.2)

By deﬁnition, the induced representation acts on a Hilbert completion of the space W of continuous functions from G ˙ g to C satisfying

´1 fppg1,X1qpg2,X2qq “ πλ,cppg2,X2q qfppg1,X1qq,

96 for every pg1,X1q P G ˙ g and pg2,X2q P Gϕ ˙ g. Using (5.1) and (5.2), we change this condition to

fpppx, y, zq, ru, v, wsqq “ fpppx, y, 0q, r0, 0, 0sqpp0, 0, zq, ru, v, wsqq

´1 “ πλ,cppp0, 0, zq, ru, v, wsq qfppx, y, 0q, r0, 0, 0sq

“ πλ,cpp0, 0, ´zq, r´u, ´v, ´wsqfppx, y, 0q, r0, 0, 0sq

“ eipλp´zq`cp´wqqfppx, y, 0q, r0, 0, 0sq

“ e´ipλz`cwqfppx, y, 0q, r0, 0, 0sq. (5.3)

Now we write σ IndG˙g π . By the construction of induced representa- “ Gϕ˙g λ,c tion, σ acts on W by

rσppg, Xqqfs ph, Y q “ f pg, Xq´1ph, Y q

“ f `pg´1, ´gXg´1qp˘ h, Y q

“ f `pg´1h, h´1p´gXg´1qh˘ ` Y q

6 rσppg, Xqqfs ph, Y q “ f `pg´1h, ´h´1pgXg´1qh ` Y q˘ , (5.4) ` ˘ for pg, Xq,ph, Y q P G ˙ g and f P W . In the Heisenberg group, we write now g “ px, y, zq, h “ px1, y1, z1q P G and X “ ru, v, ws,Y “ ru1, v1, w1s P g. Then

g´1h “ px, y, zq´1px1, y1, z1q

“ p´x, ´y, xy ´ zqpx1, y1, z1q

“ px1 ´ x, y1 ´ y, z1 ´ z ` xy ´ xy1q

“ px1 ´ x, y1 ´ y, z1 ´ z ´ xpy1 ´ yqq. (5.5)

97 And also

´ h´1pgXg´1qh ` Y

“ ´px1, y1, z1q´1 px, y, zqru, v, wspx, y, zq´1 px1, y1, z1q ` ru1, v1, w1s

“ ´p´x1, ´y1, x1`y1 ´ z1q pru, v, xv ` wsp´x,˘´y, xy ´ zqq px1, y1, z1q ` ru1, v1, w1s

“ ´p´x1, ´y1, x1y1 ´ z1qru, v, ´uy ` xv ` wspx1, y1, z1q ` ru1, v1, w1s

“ ´ru, v, ´uy ` xv ´ x1v ` wspx1, y1, z1q ` ru1, v1, w1s

“ ´ru, v, uy1 ´ uy ` xv ´ x1v ` ws ` ru1, v1, w1s

“ r´u, ´v, ´uy1 ` uy ´ xv ` x1v ´ ws ` ru1, v1, w1s

“ ru1 ´ u, v1 ´ v, w1 ´ uy1 ` uy ´ xv ` x1v ´ ws

“ ru1 ´ u, v1 ´ v, w1 ´ w ´ upy1 ´ yq ` vpx1 ´ xqs. (5.6)

Combine (5.3) ´ (5.6) altogether, we obtain rσppg, Xqqfs ph, Y q

“ fppx1 ´ x, y1 ´ y, z1 ´ z ´ xpy1 ´ yqq, ru1 ´ u, v1 ´ v, w1 ´ w ´ upy1 ´ yq ` vpx1 ´ xqsq

1 1 1 1 1 “ eítλpz ´z´xpy ýqq`cpw ´wúpy ýq`vpx ´xqqufppx1 ´ x, y1 ´ y, 0q, r0, 0, 0sq, (5.7) where g “ px, y, zq, h “ px1, y1, z1q P G, and X “ ru, v, ws,Y “ ru1, v1, w1s P g. Particularly, if g1 “ p0, 0, zq and X “ 0 (zero matrix),

´itλpz1´zq`cw1u 1 1 rσppg1,Xqqfs ph, Y q “ e fppx , y , 0q, r0, 0, 0sq. (5.8)

Also with g2 “ p0, 0, 0q (identity matrix) and X “ 0, we have

´itλz1`cw1u 1 1 rσppg2,Xqqfs ph, Y q “ e fppx , y , 0q, r0, 0, 0sq. (5.9)

But (5.4) tells us that

´1 ´1 ´1 rσppg2,Xqqfs ph, Y q “ f pg2 h, ´h pg2Xg2 qh ` Y q “ fpph, Y qq, ` ˘ 98 which means σppg2,Xqq “ Id. Therefore, by combining (5.8) and (5.9),

iλz iλz rσppg1,Xqqfs ph, Y q “ e rσppg2,Xqqfs ph, Y q “ e fpph, Y qq.

