Extremely Amenable Groups and Banach Representations

Home , Additive group, Amenable group, Hemicompact space, Operator topologies, Selection theorem, Uniform isomorphism

A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree Doctor of Philosophy

This dissertation titled Extremely Amenable Groups and Banach Representations

by JAVIER RONQUILLO RIVERA

has been approved for the Department of Mathematics and the College of Arts and Sciences by

Vladimir Uspenskiy Professor of Mathematics

Robert Frank Dean, College of Arts and Sciences 3 Abstract

RONQUILLO RIVERA, JAVIER, Ph.D., May 2018, Mathematics Extremely Amenable Groups and Banach Representations (125 pp.) Director of Dissertation: Vladimir Uspenskiy A long-standing open problem in the theory of topological groups is as follows:

[Glasner-Pestov problem] Let X be compact and Homeo(X) be endowed with the compact-open topology. If G ⊂ Homeo(X) is an abelian group, such that X has no G-ﬁxed points, does G admit a non-trivial continuous character? In this dissertation we discuss some reformulations of this problem and its connections to other mathematical objects such as extremely amenable groups.

When G is the closure of the group generated by a single map T ∈ Homeo(X) (with respect to the compact-open topology) and the action of G on X is minimal, the existence of non-trivial continuous characters of G is linked to the existence of equicontinuous factors of (X,T ). In this dissertation we present some connections between weakly mixing dynamical systems, continuous characters on groups, and the space of maximal chains of subcontinua of a given compact space. Abelian topological groups that have no non-trivial continuous characters are known as minimally almost periodic. We show a proof that the space

Lp[0, 1], 0 < p < 1 is minimally almost periodic. Also, using the theory of compactifications of G we show that for minimally almost periodic groups the greatest ambit S(G) and the WAP compactification do not match. This suggest that minimally almost periodic groups might be a good place to look for abelian topological groups for which the WAP compactification is trivial. It is still unknown whether such abelian groups exist. 4

Para Tono y Tutu 5 Table of Contents

Page

Abstract...... 3

Dedication...... 4

1 Introduction...... 7

2 Unitary representations of topological groups...... 19 2.1 Basic deﬁnition...... 20 2.1.1 Uniform Spaces...... 21 2.1.2 Actions and representations...... 26 2.1.3 Teleman’s theorem...... 28 2.1.4 Unitary representations...... 32 2.1.5 Banach algebras...... 35 2.2 Compact and locally compact groups...... 38 2.3 Free abelian topological group...... 42 2.4 Group of isometries of an inﬁnite-dimension Hilbert space...... 48 2.5 Extremely amenable groups...... 51 2.6 Glasner-Pestov problem...... 54 2.6.1 Syndetic sets and Bohr topology...... 55 2.6.2 Long almost constant pieces...... 57 2.7 A minimally almost periodic group...... 58

3 Compactifications and free actions...... 62 3.1 Uniformities, compactifications and C∗-algebras...... 62 3.2 Natural uniformities for topological groups...... 65 3.2.1 The greatest ambit...... 68 3.2.2 Universal minimal compact G-space...... 69 3.2.3 Roelcke precompactness...... 72 3.3 More compactifications on topological groups...... 74 3.3.1 WAP compactification...... 75 3.3.2 Eberlein groups...... 78 3.3.3 When is S(G) = W (G)?...... 80 3.4 Free actions on compact spaces...... 82 3.5 Applications to the Glasner-Pestov problem...... 84

4 Non-weakly mixing topological dynamical systems...... 87 4.1 Topological dynamical systems...... 87 4.1.1 Basic deﬁnitions and examples...... 87 4.1.2 Recurrence...... 92 6

4.1.3 Kronecker factor...... 96 4.1.4 Characterizations of existence of non-trivial equicontinuous factors...... 101 4.2 Non-trivial equicontinuous factors and existence of non-trivial characters106 4.3 Vietoris topology and the space of maximal chains...... 108 4.3.1 Vietoris topology...... 108 4.3.2 On the action of G on MG ...... 112 4.4 The space of maximal chains of subcontinua...... 115 4.5 Applications to the Glasner-Pestov problem...... 118

5 Future work...... 121

References...... 123 7 1 Introduction

When one goes for a walk in the domain of topological groups, it is easy to find a vast variety of them. For example, they emerge as groups of invertible bounded operators of Banach or Hilbert spaces, groups of homeomorphisms of topological spaces, groups of diffeomorphisms of manifolds, groups of ergodic maps, or groups of isometries of metric spaces. Regardless of the origin of a topological group it results of particular interest the question about what kind of topological representations such topological groups may have. After all, being able to understand a topological group as a subgroup of automorphisms of certain object might add to the tools and techniques available to understand it. In the case of topological groups a representation φ of a topological group G on a compact space X is continuous, if the associated action φ ∶ G × X → X is continuous. On the other hand, a representation φ of a topological group G on a normed space E is (strongly) continuous, if the associated action φ ∶ G × E → E is continuous. A very nice starting point in the representation theory of Hausdorff topological groups is given by Teleman’s theorem:

Theorem. (Teleman) Every Hausdorﬀ topological group admits:

1. A (strongly) continuous faithful representation on a Banach space X by linear isometries. Equivalently the topological group is topologically isomorphic to a subgroup of isometries of X, endowed with the strong-operator topology.

2. A continuous faithful representation by homeomorphisms on a compact space K. Equivalently the topological group is topologically isomorphic to a subgroup of homeomorphisms of K, endowed with the compact-open topology. 8

Given this fact it is natural to ask whether more is possible, that is, whether topological groups can be represented faithfully and continuously acting on a Hilbert space by unitary operators (which will be referred as having a faithful unitary representation). An affirmative answer to this question can lead to the study of topological groups through mechanisms from the theory of C∗−algebras and in the case of abelian topological groups it can lead to the developing of a rich duality theory through which the topological group can be better understood. It turns out that not every topological group has a faithful unitary representation, some topological group have none non-trivial unitary representation. This problem has been studied for many classes of topological groups, in the case of compact topological groups the Peter- Weyl theorem asserts that the direct sum of the finite dimensional (the Hilbert space on which it acts upon is finite-dimensional) unitary representations of a compact topological group is isomorphic to the left-regular representation of G on the Hilbert space L2(G) induced by the Haar measure on G. The left-regular representation is a unitary faithful representation of G. An irreducible unitary representation φ on a Hilbert space H is such that there is no non-trivial proper closed subspace H′ of H for which φ(H′) ⊂ H′.

Theorem (Peter-Weyl). Let G be a compact group. Then all irreducible unitary representations of G are ﬁnite-dimensional. The left-regular representation

2 πL ∶ G → U(L (G)) is isomorphic to the direct sum of irreducible representations. In

⊕dim(Vξ) fact, one has πL ≡ >ξ∈Gˆ πξ , where (πξ)ξ∈Gˆ is an enumeration of the irreducible unitary representations πξ ∶ G → U(Vξ) of G (up to unitary equivalence).

In the case of locally compact groups, Gelfand and Raikov found that the irreducible unitary representations of G separate points, which implies that the direct sum of such representation, which itself a unitary representation, is faithful. 9

Theorem (Gelfand-Raikov). If G is any locally compact group, the irreducible unitary representations of G separate points on G. That is, if x, y ∈ G and x ≠ y, there is an irreducible representation π such that π(x) ≠ π(y).

Using Schur’s lemma one can prove that all irreducible unitary representations for an abelian topological group G are one-dimensional, which means that they are continuous characters, i.e. continuous homomorphisms from the topological group G into the circle group T. This means that in the case of a abelian locally compact group G the Gelfand-Raikov theorem states that there are ‘enough’ characters of G, where ‘enough’ means that for every pair of distinct points in G there is a character for which their images under such character are diﬀerent. This fact is the starting point for the beautiful and rich theory of Pontryagin duality, which is very important in the development of the harmonic analysis of this class of groups. In the case of locally compact groups the existence of ‘enough’ unitary representations relied on the analytic techniques permitted due to the Haar measure that one can deﬁne in such groups. In contrast, members from other classes of topological groups for which faithful unitary representations exist lack a Haar measure, hence other mechanisms are used to show that such unitary representation exists. For example, in the present dissertation the proof for the following theorem given by Uspenskij using mechanism from the theory of Banach spaces and free objects is presented:

Theorem. (Uspenskij, [38]) For every Tychonoﬀ space X the free locally convex space L(X) and the free abelian topological group A(X) admit a topologically faithful unitary representation.

Another example that is explored below is the proof using mechanisms from the theory of functions of positive type for the following theorem. 10

Theorem. (V. Uspenskij, [37]) Let H be a Hilbert space, and let G = Is(H) be a topological group of all (not necessarily linear) isometries of H, endowed with the strong operator topology. Then G is isomorphic to a subgroup or a unitary group.

The three classes of topological groups mentioned above illustrate the variety of techniques needed to answer the question about whether a member from a class of topological groups has a faithful unitary representation. Coming back to the case of abelian topological groups, recall that irreducible unitary representations for such groups are all continuous characters. This means that in order to learn about the unitary representations of an abelian topological group, one may study the more basic question of whether such abelian topological group has a non-trivial continuous character. In particular, one may ask:

Question: Let X be compact (metric), f ∶ X → X an isometry diﬀerent to the identity, let G = {f n ∶ n ∈ Z} ⊂ Homeo (X) (with the compact-open topology). Does G admit non-trivial characters? Answer: Yes, Since f is an isometry then G is equicontinuous. By Arzel`a-Ascoli theorem G is compact and the Peter-Weyl theorem applies. If one disregard the hypothesis that f is an isometry:

Question: Let X be compact (metric), f ∈ Homeo (X), f not the identity map, let G = {f n ∶ n ∈ Z} ⊂ Homeo (X). Does G admit non-trivial characters? Answer: No, there exist Hausdorﬀ group topologies τ on Z such that G = (Z, τ) does not have non-trivial characters (for some examples see [2] and [9]), since by Teleman’s theorem, every Hausdorﬀ topological group is topologically isomorphic to a group of homeomorphisms on a compact space this answers the question in a negative way. A G-space is a topological space X with a continuous action of G. Let X be a

G-space, a point x ∈ X is G-ﬁxed if gx = x for all g ∈ G. In all known examples for 11 which the above question is answered in a negative way, f has a ﬁxed point. This motivates the following open question:

Problem (Glasner-Pestov Problem [9],[24]). Let X be compact (metric), f ∈ Homeo (X), f ﬁxed point free, G = {f n ∶ n ∈ Z} ⊂ Homeo (X). Does G admit non-trivial continuous characters? Or more generally: Let X be compact,

G ⊂ Homeo (X) an abelian group, such that X has no G-ﬁxed points. Does G admit non-trivial continuous characters?

One way of approaching this problem is through the theory of extremely amenable groups. A topological group G is extremely amenable, if it has a fixed point in every compact space X it acts upon. The first known examples of extremely amenable groups seem rather unnatural and fabricated (for example [2]), but recent developments show a trend for many ‘massive’ groups to be extremely amenable. Some examples are: Gromov and Milman in [11] used the concentration in measure property to show that the unitary group U(`2) with the strong topology is extremely amenable. Pestov in [25] also used the concentration in measure property and showed that the space of isometries of the Urisohn space ([24], [33]) equipped with the standard Polish topology is a Lévygroup, which implies that it is extremely amenable. Pestov also proved in [22] with a different approach that

Homeo+(I) and Homeo+(R), the groups of orientation-preserving homeomorphisms of the closed interval and the real line, equipped with the compact-open topology are extremely amenable. Example of non-extremely amenable is given by Veech’s theorem (see 3.4.1), which states that locally compact groups have a free action on a compact space, meaning that they are not extremely amenable. The connection between extremely amenability and the Glasner-Pestov problem comes from the fact that if an abelian group is extremely amenable then it has no non-trivial continuous characters. The converse of this statement is unknown 12

to be true or false and indeed generates the following rephrasing of the Glasner-Pestov problem.

Problem (Glasner-Pestov problem, extremely amenable version). Is every abelian topological group G that admits no non-trivial characters extremely amenable?

Some examples of monothetic (topologically generated by one element) minimally almost periodic (that admits no non-trivial continuous characters) groups are cited in [24] (page 61), it is unknown if they are extremely amenable or not. These are natural candidates to study when trying to solve the Glasner-Pestov problem.

Using the lack of convex neighborhoods of the topological spaces Lp[0, 1] when 0 < p < 1, it was possible to prove the following result:

Theorem. The additive group Lp(0, 1) when 0 < p < 1 with the correspondent p-metric is minimally almost periodic.

This adds to the list of minimally almost periodic abelian topological groups for which it is unknown if they are extremely amenable. Some characterizations of extremely amenability discovered by Pestov were used by the same author to rephrase the Glasner-Pestov problem, and are discussed below. One of the interesting connections of the Glasner-Pestov problem is to an open problem in combinatorial number theory and harmonic analysis.

S ⊂ Z is syndetic, if S + [0, n] = Z, for some positive n ∈ Z, i.e., S has bounded gaps. The Bohr topology in Z is generated by the basic neighborhoods:

n {n ∈ Z ∶ S1 − ai S < , i = 1, 2, ..., s}, a1, a2, ..., as ∈ T, > 0, this is the topology inherited by Z from its Bohr compactiﬁcation. The following is an open problem: Let S be a

syndetic subset of the integers. Is the set S − S a Bohr neighborhood of zero in Z? 13

The connection to the Glasner-Pestov problem comes from considering the following statements and the proposition below:

Statement (Gl). If f ∈ Homeo (X) is ﬁxed point free, then G = {f n ∶ n ∈ Z} has a non-trivial character.

Statement (SB). If S ⊂ Z is syndetic, then S − S contains a Bohr neighborhood of zero.