Therefore, we obtain

5.11 Lemma. IndG˙g π 0, 0, z , 0 eiλz, where ϕ ϕ . Gϕ˙g λ,c pp q q “ “ 00c

By applying Theorem 5.9,

5.12 Corollary. IndG˙g π is not weakly contained in α when λ 0. Gϕ˙g λ,c ‰

Having excluded certain irreducible unitary representations of G˙g from the spectrum of Apg˚q we shall now prove that certain others do belong to the spectrum. First we shall show that IndG˙g π is weakly contained in Gϕ˙g 0,c ˚ α. In order to prove this, we introduce new notation Oc Ă g . It is deﬁned by

Oc “ tlabc | a, b P Ru.

It corresponds to the coadjoint orbit of ϕ00c which is tϕabc | a, b P Ru by

Lemma 5.6 and Remark 5.4. Also the map πOc is deﬁned by using the same 2 formula as α. Therefore πOc : G ˙ g Ñ UpL pOcqq is deﬁned by

πOc pg, Xqf “ πOc |Gpgq pπOc |gpXqfq ,

2 where f P L pOcq. More explicitly, we have

ilpgXg´1q ˚ rπOc pg, Xqfs plq “ e fpAdg´1 lq.

From now on we write pa, bq for labc since c is ﬁxed. If g “ px, y, zq P G and X “ ru, v, ws P g, then by Lemma 5.5

ipau`bv`cp´uy`w`xvqq rπOc pg, Xqfs pa, bq “ e fpa ´ cy, b ` cxq. (5.10)

We shall prove the following two results.

99 5.13 Theorem. πOc is weakly contained in α.

2 Proof. Fix c0 P Rzt0u. Given h P L pOc0 q with }h}“ 1, we deﬁne its extension on g˚ by

hplq “ hplqϕ plpr0, 0, 1sq ´ c0q ,

` 2 where ϕ : Ñ hasr support on |c ´ c0| ă and ϕpxq dx “ 1. In R R R particular, if l “ labc, we have ş

hpa, b, cq “ hpa, b, c0qϕpc ´ c0q.

Here we write pa, b, cq rfor labc. We shall show that

α g, X h , h π g, X h, h , (5.11) x p q y Ñ x Oc0 p q y for all g P G and X P g, whenr r Ñ 0. Let g “ px, y, zq P G and X “ ru, v, ws P g. Similar to what we have done in (5.10),

ipau`bv`c0p´uy`w`xvqq rπOc pg, Xqhs pa, b, cq “ e hpa ´ c0y, b ` c0x, c0q.

Therefore the RHS of (5.11) is

ipau`bv`c0p´uy`w`xvqq e hpa ´ c0y, b ` c0x, c0qhpa, b, c0q da db. ĳ On the other hand,

ipau`bv`cp´uy`w`xvqq αpg, Xqh pa, b, cq “ e hpa ´ cy, b ` cx, c0qϕpc ´ c0q. ” ı by the deﬁnitionr of α and h. As a result, the LHS of (5.11) is

ipau`bv`cp´uy`w`xvqq 2 e hrpa ´ cy, b ` cx, c0qhpa, b, c0qϕpc ´ c0q da db dc. ¡ Therefore α g, X h , h π g, X h, h is x p q y ´ x Oc0 p q y

ipau`bvq r r 2 icp´uy`w`xvq e hpa, b, c0qϕpc ´ c0q e hpa ´ cy, b ` cx, c0q ¡ ic0p´uy`w`xvq ´ e “ hpa ´ c0y, b ` c0x, c0q da db dc ‰ 100 By Cauchy-Schwarz inequality,

α g, X h , h π g, X h, h 2 A B, (5.12) |x p q y ´ x Oc0 p q y| ď ¨ where r r

ipau`bvq 2 2 A “ |e hpa, b, c0q| ϕpc ´ c0q da db dc ¡ 2 icp´uy`w`xvq B “ ϕpc ´ c0q e hpa ´ cy, b ` cx, c0q ¡ ic0p´uy`wˇ `xvq 2 ´ e ˇ hpa ´ c0y, b ` c0x, c0q da db dc.