Proposition (Glasner [9]). Not (GL) Ô⇒ Not (SB)

In order to answer in the negative the Glasner-Pestov problem in its extremely amenable version, a path is to show the existence of an abelian minimally almost periodic topological group G that is not extremely amenable, then this G needs to act without ﬁxed points on a compact space, in particular G might act freely on a

compact space. For a topological space X, a compactiﬁcation of X is a pair (Y, c) where Y is a compact space and c ∶ X → Y is a homeomorphic embedding where c(X) = Y . As mentioned above a classical example of a free action of a group on a compact space is given by Veech’s theorem, in which it is shown that locally compact groups act freely on a natural compactiﬁcation of G known as the greatest

ambit SG of G. There the topological structure of SG is used to show that such action is free. A characterization of G acting freely on its greatest ambit was given in terms of inessential sets by Pestov and is presented below (see section 3.4). It is then of interest to study topological groups compactifications and the classes of groups that act freely on them. Every Hausdorff topological group is a Tychonoff space. It is known that (equivalence classes) of compactifications of a Tychonoff space X, totally bounded uniformities on X and unital C∗-subalgebras of Cb(X) (complex-valued continuous functions on X) with the property that their elements separate points and closed 14

sets in X, are partially ordered sets and the three of them are order isomorphic (see section 3.1). In the case of topological groups, some different natural compactifications are presented below and their correspondence to different compatible uniformities or C∗-algebras of functions is discussed. For example, the

greatest ambit SG and the Roelcke compactiﬁcation R(G) are some of them. In the case of abelian topological groups the natural compactiﬁcations of G

coincide with SG. Nevertheless there is a compact G-space that is obtained as the maximal ideal space of a C∗-subalgebra of Cb(G). This is the WAP compact space or W (G). G is called and Eberlein group if W (G) is a compactiﬁcation if and only if G can be represented as group of isometries (with the strong topology) of a

reﬂexive Banach space ([16]), if and only if the C∗−algebra of weakly almost periodic functions on G separates points and closed sets in G. This algebra of

functions is deﬁned as follows. A continuous bounded function f ∶ G → C is said to be weakly almost periodic if the G−orbit by left (equivalently by right) translations of f by the natural left action of G on Cb(X) is relatively weakly compact (this means that Cb(X) is endowed with the weak topology, as a Banach space). An example, where the algebra of weakly almost periodic functions does not separate points is when G =Homeo+(I) (see [17]). In this case the only weakly almost periodic functions are the constant functions. This implies that Homeo+(I) is not representable as a group of isometries on a reﬂexive space, and the W (G) is a point. In [18], Megrelishvili, Pestov and Uspenskij proved that the greatest ambit of

G coincides with W (G) if and only if G is isomorphic to a subgroup of a compact group. Using this it was possible to prove that if G is minimally almost periodic then the W (G) is diﬀerent than the greatest ambit (W (G) is always contained in the greatest ambit) and might provide interesting examples of W (G) spaces. A unanswered question by Megrelishvili asks whether there are abelian topological 15

groups for which W (G) is a point. From the discussion above some candidates are the abelian minimally almost periodic groups. In the case that W (G) is not a point then a study of the topological groups that act freely on W (G) might help solve Glasner-Pestov problem.

A topological dynamical system (X,T ) consist of a non-empty compact metrizable space X together with a homeomorphism T ∶ X → X. (X,T ) is minimal if X has no proper non-empty invariant subspaces. A factor of (X,T ) is a topological dynamical system (Y,S) together with an onto map π ∶ X → Y , such that π ○ T = S ○ π. (X,T ) is equicontinuous if the collection of maps {T nn ∈ Z} is equicontinuous for some metric d compatible with the topology of X. An interesting theorem from the theory of topological dynamics characterizes the the existence of non-trivial equicontinuous factors for a minimal topological dynamical system (X,T ) in terms of a the property of weakly mixing and the existence of non-trivial eigenvalues. All of these properties imply the existence of a non-trivial character for the group {T n ∶ n ∈ Z} with the compact-open topology.

Theorem. Let (X,T ) be a minimal topological dynamical system, then the following are equivalent:

1. (X,T ) has a non-trivial eigenvalue.

2. The Kronecker factor of (X,T ) is not trivial.

3. (X,T ) has a non-trivial equicontinuous factors.

4. The Kronecker factor has a non-trivial character.

5. (X,T ) is not topologically weakly mixing.

And all of them imply the following two equivalent statements: 16

• The group {T n ∶ n ∈ Z} with the compact-open topology has a non-trivial character.

• (MG,T ) has an equicontinuous factor, where MG is the universal minimal compact G-space and G = {T n ∶ n ∈ Z}

A continuous function f ∶ X → C is said to be an eigenfunction of (X,T ) if f ≡~ 0 and if there exist λ ∈ C such that T f = λf, for the natural action of T on C∗(X).A topological dynamical system (X,T ) is topologically weakly mixing if the product dynamical system (X × X,T × T ) is topologically transitive, that is when there is an element of the product dynamical system whose T × T -orbit is dense in X × X On the other hand Gutman in [12] studied the space M(X) of maximal chains of subcontinua of a G-space X (see section 4.4). The same author showed that

under certain circumstances the natural action of G on M(X) is minimal. This means that under these circumstances if G is monothetic, topologically generated by

T , then (M(X),T ) is a minimal topological dynamical system.

Theorem. (Theorem 6.5 Gutman [12]) Let G act locally transitively on a Peano

continuum X which is strongly arcwise-inseparable. Then the action of G on M(X) is minimal. Moreover the action of G on M(X) is not 1-transitive (for any x, y ∈ M(X) there is g ∈ G, such that gx = y).

Now one may apply the theorem above and express a suﬃcient condition for

G = {T n ∶ n ∈ Z} ⊂ Homeo (X) to have a non-trivial continuous character:

Theorem. Let G = {T n ∶ n ∈ Z} ⊂ Homeo (X) act locally transitively on a Peano continuum X which is arcwise-inseparable (SAI). If (M(X),T ) is not weakly mixing then G has a non-trivial continuous character. 17

Since the action of G on M(X) is not 1-transitive, this leaves the question open to weather this action is weakly mixing or not, if the action was 2-transitive, immediately one gets that the system is weakly mixing.

In order to study the weakly mixing property of the system (M(X),T ) one has to ask when (M(X) × M(X),T × T ) is topologically transitive. (M(X) × M(X),T × T ) is topologically transitive when there is an element

(a1, a2) ∈ M(X) × M(X) whose T × T orbit is dense in M(X) × M(X). Gutman, provides set that is dense M(X) × M(X). Thus one might study topologically transitivity through approximations to such set.

Besides the need to investigate if (M(X),T ) is weakly mixing or not the following questions arise:

Question. How can one characterize the groups G = {T n ∶ n ∈ Z} ⊂ Homeo (X) that act locally transitively on X a Peano continuum space which is arcwise-inseparable (SAI)?

Question. Can the hypothesis of locally transitive be relaxed and still get a minimal

M(X) to broaden the class of groups to which theorem 4.5.1 can be applied?

A diﬀerent approach to use theorem the theorem above about existence of characters is to look for non-trivial eigenvalues of (X,T ). In that sense one needs eigenvalues of T acting on C∗(X). One theorem that implies the existence of eigenvalues for Banach algebras under certain hypothesis is 2.1.34, so one can

consider T in the Banach algebra of bounded linear operator on C∗(X).

Theorem. Let (X,T ) be a minimal topological dynamical system, Ω an open set of C, and h a holomorphic function on Ω. If T acting on C∗(X) is such that T = h(U) where U is a linear bounded operator of the Banach algebra C∗(X), and ∅ =~ σ(U) ⊂ Ω with h(σ(U)) ∖ {1}= ~ ∅, where σ(U) is the set of eigenvalues of U, 18 then (X,T ) has a non-trivial equicontinuous factor. Thus {T n ∶ n ∈ Z} has a non-trivial character.

This theorem is abstract and to really understand the algebraic counterpart one has to deepen in the study of linear bounded operators of C∗(X). This introduction ﬁnalizes listing some of the contributions of the present dissertation:

1. A new strategy to ﬁnd non-trivial continuous characters for certain groups, in

terms of the nature of the dynamical system (M(X),T ) where M(X) is space of maximal chains of subcontinua of a G-space X.

2. A proof showing that the additive group of Lp[0, 1] 0 < p < 1 (and a larger class of related groups) is minimally almost periodic.

3. A connection between abelian minimally almost periodic topological groups

and the G-space W (G).

4. A compilation of different ways of expressing the Glasner-Pestov problem and of different techniques used by other authors to prove the existence of unitary representations for different classes of topological groups. 19 2 Unitary representations of topological groups

In the representation theory of topological groups it is known (see theorem 2.1.20) that every Hausdorﬀ topological group G acts continuously and eﬀectively by isometries on a Banach space and by homeomorphisms on a compact space. Given this fact it is natural to ask weather a class of topological groups has a rich representation theory on Hilbert spaces. In particular one may ask:

Question 1. Given a class of topological groups, do groups from the class have ‘enough’ non-trivial unitary representations? 1

An affirmative answer to this question can lead to the study of this class of topological groups through mechanisms from the theory of C∗-algebras and to the development of a rich duality theory through which the topological group can be better understood. An example of this is the class of locally compact abelian groups and the theory of Potryagin duality. In this chapter some examples for which question1 has been answered are presented. In sections 2.2, 2.3, 2.4 tools are provided to prove an affirmative answer for question1 for the cases of locally compact groups, free abelian topological groups generated by a Tychonoff space X, and the group of isometries of an infinite dimensional Hilbert space, respectively. In contrast, in section 2.5 the class of abelian extremely amenable topological groups is proven to have no irreducible unitary representations. Finally, in section 2.6 a class of topological groups for which question1 remains open is introduced. For that class of groups, question1 is known as the Glasner-Pestov problem and some implications for the case of a positive or a negative answer to the question are discussed.

1 ‘enough’ in this context means that the unitary representations of a topological group separate its elements 20

2.1 Basic deﬁnition

Some basic definitions and examples that are central for this investigation are provided below. Throughout this dissertation only Hausdorff topological groups and Tychonoff spaces are considered.

Deﬁnition 2.1.1. Let G be a group and a Hausdorﬀ topological space. G is said to be a topological group if:

1. the mapping (x, y) Ð→ xy of G × G onto G is a continuous mapping of the cartesian product G × G (with the product topology) onto G, and

2. the mapping x Ð→ x−1 of G onto G is continuous.

Example 2.1.2. 1. Any group G with the discrete topology becomes a topological group. It is said that G with this topology is a discrete group. In this case one can notice that all the information of the topological group lives inside the algebraic structure

2. All additive subgroups of Rn and Cn with the induced topology.

3. Linear groups of n × n matrices with coeﬃcients in C with matrix

2 multiplication. these groups can be regarded as subsets of Cn , from which

they inherit the topology. The group of unitary n × n matrices U(n) with complex coeﬃcients is an example of this kind of groups.

4. The circle group T consisting of complex numbers with modulus one. The operation is complex multiplication and the topology is the induced by C.

This group is topologically isomorphic to U(1) which means there is an bicontinuous isomorphism between them. 21

5. Let H be a Hilbert space, let U(H) be the group of unitary operators on H with the strong operator topology, this is the same topology as the one inherited from the product topology of HH .

2.1.1 Uniform Spaces

A natural setting for topological groups that allows to further their study uses the notion of uniform spaces.

Deﬁnition 2.1.3. A uniformity or uniform structre on a set X is a non-empty family U of subsets of X × X such that

1. Each member of U contains the diagonal ∆ = {(x, x) ∶ x ∈ X}.

2. U ∈ U then U −1 ∈ U, where U −1 = {(y, x) ∶ (x, y) ∈ U}.

3. If U ∈ U then there exist a V ∈ U such that V ○ V ⊂ U, where V ○ V = {(x, z) ∶ (x, y), (y, z) ∈ V }.

4. If U ∈ U and U ⊂ V ⊂ X × X then V ∈ U.

5. If U, V ∈ U then U ∩ V ∈ U.

(X, U) is called a uniform space and if ∩U = ∆, U is called separated. The elements of U are called entourages of the diagonal.

Deﬁnition 2.1.4. A subfamily B ∈ U is called a basis of the uniformity U if for every U, V ∈ B there is W ∈ U such that W ⊂ U ∩ V , and for every entourages P ∈ B there exist U ∈ B such that U ⊂ P . The elements of B are called basic entourages of the diagonal, and U can be recovered by considering all subsets of X × X that contain an element of B. 22

Deﬁnition 2.1.5. A uniformly continuous map between uniform spaces (X, U) and (Y, V) is a map f ∶ X → Y such that for all V ∈ V there exist U ∈ U such that (f × f)(U) ⊂ V . f is a uniform isomorphism or X and Y are said to be uniformly isomorphic if f is a bijection and f −1 is uniformly continuous.

Every uniform space induces a topology with a system of neighborhoods of x ∈ X given by {Ux}U∈U where Ux = {y ∶ (x, y) ∈ U} for U ∈ U, such topology is called the topology generated by U. Also, if A ⊂ X, UA denotes the set ⋃x∈A Ux.A uniformity U on a topological space (X, τ) is called compatible with τ if the topology generated by U matches τ. On the other hand if X is a Tychonoﬀ space, X admits at least one compatible separated uniformity, namely the one with basic entourages of the diagonal given by {(x, y) ∶ Sf(x) − f(y)S} where f is a real-valued function on a X.

Example 2.1.6. 1. A metric space (X, d) has a compatible uniformity. The

basic entourages are given by U = {(x, y) ∶ d(x, y) < }. It is clear that the concept uniform continuity of a function between metric spaces matches the

one given in deﬁnition 2.1.5 using the entourages U.

2. In the case of topological groups one can deﬁne the right uniformity on G

denoted by RG or just R with basic entourages of the diagonal

−1 VR = {(g, h) ∈ G × G ∶ gh ∈ V }, where V runs over a neighborhood system of the identity in G. This uniformity is compatible with the topology of the topological group G, which means that topological groups are uniform spaces. In section 3.2 a broader discussion about other uniformities compatible with a topological group and their properties is presented.

Proposition 2.1.7. Every element of a compatible uniformity U on a topological space X is a neighborhood of the diagonal in the product topology. Thus if X be a 23

compact Hausdorﬀ space then X admits a unique compatible uniformity, consisting of all neighborhoods of the diagonal

Proof. Let (X, U) a uniform space and let V ∈ U. Let W ∈ U such that W is

−1 symmetric (W = W ) and W ○ W ⊂ V . Notice that Wx × Wx ⊂ V . Indeed, this follows from the fact that if z1, z2 ∈ Wx then by symmetry of W (z1, x), (x, z2) ∈ W

thus (z1, z2) ∈ W ○ W ⊂ V . Recall that the interior of Wx, Int(Wx) is a non-empty

neighborhood of x, thus (x, x) ∈ Int(Wx) × Int(Wx) ⊂ V . This means that U contains only neighborhoods of the diagonal in X × X with the product topology and X with the topology induced by U. If X is compact Hausdorﬀ then

∆ = U = U U∈U U∈U .

Let O be an open neighborhood of the diagonal, ∆ ⊂ O. Then,

c c X × X = O ∪ ∆ = O ∪ U U∈U .

By compactness of X × X there exists U1, ..., Un open sets in X × X such that

c c X × X = O ∪ U1 ∪ ⋯ ∪ Un

so,

U1 ∩ ⋯Un ⊂ U1 ∩ ⋯... ∩ Un ⊂ O

hence O ∈ U.

The uniformity of a compact Hausdorﬀ space (X, U) can be used to deﬁne the topology of uniform convergence on the group of homeomorphisms Homeo(X). 24

Indeed, let C(X,X) be the set of continuous functions from X to X, consider the uniformity deﬁned on this space with basic entourages of the diagonal

∆ ⊂ C(X,X) × C(X,X) given by:

V̂ = {(f, g) ∶ (f(x), f(y)) ∈ V for every x ∈ X}

where V ∈ U. This means that a neighborhood of the identity in the topology of uniform convergence is of the form:

Ṽ = {g ∈ Homeo(X) ∶ ∀x ∈ X, (x, gx) ∈ V }

Proposition 2.1.8. Let X be a compact space, the topology of uniform convergence

is a group topology for Homeo(X), also the topology of uniform convergence coincides with the compact-open topology.

Proposition 2.1.9. Let X be a compact space. The natural action of

G = Homeo(X) on X is continuous as a map Φ ∶ Homeo(X) × X → X.

Proof. Let g0 ∈ G, x0 ∈ X and U an open set in X such that Φ(g0, x0) ∈ U. Observer −1 that x0 ∈ g0 (U) which is an open set of X. Since X is compact there exist an open −1 set V of X such that x0 ∈ V ⊂ V ⊂ g0 (U). Since X is compact then V is also compact. Consider now, the open neighborhood P (V,U) = {g ∈ G ∶ g(V ) ⊂ U}, it is

clear that g0 ∈ P (V,U) and Φ(P (V,U),V ) ⊂ U, thus Φ is continuous.

Uniform spaces need not to be compact but there is a natural compactiﬁcations for them, for its description the concepts of complete and totally bounded uniform

spaces are needed. If (X, U) is a uniform space and if F ⊂ X and V ∈ U whenever F × F ⊂ V , F is called V-small. A collection F of subsets of X is said to be a Cauchy ﬁlter F if for every V ∈ U there exist an F ∈ F such that F is V-small. 25

Definition 2.1.10. A uniform space (X, U) is complete if every Cauchy filter F of (X, U) is convergent to a point, i.e. for all neighborhood O of x ∈ X there exists F ∈ F such that F ⊂ O. Equivalentely (X, U) is complete if each Cauchy filter of closed subsets of X which has the finite intersection property, also has non-empty intersection.

Deﬁnition 2.1.11. Let (X, U) be a uniform space. U is called totally bounded if for all V ∈ U there exists F a ﬁnite subset of X such that VF = ⋃x∈F Vx = X.

Proposition 2.1.12. For every uniform space (X, U) there is exactly one (up to a uniform isomorphism) complete uniform space X,˜ U˜, such for a dense subset A of X˜ ˜ ˜ ˜ the space (X, U) is uniformly isomorphic to (A, UA). Such (X, U) is called completion of (X, U). Moreover, if U is totally bounded so is U˜.

Proposition 2.1.13. A uniform space (X, U) is compact if and only if it is complete and U is totally bounded.

Consider now a uniform space (X, U), let C∗U be the totally bounded replica of U, that is, the ﬁnest uniformity compatible with the space X which is totally bounded and coarser than U. Equivalently, C∗U is the coarsest uniformity on X with regard to which every bounded uniformly continuous function on (X, U) remains uniformly continuous. Another equivalent description of C∗U is given by proposition 2.1.14. A cover χ of the set X is called uniform with respect to U if there exist V ∈ U such that {Vx}x∈X is a reﬁnement of χ.

Proposition 2.1.14. ([24]) Let (X, U) be a uniform space, and let B be the collection of all covers of X which have a ﬁnite reﬁnement uniform with respect to

U. Then B is a basis for C∗U.

Given that C∗U is totally bounded by propositions 2.1.12, 2.1.13, the completion of (X, C∗U) is a compactiﬁcation of X, known as the Samuel 26

compactification of X and is denoted by σX. In section 3.1 a broader discussion about compactifications of Tychonoff spaces is presented. For a more detailed study of uniform spaces and proofs for propositions 2.1.8, 2.1.12 and 2.1.13 the reader can see [6].

2.1.2 Actions and representations

As mentioned above, theorem 2.1.20 describes some topological representations of Hausdorff topological groups through effective actions on spaces or compact spaces. An action of G is called effective if for every g ∈ G there exist x ∈ X such taht g ⋅ x =~ x, meaning that as maps on the phase space S, g =~ e, where e is the identity element of G. Besides the effectiveness of actions of G, it is of interest to know when the induced representations by these actions are continuous, the following propositions give an insight to these questions.

Proposition 2.1.15. An action of a group G on a compact space X is continuous as a map G × X → X if and only if the associated homomorphism from G to the homeomorphism group Homeo(X), equipped with the topology of uniform convergence, is continuous.

Proof. Let Φ ∶ G × X → X be the map deﬁned by the action of G on X, and let φ ∶ G → Homeo(X) be the associated isomorphism. It is clear that as a corollary of proposition 2.1.9 if φ is continuous so is Φ. On the other hand, let V ∈ U, where U is the uniformity on X. One has to show that there exists a neighborhood of the

identity O of G such that φ(O) ⊂ Ṽ, that is, if g ∈ O then (x, φ(g)x) ∈ V for all x ∈ X. Indeed, let U ∈ U such that U ○ U ⊂ V and U = U −1, if Φ is continuous, for every x ∈ X there exist an neighborhood of the identity Ox in G, and a neighborhood Ax of x in X such that Φ(Ox,Ax) ⊂ Ux. Consider a ﬁnite subcover of n X, Axi , for i = 1, 2, ...n, and let O = ⋂i=1 Axi . Notice that for any x ∈ X, x ∈ Axi for 27

some i, if g ∈ O then Φ(g, x) = φ(g)x ∈ Uxi which is the same as writing:

(xi, φ(g)x) ∈ U. Also, notice that since x ∈ Axi and Φ(Oxi ,Axi ) ⊂ Uxi then

Φ(e, x) = x ∈ Uxi which is the same as (xi, x) ∈ U, by symmetry of U, (x, xi) ∈ U.

Since (xi, φ(g)x), (x, xi) ∈ U and U ○ U ⊂ V then (x, φ(g)x) ∈ V and φ(O) ⊂ Ṽ.

Consider a normed vector space E the strong operator topology on the group of automorphisms GL(E) is the one inherited from the product topology of EE. The strong topology is not a group topology for GL(E) but it is for the group of isometries of E. The notation used for this topological group is Isos(E). It is easy to see that an open neighborhood of the identity in Isos(E) has the form

{u ∈ Iso(E) ∶ ∀i = 1, 2, ..., n, Yu(xi) − xiY < } for xi ∈ E and > 0.

Proposition 2.1.16. An action of a group G on a normed vector space E is continuous as a map G × E → E if and only if the associated homomorphism from G to the group of isometries Isos(E) is continuous.

Proof. Let Φ ∶ G × E → E be the map deﬁned by the action of G on E, and let

φ ∶ G → Iso(E) be the associated isomorphism. Let xi ∈ X and Bi an open neighborhood of xi, consider the sub-basic open neighborhood of the identity in

Isos(E), P (xi,Bi) = {g ∈ Iso(E) ∶ g(xi) ∈ Bi}. (Ô⇒) In order to prove the continuity of φ given a basic open set n P = ⋂i=1 P (xi,Bi) in Isos(E) one must ﬁnd a neighborhood of the identity O in G such that φ(O) ⊂ P . If Φ is continuous then there exist Oi a neighborhood of the identity of G and Ai a neighborhood of xi in X, such that Φ(Oi,Ai) ⊂ Bi. This n implies that φ(O)(xi) ⊂ Bi, thus φ(O)i ⊂ P (xi,Bi). If O = ⋂i=1 Oi then φ(O) ⊂ P

(⇐Ô) It is enough to show that the natural the action Φ ∶ Isos(E) × E → E is continuous. Let g0 ∈ Isos(E), x ∈ X and the open ball B(g0(x), ) with center in 28 x ∈ E and radius > 0. Consider the following neighborhood of g0 in Isos(E): P (g0; x, 2 ) = {g ∈ Isos(X) ∶ g(x) ∈ B(g0(x), 2 )}. Let y ∈ B(x, 2 ). Notice that:

g ∈ P (g0; x, ) if and only if Yg(x) − g0(x)Y < 2 y ∈ B(g0(x), ) if and only if Yg(x) − g(y)Y = Yx − yY < , 2

using triangle inequality: Yg(y) − g0(x)Y ≤ Yg(y) − g(x)Y + Yg(x) − g0(x)Y < , which implies that Φ(P (g0; x, 2 ),B(g0(x), 2 )) ⊂ B(g0(x), ) then Φ is continuous.

Deﬁnition 2.1.17. 1. A representation φ of a topological group G on a compact

space X is continuous, if the associated map Φ ∶ G × X → X is continuous.

2. A representation φ of a topological group G on a normed space E is

(strongly) continuous, if the associated map Φ ∶ G × E → E is continuous.

2.1.3 Teleman’s theorem

Let the right translations Rg ∶ G → G, g ∈ G be deﬁned by Rg(x) = xg, the left

−1 translations can be deﬁned in a similar way and will be denoted by Lg(x) = g x. Denote by RUCb(G) the vector space formed by all complex-valued bounded functions on G that are uniformly continuous with respect to to the right uniformity of G, i.e.

−1 ∀ > 0, ∃V ∋ e ∈ V, xy ∈ V ⇒ Sf(x) − f(y)S <

RUCb(G) becomes a Banach space if equipped with the supremum norm:

SSfSS = supx∈GSf(x)S 29

Consider the action of G on itself by left multiplication and the in induced

action of G on RUCb(G):

b b G × RUC (G) Ð→ RUC (G)

(g, f) Ð→ g ⋅ f =g f = f ○ Lg

−1 which makes gf(x) = f(g x). Notice that this action is continuous.

b Indeed, Let f ∈ RUC (G), g0 ∈ G and > 0, since f is uniformly continuous with respect of the right uniformity, there exist an open neighborhood of the identity V ⊂ G such that for every x, y ∈ G such that xy−1 ∈ V implies

b Sf(x) − f(y)S < ~2. Let h ∈ RUC (G) such that Yh − fY∞ < ~2, and let g ∈ G such

−1 −1 −1 that g ∈ V g0 (which is equivalent to g g0 ∈ V ), then:

−1 −1 Sh(g x) − f(g x)S < ~2, from Yh − fY∞ < ~2

−1 −1 −1 −1 −1 Sf(g x) − f(g0 x)S < ~2, since g xx g0 = g g0 ∈ V

Thus,

−1 −1 −1 −1 −1 −1 Sgh(x) −g0 f(x)S = Sh(g x) − f(g0 x)S ≤ Sh(g x) − f(g x)S + Sf(g x) − f(g0 x)S <

for all x ∈ G, which implies that Ygh −g0 fY∞ < , hence the above mentioned action is continuous.

Observe that the maps f Ð→g f are isometries since YfY∞ = YgfY∞. Thus, by proposition 2.1.16 G has a continuous representation by isometries on the Banach

space RUCb(G), the fact that topological groups are uniformly Tychonoﬀ implies that this representation in faithful and the second part of theorem 2.1.20 has been proven. 30

Remark 2.1.18. Similarly to the right uniformity one can deﬁne the left uniformity (see section for more details 3.2). A semi-topological group G is called uniformly Tychonoﬀ if for every open neighborhood V of the identity there exists a real valued function f such that f(e) = 0 and f(x) ≥ 1 for all x ∈ G ∖ V such that f is uniformly continuous with respect of both, right and left uniformities. In particular

this means that to be uniformly Tychonoﬀ implies that functions on RUCb(G) separate closed sets and points. The fact that a topological group is uniformly Tychonoﬀ can be established by the existence on uniformly continuous semi-norms on G that separate points and closed sets, see [6] for more details.

To establish the second part or theorem 2.1.20, one needs to consider a dual action of G. Let E be a Banach space, recall to that for every isometry u of E there

is an isometry u∗, of the dual Banach space E∗, deﬁned by u∗(ϕ) = ϕ ○ u where ϕ ∈ E∗. The map u Ð→ u∗ is anti-homomorphism from Iso(E) to Iso(E∗), this is because (u ○ v)∗(ϕ) = ϕ ○ (u ○ v) = (v∗ ○ u∗)(ϕ). Thus in order to obtain a

−1 ∗ homomorphism one need use the map u Ð→ (u ) . Recall that the unit ball of BE∗

∗ ∗ ∗ is invariant under isometries, thus u SBE∗ ∈ Homeo(BE ). If we endowed BE with the weak∗ topology by the Banach-Alaoglu theorem, it becomes a compact space,

∗ thus the map u Ð→ u SBE∗ is a monomorphism from the isometries of the Banach

space E to the homeomorphisms group of the compact space BE∗ .

∗ Proposition 2.1.19. Let E be a Banach space and BE∗ be the unit ball of E with

∗ ∗ −1 the weak topology. The monomorphism u Ð→ u SBE∗ = (ϕ → ϕ ○ u ) from Isos(E)

to Homeo(BE∗ ) (with the topology of uniform convergence) is an embedding of topological groups.

Proof. A basic open neighborhood of the identinty in Isos(E) has the form, 31

α = {u ∈ Iso(E) ∶ ∀i = 1, 2, ..., n, Yu(xi) − xiY < }

for xi ∈ E and > 0, and basic open neighborhood of the identity in

Homeo(BE∗ ) is of the form,

β = {u ∈ Homeo(BE∗ ) ∶ ∀ϕ ∈ BE∗ , (u(ϕ), ϕ) ∈ U}

∗ for U in the unique compatible uniformity U from BE∗ with the weak topology. Consider now the set

γ = {u ∶ ∀i = 1, 2, ..., n, ∀ϕ ∈ BE∗ , Sϕ(u(xi)) − ϕ(xi)S < }

1. Claim: α = γ: (⊆) Since Yϕ(u(xi)) − ϕ(xi)Y ≤ YϕYYu(xi) − xiY then for

ϕ ∈ BE∗ , YϕY ≤ 1 and if Yu(xi) − xiY < then Yϕ(u(xi)) − ϕ(xi)Y < . Thus α ⊆ γ.