2 2 ˇ Note that A “ |hpa, b, c0q| da db ϕpc ´ c0q dcˇ “ 1. We can as- ˆ8ĳ ˙ ˆż ˙ sume that h P Cc pOc0 q. Then the function

icp´uy`w`xvq c ÞÑ pa, bq ÞÑ e hpa ´ cy, b ` cx, c0q ” ı is continuous. So for any η ą 0, there is ą 0 so that if |c ´ c0| ă ,

icpúy`w`xvq ic0púy`w`xvq 2 e hpaćy, b`cx, c0qé hpać0y, b`c0x, c0q da db ă η. ĳ ˇ ˇ ˇ 2 ˇ Hence B ă η ϕpc ´ c0q dc “ η. Therefore πOc is weakly contained in α by (5.12). ż

5.14 Theorem. IndG˙g π is unitarily equivalent to π . Gϕ˙g 0,c Oc

Proof. By the definition of induced representation, the Hilbert space of 1 a 0 G˙g 2 Ind π0,c can be identified with L 0 1 b , a, b P R (equiva- Gϕ˙g ¨$¨ ˛ ,˛ & 0 0 1 . 2 2 lently L pR q). This is true since there˝ is an˝ isomorphism‚ from G˙‚g{Gϕ ˙g 1 a 0 % - onto 0 1 b , a, b P R via $¨ ˛ , & 0 0 1 . ˝ ‚ % 1 a 0 -0 0 0 1 a 0 0 1 b , 0 0 0 Gϕ ˙ g Ð 0 1 b . ¨¨0 0 1˛ ¨0 0 0˛˛ [ ¨0 0 1˛ ˝˝ ‚ ˝ ‚‚ ˝ ‚ 101 1 a 0 We write a, b for 0 1 b and σ for IndG˙g π . Let g x, y, z G p q Gϕ˙g 0,c “ p q P ¨0 0 1˛ and X “ ru, v, ws P˝g. With new‚ identification together with (5.4),

rσppg, Xqqfs pa, bq

“ f g´1pa, b, 0q, ´pa, b, 0q´1pgXg´1qpa, b, 0q

“ f `px, y, zq´1pa, b, 0q, ´pa, b, 0q´1ru, v, ´uy˘` xv ` wspa, b, 0q

“ f pp´` x, ý, xy ´ zqpa, b, 0q, ´pá, ´b, abqru, v, úy ` xv ` w˘` ubsq

“ f ppa ´ x, b ´ y, xy ´ z ´ xbq, r´u, ´v, ´w ` uy ´ ub ` av ´ xvsq .

Then we apply (5.3) with λ “ 0 to obtain

rσpg, Xqfs pa, bq “ eip´cav`cbu´cuy`cw`cxvqfpa ´ x, b ´ yq.

2 2 2 2 after simpliﬁcation. Now we deﬁne a map U from L pR q to L pR q by

pUfqpa, bq “ c´1fp´c´1b, c´1aq.

It is straightforward to check that its inverse U ´1 is

pU ´1fqpa, bq “ cfpcb, ´caq.

Moreover U is a unitary map since U ˚ “ U ´1. To see this, consider

xU ˚f, gy “ xf, Ugy

“ fpa, bqc´1gpć´1b, c´1aq da db ż “ fpcy, ćxqc´1gpx, yqc2 dx dy ż “ cfpcy, ćxqgpx, yq dx dy ż “ xU ´1f, gy.

102 Moreover U intertwines between σ and πOc since

Uσpg, XqU ´1f pa, bq “ c´1rσpg, XqU ´1fsp´c´1b, c´1aq

“ ‰ “ c´1 eipbvàućuy`cw`cxvqU ´1fpć´1b ´ x, c´1a ´ yq

“ c´1 ”eipbv`au´cuy`cw`cxvqcfpa ´ cy, b ` cxq ı

“ eipbv”`au´cuy`cw`cxvqfpa ´ cy, b ` cxq ı

“ rπOc pg, Xqfspa, bq, for all a, b 2. Consequently IndG˙g π is unitarily equivalent to π . p q P R Gϕ˙g 0,c Oc

Combine two theorems above together, we get the result which is roughly analogous to Theorem 4.2.

5.15 Corollary. IndG˙g π is weakly contained in α. Gϕ˙g 0,c

Then we shall give the relationship between two representations in Apg˚q.

5.16 Theorem. tπαβ,γδ | α, β, γ, δ P Ru is weakly contained in tπOc | c ‰ 0u and therefore it is weakly contained in IndG˙g π c 0 . t Gϕ˙g 0,c | ‰ u

2 Proof. Deﬁne a function fc P L pOcq by

2 2 2 iαb ´ iβa ´ pa´γq ´ pb´δq f pa, bq “ ¨ e c c e c c , c πc c for a, b P R. Therefore, we have

2 2 2 iαpb`cxq ´ iβpa´cyq ´ pa´cy´γq ´ pb`cx´δq f pa ´ cy, b ` cxq “ ¨ e c c e c c . c πc c