(⊇) By the Hahn-Banach theorem there exists ϕ ∈ BE∗ such that

ϕ(u(xi) − xi) = Yu(xi) − xiY, if Sϕ(u(xi) − xi)S < then Yu(xi) − xiY < , which implies that α ⊇ γ.

2. Claim β = γ: Recall that U is induced by the additive structure of E∗. A basic

∗ ∗ neighborhood of 0 ∈ E is given by V = {ϕ ∈ E ∶ ∀i = 1, 2, ..., n, Sϕ(xi)S < } for

xi ∈ E, then a basic entourage of the diagonal U ∈ U has the form

∗ U = {(ϕ1, ϕ2) ∈ E ∶ ∀i = 1, 2, ..., n, Sϕ1(xi) − ϕ2(x2)S < }, xi ∈ E. This means that a basic β can be rewriten as:

{u ∶ ∀i = 1, 2, ..., n, ∀ϕ ∈ BE∗ , Sϕ(u(xi)) − ϕ(xi)S < }, thus β = γ.

This implies that the mentioned monomorphism is an embedding of topological groups. 32

Applying proposition 2.1.19 to the continuous representation of G on the

Banach space RUCb(G) one obtains a continuous faithful representation of G on

Homeo(BRUCb(G)∗ ) with the topology of uniform convergence where BRUCb(G)∗ is endowed with the weak∗ topology. Thus the proof for the following theorem has been ﬁnished.

Theorem 2.1.20 (Teleman). Every Hausdorﬀ topological group admits a topologically faithful representation on:

1. (strongly) continuous faithful representation a Banach space X by linear isometries.

2. continuous faithful representation by homeomorphisms on a compact space.

2.1.4 Unitary representations

Deﬁnition 2.1.21. Let G be a topological group, Hπ some nonzero Hilbert space and U(Hπ) the group of unitary operators on Hπ.A (topological) unitary representation of G is a homomorphism π from G into U(Hπ) that is continuous with respect to the strong operator topology, i.e. π ∶ G → U(Hπ) which satisﬁes π(xy) = π(x)π(y) and π(x−1) = π(x)−1 = π(x)∗, and for which x → π(x)u is continuous from G to Hπ for any u ∈ Hπ. The continuity of π is equivalent to the condition that the action map G × H → H that π induces is continuous.

Hπ is called the dimension space of π and its dimension is called the dimension or degree of π.

Let M be a closed subspace of Hπ. M is called an invariant subspace for π if

π(x)M ⊂ M for all x ∈ G. If M is invariant and a non-zero subspace, the restriction of π to M deﬁnes a unitary topological representation of G on M, called a (topological) subrepresentation of π and it is denoted by πM . 33

If π admits a nonzero closed invariant subspace diﬀerent from Hπ with subrepresentation πM , π call π reducible, otherwise π is irreducible.

Example 2.1.22. An example of a faithful (embedding homeomorphism) unitary representation for a locally compact group G is the left-regular representation

2 πL ∶ G → U(L (G)), where G is equipped with its normalized Haar measure µ (and

2 the Borel σ-algebra) to form the Hilbert space L (G), and πL is the translation

−1 operation πL(g)f(x) ∶= f(g x). See section 2.2 for more details.

One is interested in irreducible unitary representations because in many cases they constitute the building blocks of unitary representations. For example in the case of compact groups (a topological group which is compact as topological space) every unitary representation is the direct sum of irreducible representations (see section 2.2), where the deﬁnition of direct sum is given as follows:

Deﬁnition 2.1.23. Let {πi}i∈I be a family of unitary representations, their direct sum ⊕πi is the representation π on H = ⊕Hπi deﬁned by π(x)(∑ vi) = ∑ πi(x)vi for

vi ∈ Hπi .

With this in mind it is natural to reﬁne question1 and ask about more basic unitary representations instead of asking about the whole class of unitary representations.

Question 2. Given a class of topological groups, do groups from this class have ‘enough’ irreducible unitary representations?

Remark 2.1.24. By means of deﬁnition 2.1.21 one can clarify what it means to have ‘enough’ representations. A topological group G has ‘enough’ (irreducible) unitary representations if this class of representations separates points on G. That is, if

x, y ∈ G and x ≠ y, there is an irreducible representation π such that π(x) ≠ π(y) 34

An important fact about irreducible unitary representations is the following version of Schur’s lemma, preceeded by the following deﬁnition:

Deﬁnition 2.1.25. Let π1 and π2 be unitary representations of G, and intertwining operator for π1 and π2 is a bounded linear map T ∶ Hπ1 → Hπ2 such that T π1(x) = π2(x)T for all x ∈ G. The set of such operators is denoted by

C(π1, π2). π1 and π2 are (unitarily) equivalent or ismorphic if C(π1, π2)

−1 contains a unitary operator U for which π1(x) = U π2(x)U

Lemma 2.1.26 (Schur’s Lemma). (See [8])

• Suppose that π1 and π2 are irreducible unitary representations of G a

(topological) group. If π1 and π2 are equivalent then C(π1, π2) is

one-dimensional; otherwise, C(π1, π2) = {0}

• A unitary representation π of G a (topological) group is irreducible if and only

if C(π, π) contains only scalar multiples of the identity.

Remark 2.1.27. Schur’s lemma is true in a more general setting than topological groups, a restricted version is given here for the purpose of being consistent with the deﬁnitions given above.

An application of Schur’s Lemma to irreducible representations of abelian topological groups is the following proposition

Proposition 2.1.28. If G is an abelian topological group, then every irreducible representation of G is one-dimensional

Proof. Let π be a representation of G then the operators π(x) commute with one another satisfying π(x1)π(x2) = π(x2)π(x1) for all x1, x2 ∈ G. Thus if π is irreducible by Schur’s lemma one has that π(x) = cxI. This means that every one-dimensional subspace of Hπ is invariant so dimHπ = 1 35

As mentioned in example 2.1.2, the circle group T is topologically isomorphic to

U(1) the group of unitary operators of a one-dimensional Hilbert space. Considering this together with proposition 2.1.28 question2 is rewritten for abelian topological groups in a diﬀerent way using the following deﬁnition:

Deﬁnition 2.1.29. If G is an abelian topological group then a continuous homomorphism γ ∶ G → T is said to be a character. The collection of all the characters of G is called the character group or dual group of G and is denoted by Gˆ. The dual group is a topological group with the compact-open topology.

Question 3. Given a class of abelian topological groups, do groups from this class have ‘enough’ characters?

Example 2.1.30. In the case of a discrete abelian group G one has that the circle group is divisible, hence it is injective, which implies that the characters of G separate elements of G. This answers in a positive way question3.

2.1.5 Banach algebras

A Banach algebra A is a complete normed complex algebra that satisfies the condition: YxyY ≤ YxYYyY , for all x, y ∈ A. A is unital if there is an (unit) element e ∈ A such that,xe = ex = x, for all x ∈ A, and YeY = 1. A homomorphism between Banach algebras A, B is a bounded linear map φ ∶ A → B that satisfies φ(xy) = φ(x)φ(x) for al x, y ∈ A. When A, B are unital we say φ is unital if φ(1) = 1. An involution on an algebra A is a mapping x → x∗, that satisfies for all x, y ∈ A and α ∈ C, (x + y)∗ = x∗ + y∗, (αx)∗ = αx¯ ∗, (xy)∗ = y∗x∗, (x∗)∗ = x.A C∗-algebra A is a Banach algebra with an involution and the condition that,

2 Yxx∗Y = YxY for all x ∈ A.A C∗-algebra homomorphism is a Banach algebra 36

homomorphism that also satisﬁes φ(x∗) = (φ(x))∗. An example of a C∗-algebra is the set of all complex-valued functions on X a Hausdorﬀ compact space with the

sup norm, this is denoted by C∗(X). If A is a unital Banach algebra with unit element e x ∈ A is invertible if there exist x−1 ∈ A such that xx−1 = x−1x = e. The set of all invertible elements of A is denoted by G(A) and it is easy to see that it is a group. Let x ∈ A, the spectrum of x is the set σ(x) = {λ ∈ C ∶ λe − x is not invertible}. The spectral radius of x is the

n 1~n number ρ(x) = sup{SλS ∶ λ ∈ σ(λ)} = limn→∞Yx Y ≤ YxY. If A is an abelian Banach algebra (which means that xy = yx), a multiplicative linear functional on A is a non-zero (Banach algebra) homomorphism. The set of all multiplicative linear functionals of A is called the maximal ideal space of A, and is denoted by MA. It happens that multiplicative functionals on a unital abelian Banach algebra A are continuous of norm 1, and the map taking each multiplicative functional to its kernel is a bijection onto the set of maximal ideals of the algebra A. The maximal ideal space of a unital abelian Banach algebra, endowed with the weak∗-topology is a compact Hausdorﬀ space

(this is implied by the Banach-Anaoglu theorem, since MA is a closed subset of the unit ball in the dual space of A). If A is abelian but not unital, then MA endowed with the weak∗-topology is locally compact.

The Gelfand transform Γ ∶ A → C0(MA), from a commutative Banach algebra A into the complex-valued continuous functions on MA that vanish at inﬁnity C0(MA). Where Γ(a) = aˆ, wherea ˆ(φ) = φ(a). The Gelfand transform is a contractive (since YxˆY = ρ(x)) algebra homomorphism, and the image algebra separates points sets in MA. The radical rad(A) of a Banach algebra A is the intersection of all maximal ideals of A. If radA = {0}, A is called semisimple. For a unital commutative Banach algebra A the kernel of the Gelfand transform is 37

radA, thus the Gelfand transform is an isomorphism if and only if A is semisimple. In the case that A is a C∗-algebra, the Gelfand transform is an isometric C∗-algebra

homomorphism of A onto C0(MA).

Example 2.1.31. In particular, if X is a compact Hausdorﬀ space. For each x ∈ X,

∗ if one deﬁnes hx ∶ C (X) → C by hx(f) = f(x), then the map x → hx is a homeomorphism from X to MC∗(X) if one identiﬁes x ∈ X with hx ∈ MC∗(X), the Gelfand transform on C∗(X) becomes the identity map.

Below some elements of the theory of functional calculus for Banach alegebras are presented. Let A be a Banach algebra, Ω is an open set in C, and H(Ω) is the algebra of all complex holomorphic function in Ω. Deﬁne AΩ = {x ∈ A ∶ σ(x) ⊂ Ω}.

Theorem 2.1.32. ([29]) Suppose A is a Banach algebra, x ∈ A, Ω is an open set in C, and σ(x) ⊂ Ω. Then there is δ > 0 such that σ(x + y) ⊂ Ω for every y ∈ (A) with

YyY < δ Thus AΩ = {x ∈ A ∶ σ(x) ⊂ Ω} is an open set.

Let A be a Banach algebra, Ω is an open set in C, and H(Ω) is the algebra of ˜ all complex holomorphic function in Ω. Deﬁne H(AΩ) to be the set of all A-valued ˜ functions f, with domain AΩ, that arises from an f ∈ H(Ω) by the formula

1 −1 f˜(x) = f(λ)(λ − x) 2πi Sγ

where γ is the contour that surrounds σ(x) ∈ Ω

Theorem 2.1.33. ([29]) Suppose that x ∈ AΩ and f ∈ H(Ω), then

(a) f˜(x) is invertible in A if and only if f(λ)= ~ 0 for every λ ∈ σ(x).

(b) σ(f˜(x)) = f(σ(x)). 38

Let X be a Banach space and T ∈ B(X), the point spectrum σp(T ) of T is the set of eigenvalues of T . This means that λ ∈ σ(T ) if and only if ker(T − λI) has positive dimension.

Theorem 2.1.34. [29] Suppose that T ∈ B(X), and Ω is open in C, σ(T ) ∈ Ω, and f ∈ H(Ω), then

(a) If x ∈ X, λ ∈ Ω, and T x = λx, then f˜(T ) = f(λ)x

˜ (b) f(σp(T )) ⊂ σ(f(T ))

then λ ∈ f(σp(T )).

˜ (d) If f is not constant in any component of Ω, then f(σp(T )) = σp(f(T ))

2.2 Compact and locally compact groups

The ﬁrst example of classes of topological groups for which question1 is answered in the positive are compact groups and locally compact group through theorems 2.2.1 and 2.2.2.

Theorem 2.2.1 (Peter-Weyl). Let G be a compact group. Then all irreducible unitary representations of G are ﬁnite-dimensional. The left-regular representation

2 πL ∶ G → U(L (G)) is isomorphic to the direct sum of irreducible representations. In

⊕dim(Vξ) fact, one has πL ≡ >ξ∈Gˆ πξ , where (πξ)ξ∈Gˆ is an enumeration of the irreducible unitary representations πξ ∶ G → U(Vξ) of G (up to unitary equivalence).

Theorem 2.2.2 (Gelfand-Raikov). If G is any locally compact group, the irreducible unitary representations of G separate points on G. That is, if x, y ∈ G and x ≠ y, there is an irreducible representation π such that π(x) ≠ π(y). 39

Remark 2.2.3. If G is locally compact and abelian by proposition 2.1.28 irreducible unitary representations are one-dimensional. Thus, theorems 2.2.1 and 2.2.2 imply an aﬃrmative answer to question3 for this class of topological. groups.

Below an outline of some of the tools and techniques from abstract harmonic analysis and C∗-algebras, that are involved in a proof of theorem 2.2.1.

If G is locally compact group, G has a regular Borel measure µG which is invariant under the left-action of G. This means that µG(gE) = µG(E) for g ∈ G and E a Borel subset of G. This measure is unique up a scalar multiple and it is known as the left Haar measure of G. If G is compact then µG is ﬁnite, in which case the measure is normalized to get µ(G) = 1. Every left Haar measure can be made into a right Haar measure µ̃G (where µ̃G(Eg) = µ̃G(E) for g ∈ G and E a Borel

−1 subset of G) by making µ̃G(E) = µG(E ). Nevertheless, the left Haar measure does not need to be invariant under right translations of G. One can deﬁne the modular function Λ ∶ G → (0, ∞) satisfying µG(Eg) = Λ(g)µG(E), as a way to ‘measure’ how much µG fails to be right translation invariant. It is not diﬃcult to prove that the modular function is a continuous homomorphism (considering (0, ∞) with its multiplicative structure and natural topology). G is called unimodular when

Λ(g) ≡ 1, that is when the left Haar measure is also a right Haar measure. For example, abelian groups are unimodular. Also if G is a compact group then G is unimodular. Indeed, since Λ is homomorphism and G is compact Λ(G) must be a compact subgroup of (0, ∞), thus Λ(G) ≡ 1.

Deﬁnition 2.2.4. A topological group is called amenable if it has an invariant mean, i.e. a linear real-valued functional φ on the Banach Space RUCb(G), which is positive (φ(f) ≥ 0 whenever f is a non-negative function), of norm one, and invariant under the left action of G, that is φ(f) = φ(Lg(f)) for all g ∈ G and f ∈ RUCb(G). 40

Example 2.2.5. If G is compact then G is amenable. In the case of G compact,

one has that RUCb(G) = C∗(G), since G has a unique uniformity. An invariant mean m on C∗(G) is in correspondence with exactly one invariant regular Borel

measure on G with m(1G) = 1. Thus the normalized Haar measure is the unique invariant measure induces that induces a unique invariant mean on C∗(G) through φ f f x dµ the Haar integral ( ) = ∫G ( ) .

If G is locally recall that L1(G) is a Banach ∗−algebra under the convolution product and the involution f ∗(x) = Λ(x−1)f(x−1) where Λ is the modular function of G. If, B is a Banach ∗−algebra, a ∗−representation of B on a Hilbert space H is a ∗−homomorphism φ ∶ B → L(H), where L(H) is the space of all bounded linear YY operators on H, notice that the norm closure of C = φ(B) is a C∗-subalgebra of L(H). One says that φ is nondegenerate if C is nondegenarete, that is, if there does

not exist v ∈ H, v =~ 0 such that φ(x)v = 0, ∀x ∈ B or equivalentely ⋂x∈G Kerφ = {0}. It turns out that when G is locally compact there is a one-to-one correspondence between the unitary representations of G and the nondegenerate

∗−representations of L1(G). For a complete proof of such result theorems the reader can go to [8].

Theorem 2.2.6. Let π be a unitary representation of G. The map f → π(f) is a

1 nondegenerate ∗−representation of L (G) on L(Hπ).

The induced ∗−representation of L1(G) comes from the use of tools from the integral calculus for operators on Hilbert spaces, namely, if π is aunitary

1 representation of G on L(Hπ) for f ∈ L (G) let

π(f) = S f(x)π(x)dx

or, 41

< π(f)u, v >= S f(x) < π(x)u, v > dx

where the integrals are taken with respect of the Haar measure of G and the

<, > is the dot product of the Hilbert space where π is deﬁned. One can proof that the operator π on L1(G) is a multiplicative ∗−homomorphism and non-degenerate, using analytic tools.

Theorem 2.2.7. Suppose π is a nondegenerate ∗−representation of L1(G) on a Hilbert space H. Then π arises from a unique unitary representation of G on H in the sense of theorem 2.2.6.

This theorem can be proven using an approximate unit in L1(G) to deﬁne a unitary representation of G. Every locally compact group G has the left regular representation on

2 L (G), deﬁned by λ(g)f(x) =g f(x) = Lg(f)(x). This map is the same used in the proof of the ﬁrst part theorem 2.1.20 just that on L2(G) instead of RUCb(G). λ is unitary since as shown above λ acts by isometries and since the Haar measure is

2 translation invariant, the dot product in L (G) gives < λf1, λf2 >=< f1, f2 >. In view of theorem 2.2.6, one can deﬁne the following C∗-algebras in terms of unitary representations of G. The reduced group C∗-algebra of G is deﬁned to

∗ 1 ∗ be Cr (G) = λ(L (G)). The group C -algebra of G is the closure of the universal

1 representation of L (G). This means, take πu to be a direct sum of all irreducible representation (up to unitary equivalence) of G. Then C∗(G) is the norm closure of

1 πu(L (G)).

Theorem 2.2.8. If G is a locally compact group, then the left regular representation

∗ ∗ λ of C (G) onto Cr (G) is an isomorphism if and only if G is ameanable. 42

Some tools from the theory of positive deﬁnite functions on G and from the theories of abstract harmonic analysis and C∗-algebras are used to proof this theorem, for more details the reader can go to [5], [8].

2.3 Free abelian topological group

In this section a discussion is presented about question1 for the class of free

abelian topological groups A(X) and the free locally convex spaces L(X) generated by a Tychonoﬀ space X. This question was answered aﬃrmatively by V. Uspenskij in [38]. The following lines include some results, proofs and questions that article.

Deﬁnition 2.3.1. Let X be a topological space. The free topological group over X is a triple consisting of X, a topological group G and a continuous map

σ ∶ X → G from X to a Hausdorﬀ topological group G that satisﬁes the following:

1. The image of σ(X) topologically generates the group G, i.e. the subgroup ⟨σ(X)⟩ is dense in G.

2. For every continuous mapping f ∶ X → H from X to a topological group H, there exist a continuous homomorphism fˆ∶ G → H such that f = fσˆ .

The triple (X, G, σ) is denoted by F (X)

If one lets all the groups in the above deﬁnition to be abelian, then the triple

(X, G, σ) is called the free abelian topological group and it is denoted by A(X). It can be proven that F (X) exists and is unique up to topological isomorphisms when X is Tychonoﬀ. A proof of the following two classical theorems can be found in [1].

Theorem 2.3.2. The free topological group F (X) and the free abelian topological group A(X) on a Tychonoﬀ space X are unique up to a topological isomorphism 43

that ‘ﬁxes’ points of X. This means that if (X,G1, σ1) and (X,G2, σ2) are free (abelian) topological groups over X, then there exists a topological isomorphism

φ ∶ G1 → G2 such that σ2 = φσ1

Theorem 2.3.3 (Markov). The free topological group F (X) = (X, G, σ) over X exists for every Tychonoﬀ space X, and the mapping σ ∶ X → G is a topological embedding. In addition the image σ(X) topologically generates G. The same is true for the free abelian topological group A(X)

Remark 2.3.4. In the proof presented in [1] for theorem 2.3.3, X is embedded in the product of a family of topological groups through the diagonal product of a family of continuous functions that go from X to each of the factors, this diagonal product becomes σ. Some restrictions are applied so that this family remains a set. Then G

is deﬁned as the group algebraically generated by σ(X). The fact that X is Tychonoﬀ is used to make sure that the family of functions whose diagonal product is σ generates the topology in X which is used to show that σ is a homeomorphism.

Deﬁnition 2.3.5. Let X be a topological space. The free locally convex space

over X is a pair consisting of a locally convex space L(X) and a topological embedding X → L(X) such that every continuous mapping f from X to a locally convex space E extends to a continuous linear operator fˆ∶ L(X) → E.

It can be proven that if X is a Tychonoﬀ space then the free locally convex

space L(X) exists and the set X is an basis (as vector space) of L(X). This gives the equivalent definition for L(X) in the case of X a Tychonoff space. Let X be a Tychonoff space, the free locally convex space L(X) is the defined as follows: Let X be an (algebraic)basis of L(X) and let its topology T0 be the coarsest topology (the one with fewest open sets) that makes continuous all the linear 44

extensions of continuous mappings f ∶ X → E where E is a Hausdorﬀ locally convex space.

Theorem 2.3.6 (See [31] and [32]). The canonical homomorphism i ∶ A(X) → L(X) is an embedding of A(X) into the additive topological group of the locally convex space L(X) as a closed additive topological subgroup.

Let G be a topological group G is unitarily representable if G is topologically isomorphic to a subgroup of U(H) for some Hilbert space H. A group G that is unitarily representable has ‘enough’ unitary representations since the one unitary representation that makes it unitarily representable separates all elements in G.

Proposition 2.3.7. Every Banach space X is a Banach quotient of a Banach space of the form l1(A). Where Banach quotient means a Banach space of the form E~F with norm SSx + F SS = inf{SSySS ∶ y ∈ x + F }.

Proof. Let A be a dense subset of the unit ball of X, take p ∶ l1(A) → X in the natural way. This is a quotient map.

Proposition 2.3.8 (the Bartle-Graves theorem). Every linear onto map p ∶ E → F between Banach spaces (or locally convex Free´chet spaces) has a (possibly

non-linear) continuous right inverse s ∶ F → E, i.e. such a map that ps = 1F

Proof. This result is a consequence of Michael’s Selection Theorem for convex-valued maps: Suppose X is paracompact, E is a locally convex Fr´echet

space, and for every x ∈ X a closed convex non-empty set Φ(x) ⊂ E is given. Suppose that Φ is lower semicontinuous: for every U open in E,

Φ−1(U) = {x ∈ X ∶ Φ(x) ∩ U ≠ ∅} is an open set in X. Then Φ has a continuous selection s ∶ X → E such that s(x) ∈ Φ(x) for every x ∈ X 45

Take X = F , and Φ(x) = p−1(x),Φ(x) is closed and convex and the lower semicontinuity is given by the Open Mapping theorem, which says that p is an open map.

Proposition 2.3.9. The additive group of the space of complex (equivalence classes

of) integrable functions L1(µ) is unitarily representable for every measure space (Ω, µ). In particular if µ is the counting measure for a set A, the Banach space l1(A) is unitarily representable.

By theorem 2.3.6 it is enough to consider if L(X) is unitarily representable since A(X) is topologically embedded with the canonical homomorphism in L(X).

Theorem 2.3.10 (Uspenskij). For every Tychonoﬀ space X the free locally convex

space L(X) and the free abelian topological group A(X) admit a topologically faithful unitary representation.

Since the product of a family of unitarily representable topological groups is unitarily representable (just take the direct sum of representations), by proposition 2.3.9 one sees that theorem 2.3.10 is a direct consequence of the following theorem.

Theorem 2.3.11 (Uspenskij). For every Tychonoﬀ space X the free locally convex

space L(X) is (topologically) isomorphic to a subspace of a power of the Banach space l1(A) for some A.

Proof. By deﬁnition (L(X),T0) can be topologically embedded in a product of

Hausdorﬀ locally convex spaces (see remark 2.3.4). Let T1 be the coarsest topology

on L(X) that makes continuous all the linear extensions of continuous mappings of

1 the form f ∶ X → l (A), for some A. If it is showed that T0 = T1 then one gets that

1 (L(X),T0) can be embedded in the product of spaces of the form l (A). In order to 46

show that T0 = T1, it will be showed that for every map f ∶ X → F where F is a ¯ Hausdorﬀ locally convex space, the linear extension f ∶ L(X) → F is T1 continuous. Without loss of generality, since any F Hausdorﬀ locally convex space can be embedded in a product of Banach spaces, one assumes that F is a Banach space.

From proposition 2.3.7 let p ∶ l1(A) → F be a quotient map.

`1(A) `1(A) g < T g¯ : p s p X / F L X / F f ( ) f¯

By proposition 2.3.8 p must have a continuous right inverse s ∶ F → l1(A). This

1 implies that g ∶ X → l (A), g = sf lifts f or f = pg. By deﬁnition of T1, the linear ¯ extensiong ¯ ∶ L(X) → F is T1 continuous which implies that f = sg¯ is T1 continuous.

Corollary 2.3.12. Every Polish abelian group is the quotient of a closed abelian subgroup of the unitary group of a separable Hilbert space

In [38] the following questions remain open:

Question 4. In the non-abelian case let F (X) be the free group generated by X a compact space. Is F (X) unitarily representable?

An aﬃrmative answer to question4 would imply a positive answer to a problem posted by Kechrich: Is every Polish group a quotient of a closed subgroup of the unitary group of a separable Hilbert Space.

Proposition 2.3.13 (Uspenskij). Let N be the Baire space (NN with product topology) . If the group F (N ) is unitarily representable, then every Polish group is a quotient of a closed subgroup of the unitary group of a separable Hilbert space. 47

Remark 2.3.14 ([38]). A topological group is uniformly Lindelo¨f if for every neighborhood U of the unity the group can be covered by countably many left (equivalently right) translates of U. If G is a uniformly Lindel¨of group of isometries of a metric space M then the orbit Gx for x ∈ M is separable (Guran’s Theorem). If G is a uniformly Lindel¨of subgroup of U(H) for a (non-separable Hilbert) space, it follows from Guran’s theorem that H is covered by separable closed

G − invariant linear subspaces and then G can be embedded in a product of unitary groups of separable Hilbert spaces.

The group F (N ) is uniformly Lindel¨of, since it is a separable topological group (N topologically generates F (N )). If F (N ) is unitarily representable, ir follows from what was mentioned above that F (N ) is topologically isomorphic to a subgroup of U(H) where H is a separable Hilbert space.

Proof. Let G be a Polish group. Then by lemma 2.3.15 and the ‘universal’ nature of the free group, there exists a quotient onto map F (N ) → G. Using this maps and a factorization argument one can see that there is a subgroup N of a countable power or U(H) and G is a quotient of N. One can assume that N is closed since the quotient homomorphism can be extended to the closure of N.

Lemma 2.3.15. Every Polish space X admits a continuous and open surjection from the Baire Space (NN)

Proof. The ﬁrst claim is that for every open set U ⊂ X and every > 0 there exit

U0,U1,U2,... ⊂ U such that diam(Un) < for all n, and U = ∪nUn = ∪nUn. Indeed, let

D be a dense countable set of X. Let U0,U1,... list all the sets B 1 d for every n ( ) 1 d D and , notice that B 1 d U. If x U there is an m small enough so that ∈ n < n ( ) ⊂ ∈ 48

1 and B 1 x U. Since D is dense, there is d D such that x B 1 d U and m < m ( ) ⊂ ∈ ∈ 4m ( ) ⊂

B 1 d is one of the Uk. 4m ( ) Let σ be a ﬁnite sequence, notice that the sets

Nσ = {f ∈ N ∶ f extends σ(σ ⊂ f)} form a cloopen basis for N . One uses the claim above proven to build inductively a collections of open sets in X, {Uσ ∶ σ ∈ N , σ a finite sequence} that satisﬁes the following:

1. U∅ = X

1 2. diam(Uσ) < SσS

3. Uτ ⊂ Uσ when σ ⊂ τ

∞ 4. Uσ = ∪t=0Uσ∧t where σ ∧ t is the sequence with ﬁrst components σ and then adding as last element t.