103 Let g “ px, y, zq P G and X “ ru, v, ws P g. Then

xπOc ppg, Xqq fc, fcy (5.13)

“ rπOc pg, Xqfcs pa, bqfcpa, bq da db ĳ ipau`bv`cp´uy`w`xvqq “ e fcpa ´ cy, b ` cxqfcpa, bq da db

ĳ 2 2 2 ipαx`βyq icp´uy`w`xvq 2 iau ´ pa´γq ´ pa´cy´γq “ e e e e c c da πc ˜c ¸ ˆż ˙ 2 2 ibv ´ pb´δq ´ pb`cx´δq e e c c db . (5.14) ˆż ˙ By using similar arguments in the proof of Lemma 4.6, we obtain

2 2 2 2 iau ´ pa´γq ´ pa´cy´γq πc ´ cy ´ u c ` icuy `iγu e e c c da “ ¨ e 2 8 2 , (5.15) 2 ż c and

2 2 2 2 ibv ´ pb´δq ´ pb`cx´δq πc ´ cx ´ v c ´ icvx `iδv e e c c db “ ¨ e 2 8 2 . (5.16) 2 ż c Now we substitute (5.15) and (5.16) in (5.14),

xπOc ppg, Xqq fc, fcy 2 2 2 2 ipαx`βyq icpúy`w`xvq ´ cy ´ u c ` icuy ìγu ´ cx ´ v c ´ icvx ìδv “ e e e 2 8 2 e 2 8 2 2 2 2 2 ipαx`βyq ipγu`δvq icpúy`w`xvq´ cy ´ u c ` icuy ´ cx ´ v c ´ icvx “ e e e 2 8 2 2 8 2 .

ipαx`βyq ipγu`δvq It converges to e e “ xπαβ,γδpg, Xq1, 1y, as c Ñ 0.

Consequently, we obtain the following theorem for Apg˚q which is analogous to Theorems 4.9 and 4.10 for ApGq.

5.17 Theorem. The spectrum of A g˚ is the union of IndG˙g π c p q t Gϕ˙g 0,c | ‰

0u and tπαβ,γδ | α, β, γ, δ P Ru.

104 5.18 Theorem. The spectrum of Apg˚q is the closure of the set

IndG˙g π c 0 . t Gϕ˙g 0,c | ‰ u

5.19 Remark. The underlying idea is that

π π˚ A G corresponds to IndG˙g π A g˚ ; c b c P p q Gϕ˙g 0,c P p q

˚ ˚ παβ b πγδ P AzpGq corresponds to παβ,ab P A{pg q.

z {

105 Chapter 6

Summary and directions for future work

In this thesis we have formulated and examined a conjecture in C˚-algebras and representation theory that is inspired by Kirillov’s orbit theorem for irreducible unitary representations of nilpotent groups, but diﬀerent from it. We constructed two C˚-algebras, one related to the decomposition of L2pGq into irreducible representations of G ˆ G, and one related to the decomposition of g˚ into coadjoint orbits. Our conjecture is that the two C˚-algebras are isomorphic. We proved the conjecture for the Heisenberg group, and examined it further detail in that case. When G is the Heisenberg group there is a natural, Lie-theoretic proof of the conjecture using the exponential map. This seems to be very special to the Heisenberg group, and so we began the program of examining the conjecture from a more detailed representation-theoretic perspective. We explained how (in the case of the Heisenberg group) both C˚-algebras decompose into simpler parts through C˚-algebra extensions. Naturally we expect to see the same structure in more general cases, and the ﬁrst example to be considered in future work will be the four-dimensional group K3

106 introduced in Chapter 2. After these calculations are done, we can hope to use the theory of C˚- algebra extensions (initiated by Brown Douglas and Fillmore and developed much more fully by Kasparov) to settle our isomorphism conjecture in at least some new cases. The idea here is that there are very few ways of re- assembling a C˚-algebra from the component parts that appear in extensions of the sort that are seen in the analysis of ApGq and Apg˚q. But we have scarcely begun to examine the issues that are involved here in any detail. Much work remains to be done.

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111 Vita

Pichkitti Bannangkoon

Born: August 17, 1984, in Phatthalung, Thailand

Scholarship: The Development and Promotion of Science and Technology Talents Project (DPST) from 1999-2014

Education:

(2015) Ph.D. (Mathematics), Pennsylvania State University Adviser: Prof. Nigel Higson

(2009) M. Sc. (Mathematics), Chulalongkorn University, Thailand Advier: Prof. Wicharn Lewkeeratiyutkul Co Adviser: Prof. Paolo Bertozzini

(2006) B. Sc. (Mathematics), Prince of Songkla University, Thailand Adviser: Prof. Jantana Ayaragarnchanakul Co Adviser: Prof. Sutep Suantai