If f ∈ N using the completeness of X we define φ ∶ N → X as follows ∞ ∞ ∩n=0UfSn = ∩n=0UfSn = {φ(f)}, where fSn is the finite sequence of the first n elements of f.

φ is surjective since we can always build sequences Uσ0 ⊂ Uσ1 ... for which

Uσ0 = X, and for x ∈ Uσn there is always a j ∈ N such that x ∈ Uσn∧j = Uσn+1 . 1 Let φ(f) = x, if gSn = fSn then φ(g) ∈ UfSn thus d(φ(g), φ(f)) < n then φ is continuous and open.

2.4 Group of isometries of an inﬁnite-dimension Hilbert space

Below it shown, following V. Uskpenkij [37], that the group of isometries (not necessarily linear) of a inﬁnite-dimensional Hilbert space endowed with the strong operator topology is topologically isomorphic to a group of unitary operators of a Hilbert space. 49

∗ Let A = (aij) be a complex matrix and denote by A = (aji). A is called Hermitian if A = A∗. A Hermitian matrix A is positive if all igenvalues of A are positive, equivlentely, if A = B2 for some Hermitian B. Another useful characterization states that A is positive if for any n-dimensional vector z, z∗Az ≥ 0, n n that is, for any αi ∈ C, i = 1, 2, ..., n, ∑i=1 ∑j=1 αiαjaij ≥ 0. A complex function p on a −1 group G is positive-deﬁnite if for every g1, g2, ..., gn ∈ G the n × n-matrix p(gi gj) is Hermitian and positive.

Notice that if ρ ∶ G → U(H) is a unitary representation of G, then for every

vector v ∈ H the function pv using the scalar product <, > of H as pv =< ρ(g)v, v > is positive-deﬁnite. Indeed,

−1 −1 −1 pv(gi gj) =< ρ(gi gj)v, v >=< ρ(gj)v, ρ(gi )v >

thus for any αi ∈ C, i = 1, 2, ..., n,

n n n n n n −1 Q Q αiαjp(gi gj) = Q Q αiαj < ρ(gj)v, ρ(gi)v >= Q Q < αiρ(gj)v, αjρ(gi)v >= i=1 j=1 i=1 j=1 i=1 j=1

n n n n =< Q Q αiρ(gj)v, Q Q αjρ(gi)v >≥ 0 i=1 j=1 i=1 j=1

On the other hand, if p is a positive-deﬁnite function on G then p = pv where pv arises from a unitary representation ρ ∶ G → H, as described above. A route to get an appropriate H is to consider the group algebra C[G] equipped with the scalar product < g, h >= p(h−1g). Take the quotient over the kernel of such scalar product and ﬁnally take the completion on that resulting space to get H. The regular left regular representation of G extended to act in the natural way on H induces a 50

unitary representation of G on H, this can be proven using the following proposition for which a proof can be found in [37].

Proposition 2.4.1. A topological group G admits a topologically faithful unitary representation, that is, G is isomorphic to a subgroup of Us(H) for some Hilbert space H, if and only if for every neighborhood U of the neutral element e of G there

exist a continuous positive-deﬁnite function p ∶ G → C and a > 0 such that p(e) = 1 and S1 − p(g)S > a for every g ∈ G ∖ U.

Using this previous proposition, V Uspenskij proved the next theorem, which together with the following lemma prove theorem 2.4.4.

Lemma 2.4.2. (V. Uspenskij, [37]) Let H be a Hilbert space, and let x1, ..., xn be

2 points in H. Then the symmetric n × n−matrix (exp(−Yxi − xjY )) is positive.

Theorem 2.4.3. (V. Uspenskij, [37]) Let (M, d) be a metric space, and let G = Is(M) be its group of isometries. Suppose that there exists a real-valued positive-deﬁnite function p ∶ R → R such that:

1. p(0) = 1, and for every > 0 we have sup{p(x) ∶ SxS ≥ } < 1

2. for every points a1, ..., an ∈ M the symmetric real n × n-matrix p(d(ai, aj)) is positive

Then the topological group G is isomorphic to a subgroup of unitary group.

Theorem 2.4.4. (V. Uspenskij, [37]) Let H be a Hilbert space, and let G = Is(H) be a topological group of all (not necessarily linear) isometries of H. Then G is isomrphic to a subgroup or a unitary group.

Proof. This case can be studied from theorem 2.4.3, when M = H. By lemma 2.4.2 p(x) = exp(−x2) satisﬁes the condition in 2.4.3 and the result is proven. 51

2.5 Extremely amenable groups

For the class of abelian topological groups one can answer in the negative question2 in this section this argument is presented. In section 2.2 amenability for a group was deﬁned, the following classical theorem provides a useful characterization for a proof see [3].

Deﬁnition 2.5.1. A G-space is a topological space X with a continuous action of

G. Let X be a G-space, a point x ∈ X is G-ﬁxed if gx = x for all g ∈ G.

Theorem 2.5.2. For a topological group G, the following properties are equivalent:

1. G is amenable

2. (Fixed point property): any continuous aﬃne action of G on a non-empty compact convex subset X of a locally convex topological vector space has a

G−ﬁxed point

This theorem motivates the following deﬁnition

Deﬁnition 2.5.3. A topological group G is said to be extremely amenable if G has a G-ﬁxed point in every compact set X it acts upon.

Remark 2.5.4. f ∈ RUCb(G) extends to a unique continuous function f¯ on a compactification given by the maximal ideal space of RUCb(G). Let G be extremely amenable, G acts on this compactification by left multiplication in a natural way. Let x∗ be a G-fixed point of this action (notice that x∗ is a state of RUCb(G)). Define φ(f) = f¯(x∗), φ is an invariant mean of G. Thus G is amenable in the sense of the initial definition given in section 2.2.

Many examples of extremely amenable groups have arisen in the throughout the years, many of them being ‘massive’ groups. In [24], Pestov develops some of 52

the theory of these groups and provides characterizations for extremely amenability

in terms of the Ramsey-Dvoretzky-Milman phenomenon also known as the ﬁnite oscillation stable property (see 2.6.2). An example of a class of groups that is extremely amenable is the class of Le´vy groups. A space with metric and measure, or mm-space is a triple (X, d, µ), consisting of a set X, a metric d on X, and a probability Borel measure µ on the metric space (X, d). For a subset A of a metric space X and an > 0, denote by A

the −neighborhood of A in X. A family X = (Xn, dn, µn)n∈N of mm-spaces is called a Le´vy family if, whenever Borel subsets An ⊂ Xn satisfy, liminfn→∞µn(An) > 0,

one has that for every > 0 limn→∞µn((An)) = 1. A metrizable topological group G is called textbfL´evy group if it contains an increasing chain of compact subgroups

G1 < G2 < ... < Gn < ... that has a dense union in G and such that for some right invariant compatible metric d on G the groups Gn, equipped with the normalized Haar measure and the restrictions of the metric d, form a Lévy family. In [11] it was established that every Lévy group is extremely amenable. In the following lines some examples of Lévy groups and extremely amenable groups are given:

Example 2.5.5. It was established by Pestov in [25] that the group of isometries of the the Uryshon space, denoted by Iso(U) [24] is a L´evy group thus it extremely amenable.

The Urysonhn metric space U is deﬁned by the following three conditions.

1. U is a complete separable metric space

2. U is ultrahomogeneous, that is, every isometry between two ﬁnite metric subspaces of U extend to a global isometry of U into itself.

3. U is universal, that is contains an isometric copy of every separable metric space. 53

Also Iso(U) satisﬁes the following condition: (Uspenskij see [33]) The Polish group Iso(U) is a universal second-countable topological group. In other words, every second-countable topological group G embeds into Iso(U) as a topological subgroup.

Example 2.5.6. It was proven by M. Gromov and V.D. Milman that the unitary

gropu U(l2), equipped with the strong operator topology is a L´evy group, thus extremely amenable.

Example 2.5.7. V. Pestov proved (see [24]) that the group Aut(Q, ≤) of order-preserving self-bijections of the set Q equipped with the natural Polish

topology, is extremely amenable. Observe that if h ∶ G → H is a continuous homomorphism between topologicla groups, having dense image, and G is extremely

amenable, then so is H. This together with the fact that Aut(Q, ≤) is extremely

amenable implies that Homeo +(R) with the compact-open topology is extremely

amenable. Since Homeo +(R) = Homeo + [0, 1] then one also concludes that Homeo + [0, 1] is extremely amenable.

An non-example of extremely amenable groups are locally compact groups. It follows from Veech’s theorem that no locally compact group is extremely amenable (See theorem 3.4.1).

Proposition 2.5.8. Let G be an abelian extremely amenable group then G admits no non-trivial characters.

Proof. By contrapositive, let φ ∶ G → T be a non-trivial character. Let (g, z) → φ(g)z be an action of G on T . Since φ is non-trivial then this action has no G-ﬁxed point. The action is also continuous. Thus G has a ﬁxed-point-free action on T which implies that G is not extremely amenable. 54

2.6 Glasner-Pestov problem

When studying question3 one is asking for an abelian topological group G to have ‘enough’ characters to separate its elements, perhaps a more basic question is:

Question 5. Does the abelian topological group G have any non-trivial characters at all?

In this section question5 is discussed for some speciﬁc classes of topological abelian groups. These problems were posted by E. Glasner and V. Pestov. A the end some of the implications and connections of these problems are presented.

Question. Let X be compact metric, f ∶ X → X an isometry diﬀerent to the identity, let G = {f n ∶ n ∈ Z} ⊂ Homeo (X). Does G admit non-trivial characters?

Answer. Yes, Since f is an isometry then G is equicontinuous. By Arzel`a-Ascoli theorem G is compact and Peter-Weyl theorem applies.

Recall that, when dealing with Homeo (X) one endows such group with the topology of uniform convergence, unless otherwise mentioned. So the topological groups mentioned below inherit such topology.

Question. Let X be compact metric, f ∈ Homeo (X) (), f not the identity map, let G = {f n ∶ n ∈ Z} ⊂ Homeo (X). Does G admit non-trivial characters?

Answer. No, there exist group topologies T on Z such that G = (Z, T ) does not have non-trivial characters, since every Hausdorﬀ topological group is topologically isomorphic to a group of transformations on a compact space (Teleman’s theorem see 2.1.20) this answers the question in a negative way. For an example of this see [2]. In known examples f has a ﬁxed point, that is the group generated by f is extremely amenable. 55

The fact that the previous question can answered in a negative with examples where f has ﬁxed point motivates the following open question:

Question 6 (Glasner’s Problem). Let X be compact (metric), f ∈ Homeo (X), f ﬁxed point free, G = {f n ∶ n ∈ Z} ⊂ Homeo (X). Does G admit non-trivial characters?

A generalization of Glasner’s problem, which remains unsolved, occurs using the following notion.

Question 7 (Pestov’s problem). Let X be compact, G ⊂ Homeo (X) an abelian group, such that X has no G-ﬁxed points. Does G admit non-trivial characters?

Together questions6 and7 are known as the Glasner-Pestov Problem.

Deﬁnition 2.6.1. An abelian group G is called minimally almost periodic if G has no non-trivial characters. These class of groups is associated with the collection of almost periodic functions, for more details on this see example 3.3.3.

The converse of proposition 2.5.8remains open and is a reformulation of the Glasner-Pestov problem:

Question 8. Are all abelian topological groups G that do not admit a non-trivial

character extremely amenable? What if G = Z? or which is the same, is every minimally almost periodic abelian group extremely amenable?

2.6.1 Syndetic sets and Bohr topology

In the following lines the connection between Glasner’s problem, syndetic sets

and the Bohr topology of Z is discussed in proposition 2.6.8

Deﬁnition 2.6.2. Let S ⊂ Z, S is syndetic if S + [0, n] = Z for some positive n ∈ Z. This is S is syndetic if it has bounded gaps. 56

Deﬁnition 2.6.3. The Bohr topology in Z is generated by the basic neighborhoods:

n {n ∈ Z ∶ S1 − ai S < , i = 1, 2, ..., s}, a1, a2, ..., as ∈ T , > 0.

Remark 2.6.4. The Bohr topology can be deﬁned for any non-compact locally compact abelian group G in the following way. Consider the set of all continuous characters and provide it with the discrete topology, denote this set by Gˆdis. We consider the dual group of Gˆdis which is denoted by bG (set of all homomorphisms from Gˆdis to T ), since Gˆdis is discrete then bG is compact. The evaluation map from

G to the dual group of Gˆdis is continuous but not a homeomorphism and the isomorphic image of G under this map is dense on bG. The topology inherited from bG to the isomorphic image of G is the Bohr topology of G and bG is called the Bohr compactiﬁcation of G. The Bohr compactiﬁcation has the following ‘universal’ property:

If K is a compact group and ρ ∶ G → K is a continuous homomorphism, then ρ extends to a continuous homomorphism from bG to K. For more details on the Bohr topology and Bohr compactiﬁcation the reader can see example 3.3.3 or go to [8] and [24]

Deﬁnition 2.6.5. Let G be a topological group, continuously acting on a compact space X. Such an X is called a minimal G-space if it contains no proper compact

G-subspace. Equivalently if the orbit of every point x ∈ X is dense in X.

Applying Zorn’s lemma one can see that every G-space contains a minimal G-subspace. For example for a space X with a G-ﬁxed point, the minimal G-spaces are singletons.

Proposition 2.6.6 ([24]). A topological group (Z, τ) has no non-trivial continuous characters if and only if every Bohr neighborhood of the identity in (Z, τdiscrete) is dense in (Z, τ). 57

Proposition 2.6.7. Let X be compact topological space and f ∈ Homeo (X) and (X, f) is minimal then for every non-empty open set U ⊂ X and p ∈ X the set S = {n ∈ Z ∶ f n(p) ∈ U} is syndetic.

Proof. Since f ∈ Homeo (X) and (X, f) is minimal then F = {f n(U) ∶ n ∈ Z} is an open cover of X. Indeed, suppose that y is not in any of the elements of F, then the orbit of y would miss V which contradicts the fact that (X, f) is minimal. Since X

N −k is compact there is N ∈ N such that X = ⋃k=0 f (U), which means that S + [0,N] = Z.

Consider the following statements:

Statement (Gl). If f ∈ Homeo (X) is a ﬁxed point free, then G = {f n ∶ n ∈ Z} has a non-trivial character

Statement (SB). If S ⊂ Z is syndetic, then S − S contains a Bohr neighborhood of zero.

Proposition 2.6.8 (Glasner, Pestov). Not (GL) Ô⇒ Not (SB)

Proof. Let f ∶ X → X be ﬁxed point free, such that G = {f n ∶ n ∈ Z} has no non-trivial characters. Assume (X, f) is minimal. If U is open in X and p ∈ X, then the set S = {n ∈ Z ∶ f n(p) ∈ U} is syndetic and S − S ⊂ {n ∈ Z ∶ f n(U) meets U} not ¯ ¯ dense in (Z, T1), (where T1 is induced by G) if U is small (fU ∩ U = ∅). Thus by proposition 2.6.6 and the fact that (Z, T1) admits no non-trivial characters then S − S is not a Bohr neighborhood of 0.

2.6.2 Long almost constant pieces

Another way of looking at Glasner’s problem is by using 2.6.9 as follows: 58

Proposition 2.6.9 (Pestov’s, see [26]). A topological group is extremely amenable if and only if for all left-invariant pseudometric d on G the G-space (G, d) has the Ramsey-Dvoretzky-Milnam property.

Let G act on a metric space (X, d) by isometries. X has the Ramsey-Dvoretzky-Milnam property or is ﬁnitely oscillation stabe, if for all bounded uniformly continuous functions f ∶ X → R, for all ﬁnite A ⊂ X and for all > 0, there exists g ∈ G such that diam(f(gA)) < .

Definition 2.6.10. Let (K, d) compact metric, xn ∈ K, n ∈ Z. The sequence (xn) is almost flat if it contains long almost constant pieces, i.e., if for all > 0 and for all n ∈ Z there exists k ∈ Z such that diam{xk+1, . . . , xk+N } < . Define the seminorm on

Z associated to the almost ﬂat sequence {xn} as follows p(n) ∶= sup{d(xk, xn+k) ∶ k ∈ Z}.

By Pestov’s criterion (2.6.9), If τ is a topology on Z, (Z, τ) admits a ﬁxed-point-free continuous action on X if and only if there exists a not almost ﬂat sequence {xn} in [0, 1] and the associated seminorm p(n) on Z is continuous on (Z, τ). In this sense one can rewrite Glasner’s problems as follows:

Question 9. Suppose {xn} is not almost ﬂat. Does the group (Z, p) admit a non-trivial character?

2.7 A minimally almost periodic group

Some examples of minimally almost periodic groups are cited in [24] (page 61), it is unknown if they are extremely amenable or not. These are natural candidates to study when trying to solve the Glasner-Pestov problem. In this section another example of this kind is introduced. It turns out that if (Ω, M, µ) is a measure space which contains no atom with ﬁnite measure, the additive group of the topological 59

vector space Lp(Ω, µ) for 0 < p < 1 is minimally almost periodic. Below a complete proof for that claim for Lp[0, 1] is given to provide the reader with an idea of the techniques used to prove the general case.

In this section let 0 < p < 1. Recall that if a, b ≥ 0 then:

p p p (a + b) ≤ a + b

Inded, this follows from:

(ap + bp)(a + b)1−p = ap(a + b)1−p + bp(a + b)1−p ≤ ap(a)1−p + bp(b)1−p = a + b The elements of Lp[0, 1] are equivalence classes of Lebesgue measurable functions f on [0, 1] that coincide almost everywhere, for which:

1 p ν(f) = Sf(t)S < ∞ S0 from the inequality mentioned above that,

ν(f + g) ≤ ν(f) + ν(g)

and

d(f, g) = ν(f − g)

deﬁnes metric on L[0, 1] that is invariant under addition. The completeness of of d is proven similarly to the case when 1 < p using the convergence theorem for the

p Lebesgue integral. Notice that the open balls Br = {f ∈ L ∶ ν(f) < r} form a local

p base for the topology in L [0, 1]. Since B1 = r−1~pBr for all r > 0, then B1 bounded in the topological vector space structure. This means that (Lp[0, 1], d) is a locally bounded F -space (complete with an additive invariant metric), with In particular,

(Lp[0, 1], d) is topological group with the additive structure. 60

Proposition 2.7.1. (Lp[0, 1], d) contains no convex open sets, other than ∅ and the whole space.

Proof. Suppose that V =~ ∅ is an open convex set of Lp[0, 1]. Without lost of

generality, let 0 ∈ V . Then there exists r > 0 such that Br ⊂ V . One can see that there is an n ∈ Z+ such that np−1ν(f) < r. h x x f t dt h x Consider the indeﬁnite integral ( ) = ∫0 S ( )S , Since ( ) is continuous then

there exist points 0 = x0 < x1, ..., < xn = 1 such that x +1 i f t pdt n−1ν f , i 0, 1, ..., n. ∫xi S ( )S = ( ) =

Let gi(t) = nf(t) if xi−1 < t ≤ xi, gi(t) = 0 otherwise. Then gi ∈ V since

p−1 ν(gi) = n ν(f) < r, i = 0, 1, ..., n, and Br ⊂ V . Since V is convex and

1 p f = n (g1 + ... + gn) then f ∈ V . Thus V = L [0, 1].

Proposition 2.7.2. If the only convex open sets in a topological vector space X are

∅ and X then X has no non-trivial continuous linear functionals.

Proof. Suppose that Λ ∶ X → Y is a continuous linear mapping into a locally convex space Y . Let B a convex local base for Y . If W ∈ B, then Λ−1(W ) is convex, open and non-empty, hence Λ−1(W ) = X. This implies that Λx = 0 for all x ∈ X.

This means (Lp[0, 1], d) has no non-trivial continuous linear functionals. More generally, with similar techniques the following can be proven that:

Theorem 2.7.3. (Farkas [7]) Let Lp(Ω, M, µ, R), 0 < p < 1, be the set of M-measurable functions f → R which satisfy ν(f) < ∞. Then the dual space of Lp(Ω, M is trivial if (Ω, M, µ) contains no atom with ﬁnite measure, i.e. for every A ∈ M which satisﬁes 0 < µ(A) < ∞, there exists a set b ∈ M such that 0 < µ(B) < µ(A) < ∞.

Proposition 2.7.4. Let X be topological vector space, suppose that additive group of X has a non-trivial character, then X a non-trivial functional. 61

Proof. From the theory of covering spaces recall that p ∶ R → T, p(t) = (cos(2πt), sin(2πt)), is a covering map from R to T. Let γ be a non-trivial character for the additive group of X. Since every topological space is simply connected, then there exist a unique continuous mapγ ¯ ∶ X → R, such thatγ ¯(0) = 0 and makes the following diagram commute:

> R γ¯ p X γ / T

Since γ(x + y) = γ(x)γ(y) = p(γ¯(x), γ¯(y)) and the nature of p, one knows that γ¯(x + y) = γ¯(x) + γ¯(y) + K, where K ∈ Z. Make y = 0, then K = 0 sinceγ ¯(0) = 0 . Thusγ ¯ is a continuous homomorphism from X to R. It is easy to show by induction that for n ∈ Z and any x ∈ X,γ ¯(nx) = nγ¯(x), thus for 0 =~ m ∈ Z n nx n n mγ¯( m x) = γ¯(m( m )) = γ¯(nx) = nγ¯(x), thusγ ¯( m x) = m γ¯(x). By continuity ofγ ¯ this implies thatγ ¯(αx) = αγ¯(x), for all α ∈ R and for all x ∈ X. Thusγ ¯ is a continuous functional.γ ¯ is non-trivial since γ is non-trivial.

The following corollary follows immediately

Corollary 2.7.5. Let X = Lp(Ω, M, µ, R), 0 < p < 1, be the set of M-measurable functions f → R which satisfy ν(f) < ∞. If (Ω, M, µ) contains no atom with ﬁnite measure, then the additive topological group of X is minimally almost periodic. 62 3 Compactiﬁcations and free actions

In this chapter a route to solve the Glasner-Pestov problem by studying the actions of topological groups on some of their natural compactifications is presented. In section 3.1 a well known relationship between uniformities, compactifications and C∗-algebras related to a Tychonoff space is introduced. In section 3.2 some of the natural compactifications for topological groups are discussed and in section 3.3a special focus is given to the WAP compactification. In section 3.4, some known results about when a group G acts freely on a compact space are given. In the final section some applications to the Glasner-Pestov problem are presented.

3.1 Uniformities, compactiﬁcations and C∗-algebras

Let X be a Tychonoff space, a compactification of X is a pair (Y, c) where Y is a compact space and c ∶ X → Y is a homeomorphic embedding where c(X) = Y . Y is denoted by cX. One can define an order relation in the family of all

compactiﬁcations of X. If (c1X, c1), (c2X, c2) are compactiﬁcation of X, and if there

exists f ∶ c1X → c2X, f continuous, such that f ○ c1 = c2 one says that c2 ≤ c1. Note

that the condition f ○ c1 = c2 implies that f is an onto map. It is easy to see that

transitivity of such relation and also that if c2 ≤ c1 and c1 ≤ c2 then c2 and c1 are homeomorphic. Denote by C the collection of all compactiﬁcations of X up to

homeomorphisms, thus (C, ≤) is a partial order. Denote by T the collection of all totally bounded uniformities compatible with

the the topology of X, it is clear that the (T, ≤) is partially ordered by set inclusion. Let A be a unital subalgebra of Cb(X) the set of bounded complex-valued functions on X, A is said to have regular separation property if points lying outside of closed subsets of X can be separated with a function from A. Denote by 63

A the collection of all unital C∗-subalgebras of bounded functions with the regular separation property. (A, ≤) with the order of set inclusion is a partial order. Consider the following maps:

1. α ∶ A → C, where α(A) = MA where MA is the maximal ideal space of A. MA

together with the map t ∶ x → hx, where hx(f) = f(x) is a compactiﬁcation of

∗ X. Indeed, It is clear that hx is a multiplicative functional of A ⊂ C (X), also

if hx = hy for x, y ∈ X then for all f ∈ A, f(x) = f(y), since elements in A separate points in X then x = y, which means that t is injective. On the other

hand, if a net xα → x in X, then f(xα) → f(x) for all f ∈ A, but this is

∗ hα → hx in MA with the weak topology. Thus the map t is continuous, and

because MA is compact, t must be a homeomorphic embedding. In order to

show that t(X) is dense in MA, suppose by contradiction that it is not. Then

there exists x0 ∈~ t(X), since MA is a compact Hausdorﬀ space then it is ˆ ∗ normal and by Urysohn’s lemma there exists f ∈ C (MA) such that ˆ ˆ f(t(X)) = {0} and f(x0)= ~ 0, since the Gelfand transform is an isomorphism for C∗-algebras then f ∈ A for which fˆ= Γf must be the 0 function, in which case fˆ≡ 0, which is a contradiction.

∗ Suppose that A1 ⊂ A2 thus there is a C -algebra embedding ι ∶ A1 → A2, and

let η ∶ MA2 → MA1 , such that η(h) = h ○ ι. It is easy to see that η is a continuous map since the weak∗ topology is being used for both spaces and ι

is and embedding. Also it is easy to see that t2 ○ η = t1, where t1, t2 are coming

from the compactiﬁcations MA1 and MA2 of X, thus MA1 ≤ MA2 and α is a order preserving map.

2. β ∶ C → T, where β(c) is the uniformity inherited by X from the Samuel compactiﬁcation cX via the embedding c. It is clear that since X can be 64

regarded as a subspace of cX the totally bounded uniformity of cX which is inherited to X is compatible with X, then β is well deﬁned. On the other

hand, suppose that (c1X, c1) and (c2X, c2) are compactiﬁcations of X, such

that β(c1X) = U1 and β(c2X) = U2, and (c1 ≤ c2). Then there exists

f ∶ c1X → c2X such that f ○ c2 = c1. Since f is a map between compact spaces f

is uniformly continuous. If X is regarded as a subspace of c1X and c2X, then

the restriction of fSX into c2X is just the identity map. The fact that f is

uniformly continuous implies that for every V ∈ U1 there exist U ∈ U2 such that

U ⊂ V , thus U1 ≤ U2. Hence β is a order preserving map.

3. γ ∶ T → A, deﬁned by γ(U) = UCB(X), where UCB(X) is the set of all bounded complex-valued uniformly continuous functions on X with respect of

U. It is easy to see that UCB(X) is a closed unital subalgebra of C∗(X).

A pseudometric d on (X, U) is uniformly continous if for every > 0 there is a V ∈ U such that if (x, y) ∈ V then d(x, y) < . Every uniform space admits a family of uniformly continuous bounded pseudometrics that determine U in the following sense: for every V ∈ U there is a bounded uniformly continuous pseudometric d such that {(x, y) ∈ X × X ∶ d(x, y) < 1} ⊂ V (see [6] theorem

8.1.10). If one ﬁxes x0 in d(x0, y) = f(y), f is a bounded uniformly continuous

function that separates x from the closed set X ∖ (Vx). This means that UCB(X) has the regular separation property. Thus UCB(X) ∈ A.

Let U1, U2 ∈ T, such that A1 = γU1 and A2 = γU2. It is clear that if f ∈ A1 then

f ∈ A2 thus A1 ⊂ A2. This means that γ is a order preserving map.

Theorem 3.1.1. The maps α, β, γ are order isomorphisms, thus A, T, C are order isomorphic. 65

Proof. Recall that the totally bounded replica of U for the uniform space (X, U) deﬁned in section 2.1.1, is the coarsest uniformity on X with regard to which every

bounded uniformly continuous function on (X, U) remains uniformly continuous. In the case of the replica of a uniformity inherited from the unique compactiﬁcation cX of X, such uniformity Uc is already totally bounded, therefore coincides with the replica. This means that C∗(cX) is isomorphic to the algebra of functions

UCBc(X) that are uniformly continuous in X with respect of the uniformity Uc

∗ inherited from cX. This means that γ(β(cX)) = γ(Uc) is isomorphic to C (cX). Consider a compactiﬁcation cX of X, let

∗ Ac = {f ∈ C (X) ∶ f extends continuously to cX}. It is clear that Ac is isomorphic

∗ ∗ to C (cX), moreover using the Gelfand transform Ac is isomorphic to C (MAc ).

By example 2.1.31 then MAc and cX are homeomorphic. Moreover, this is an equivalence of compactiﬁcations of X. This means that α(C∗(cX)) = α(γ(β(cX)) is equivalent to cX. Which means that α ○ γ ○ β is the identity in C. To show that the other permutations of this functions are bijection one can use

∗ the fact that the algebra of functions C (cX), Ac, and UCBc(X) are isomorphic.

3.2 Natural uniformities for topological groups

The class of topological groups as uniform spaces has many natural uniformities that are compatible with a topological group G. The uniformities (1), (2) and (3) below are classically known and broadly studied. Uniformity (4) has become more signiﬁcant for some “massive” groups, this fact was broght to attention ﬁrst by Roelcke [28] and then by the work of Uspenskij. A proof of the compatibility of these four uniformities with the topology of G and some of their properties can be found in [1]. 66

1. In example 2.1.6 the right uniformity R of G was deﬁned, its basic entourages

−1 are given by VR = {(x, y) ∈ G × G ∶ xy ∈ V } where V is a neighborhood of the identity e of G. As mentioned in section 2.1.3 the C∗-algebra of bounded complex-valued right uniformly continuous functions on G is denoted by

RUCb(G). The maximal ideal space of RUCb(G) and the Samuel compactiﬁcation of R is called the greatest ambit of G and is denoted by

S(G) = σGR.

2. Similarly one can deﬁne the left uniformity of G, denoted by LG or just L,

−1 its basic entourages are given by VL = {(x, y) ∈ G × G ∶ x y ∈ V } where V is a neighborhood of the identity e of G. One can deﬁne the C∗-algebra of bounded complex-valued left uniformly continuous functions on G as the

algebra of uniformly continuous functions from (G, L) to C. This C∗-algebra is denoted by LUCb(G). The maximal ideal space of LUCb(G) and the

Samuel compactiﬁcation will be denoted by σGL.

3. The two-sided group uniformity denoted by L ∨ R is deﬁned as the least upper bound of the uniformities L and R, with basic entourages of the

diagonal given by V∨ = VL ∩ VR where V is a neighborhood of the identity e of G. One can deﬁne the C∗-algebra of bounded complex-valued two-sided uniformly continuous functions on G as the algebra of uniformly continuous

functions from (G, L ∨ R) to C. This C∗-algebra will be denoted by TUCb(G). The maximal ideal space of TUCb(G) and the Samuel compactiﬁcation will be

denoted by σG∨.

4. The Roelcke uniformity or lower uniformity denoted by L ∧ R has as

basic entourages V∧ = {(x, y) ∈ G × G ∶ x ∈ V yV } where V is a neighborhood of the identity e in G. In general the greatest lower bound of two compatible 67

uniformities on a topological space need not to be compatible, although the Roelcke uniformity is compatible with the topology of G and it is indeed the

lower bound of L and R. The C∗-algebra of all bounded complex-valued functions on G which are both right, and left uniformly continuous from

∗ b (G, L ∧ R) to C. This C -algebra will be denoted by UC∧(G). The maximal

b ideal space of UC∧(B) and its Samuel compactiﬁcation will be called Roelcke compactiﬁcation and will be denoted by R(G).

In view of the result proven in section 3.1 the partial order isomorphic diagram of the above uniformities, compactiﬁcations and C∗-algebras is as follows: b σ(L ∨ R) σG∨ TUC (G)

b b σ(L) σ(R) σGL S(G) LUC (G) RUC (G)

b σ(L ∧ R) R(G) UC∧(G)

Deﬁnition 3.2.1. A topological group G is called balanced if it has a neighborhood basis at the identity element e ∈ G for which each neighborhood is invariant. A subset A of G is called invariant is xAx−1 = A, for each x ∈ G.

It is clear that abelian groups and compact groups are examples of balanced topological groups. Using the deﬁnitions of the objects which appear in he following proposition the result follows.

Proposition 3.2.2. The following conditions are equivalent for a topological group G

1. L = R

2. L ∨ R = L ∧ R 68

3. L = R = L ∨ R = L ∧ R

4. the group G is balanced

Remark 3.2.3. This proposition implies that the above diagrams become one point if and only if G is balanced. In particular this happens when G is abelian.

3.2.1 The greatest ambit

Recall that a G-space is a topological space X with a continuous action of G.A

G-map or equivariant map is a map f ∶ X → Y between G-spaces X and Y , such that f(gx) = gf(x) for all x ∈ X and g ∈ G. A compact G-space X together with a distinguished point x∗ ∈ X such that the G-orbit of x∗ is dense in X is called and ambit. A morphism between ambits (X, G, x∗) and (Y, G, y∗) is a G-map between X and Y such that f(x∗) = y∗. Consider G as a dense subset of the greatest ambit S(G). The action G × S(G) → S(G) extends the multiplication G × G → G. It is easy to see that for the G-space S(G) the identity e of G is a distinguished point and (S(G), e) is an ambit. Moreover it has the following property: For every G- ambit (X, x∗) there is a unique (G-ambit) morphism f ∶ S(G) → X. Indeed, the map g → gp from G to X is R-uniformly continuous, thus can be extended to S(G). The greatest ambit S(G) also has an topological algebraic structure. A semigroup is a set with an associative multiplication. A semigroup X is left-topological if it is a topological space and for every y ∈ X the self map x → xy of X is continuous.

Theorem 3.2.4. For every topological group G the greatest ambit X = S(G) has a natural structure of left-topological semigroup with a unity such that the multiplication X × X → X extends the action G × X → X. 69

Proof. Let x, y ∈ X. Using the universal property of X one gets that there exists a

unique G-map ry ∶ X → X (to the ambit (X, y)) such that ry(e) = y. Deﬁne the

multiplication in X as xy = ry(x). If one ﬁxes y and gets the map x → xy the map

matches ry, thus it is continuous. If y, z ∈ X, notice that ryz(e) = rz(ry(e)) = yz by

uniqueness of the G-map between X and the ambit (X, yz), ryz = rz ○ ry. This

implies that x(yz) = ryz(x) = rz(ry(x)) = (xy)z, the multiplication is associative.

Observe that ex = rx(e) = x and ex = re(x) = x since re is the identity. It remains to show that the action of G on X, matches the deﬁned multiplication. Let g ∈ G and x ∈ X, notice that rx(ge) = grx(e) since rx is a G-map, the right hand side of the equality referst to the action of G on X. Then one concludes that gx = rx(g) = rx(ge) = grx(e) = gx, where the last three terms refer to the action of G on X.

Remark 3.2.5. Using the maps ra one can conclude more about G-subspaces of

S(G). A subset I of a semigroup K is a left ideal if KI ⊂ I. It turns out that all closed G-subspaces of S(G) are left ideals. Let Y be a G-subspace and let a ∈ Y ,

then ra(G) = {ga ∶ g ∈ G} which is a subset of Y since Y is a G-subspace. Since ra is

continuous, G is dense in S(G), and Y is closed then ra(S(G)) ⊂ Y and S(G)Y ⊂ Y .

3.2.2 Universal minimal compact G-space

A G-space is minimal if it has no proper G-invariant closed subset,

equivalently, if the G-orbit Gx is dense in X for every x ∈ X.

Deﬁnition 3.2.6. The universal minimal compact G-space MG is a minimal compact G-space that has the following property: for every compact minimal

G-space X there exists a onto G-map f ∶ MG → X.

Notice that by Zorn’s lemma any G-space has a minimal compact G-subspace.

The existence or MG is given by taking any minimal closed G-subspace of S(G). 70

The universal property of (S(G), e) implies the universal property of MG. On the

other hand MG is unique up to G-isomorphisms. V. Uspenskij in [36] gives a proof of this fact, the next lines follow his argument

As in section 3.2.1, for a ∈ X let ra be the map x → xa of X to itself.

Proposition 3.2.7. If f ∶ X → X is a G-map and a = f(e), then f = ra.

Proof. Notice that, if g ∈ G f(g) = f(ge) = gf(e) = ga = ra(g) for all g ∈ G, since f is

continuous then f = ra.

An element x of a semigroup is an idempotent if x2 = x.

Theorem 3.2.8. (Ellis theorem) Every non-empty compact left-topological semigroup K contains an idempotent.

Proof. By Zorn’s lemma, K contains a minimal closed non-empty subsemigroup K0.

It happens that every element in K0 is an idempotent, that is K0 is the trivial semigroup). Indeed, let a ∈ K0, K0a = K0 by minimality of K0, thus the closed

subsemigroup {k ∈ K0 ∶ ka = a} is non-empty and must be the whole K0, thus a2 = a

Let M be a minimal closed G-subspace of S(G), by remark 3.2.5 M is a minimal closed left ideal of S(G). Since M is itself a closed (thus compact) subsemigroup of S(G) by the theorem just proved, there exists and idempotent p ∈ M. Observe that Xp is a closde left ideal contained in M, my minimality of M then Xp = M. This means that xp = x for all x ∈ M, which is equivalent to rpSM = idM . Thus the G-map rp ∶ S(G) → M is a G-map of X onto M.

Proposition 3.2.9. Every G-map f ∶ M → M has the form f(x) = xy for some y ∈ M. 71

Proof. Notice that the map h = f ○ rp ∶ S(G) → M is a G-map of S(G) into itself then by proposition 3.2.7 it has the form ra where h(e) = a ∈ M. Since rpSM = idSM then f = hSM = rySM.

Proposition 3.2.10. Every G-map f ∶ M → M is bijective.

Proof. From 3.2.9, there is a ∈ M such that f(x) = xa for all x ∈ M. Again since, Ma is a closed ideal of X contained in M then Ma = M by minimality of M. This means that there exists b ∈ M such that ba = p. Leg g ∶ M → M be the G-map deﬁned by g(x) = xb, then fg(x) = xba = xp = x for every x ∈ M, thus fg = idSM . This implies that in semigroup S of all G-self-maps of M, every element has a right inverse thus it is a group.

Theorem 3.2.11. Every universal minimal G-space is isomorphic to M

Proof. As stated after deﬁnition 3.2.6, M itself is universal. Let M ′ be a universal compact minimal G-space. There are G-maps f ∶ M → M ′ and g ∶ M ′ → M, by minimality f are surjective, and gf ∶ M → M is an G-map from M to M, by proposition 3.2.10 gf is bijective. Which implies that f is injective and hence a G-isomorphism.

Example 3.2.12. A group G is extremely amenable if and only if MG = {∗} is the one point space.

Theorem 3.2.13. (Pestov, theorem 6.2.1 [24]) The circle T forms the universal

minimal compact G-space for the Polish group Homeo (T), when equipped with the standard action by homeomorphisms.

In section 4.3 and 4.4 more properties on the nature of the action of G on MG are discussed. 72

3.2.3 Roelcke precompactness

Deﬁnition 3.2.14. A group G is precompact if one of the following equivalent

properties holds: L is totally bounded; R is totally bounded; G is a subgroup of a compact group.

Deﬁnition 3.2.15. A group G is Roelcke precompact if the Roelcke uniformity

L ∧ R is totally bounded. Equivalentely if for every neighborhood of the identity U there exists a ﬁnite set F such that G = UFU

There are non-abelian non-precompact groups which are Roelcke precompact. Some examples are presented below:

Example 3.2.16. ([36])

1. Let E be a discrete space, the symmetric group Symm(E) of all permutations of E is Roelcke precompact.

2. Let H be an inﬁnite-dimensional Hilbert space, the full unitary group

G = U(H)s with the strong operator topology is Roelcke precompact.

Moreover, the Roelcke compactiﬁcation of U(H)s can be identiﬁed with the unit ball Θ in the Banach algebra of bounded linear operators B(H) of H, with the weak topology. In this case R(G) has a natural structure of a semitopological semigroup.

3. Let G = Homeo+(I) be the group of all orientation-preserving homeomorphisms of the closed interval I = [0, 1]. V. V. Upenskij proved in [34] that the map h → Γ(h), that identifies each h ∈ G with its graph Γ(h) in I × I is a homeomorphism, given that Γ ∈ Exp(I × I) (see section 4.3 for a definition of Exp(I × I)). This means that R(G) can be identified with the closure of the space of graphs Γ(G) which is the collection c of all curves C in I × I starting 73

at the lower left corner (0, 0) and ending at the right upper corner (1, 1) and never going left or down. That is, this C look like increasing functions with the exception that C may include vertical and horizontal segments.

Uspenskij [34], has studied a way of describing the Roaelcke compactiﬁcation of some groups using a special construction that involves object from section 4.3.1, it

goes as follows. Let G act on a compact space X. For g ∈ G let Γ(g) ⊂ X2 be the graph of the map x → gx. The map g ∶ G → Exp (X2) which takes g → Γ(g) is both, left and right uniformly continuous (Here Exp (X2) is the compact space of closed relations of X2 endowed with the Vietoris topology, this means it has a unique

2 uniformity). This means that such map extends to a map fX ∶ R(G) → Exp(X ). If the action of G on X is topologically faithful, then the map fX sometimes becomes and embedding, in which case R(G) can be identiﬁed with the set {Γ(g) ∶ g ∈ G} ⊂ Exp (X2). An instance of this is when X = S(G). This is useful since the structure of Exp (X2) includes that the relations can be composed, reversed and compared by inclusion. It is clear that in the example presented above for G = Homeo+(I) this structure was used describe R(G). In particular the main theorem proved by Uspenskij in [34] is as follows:

Theorem 3.2.17. Let X be an h-homogeneous compact space, and let G = Aut (X) be the topological group of all self-homeomorphisms of X. Then”’

1. G is Roelcke precompact: the Roelcke compactiﬁcation of G can be identiﬁes

with the semigroup E0(X) (composition as the operaton) of all closed relations R on X such that DomR=RanR=X.

2. G does not admit a coarser Hausdorﬀ group topology, and G has no closed

normal subgroups besides G and {1}. 74

Where a compact space X is h − homoegeneous if X is zero-dimensional and all non-empty clopen subsets of X are homeomorphic to each other. An example of this class of spaces is the Cantor set.

3.3 More compactiﬁcations on topological groups

In this section a less restrictive definition for compactification is considered: A compactification of a topological space X is a compact Hausdorff space K

together with a continuous map j ∶ X → K with a dense range. In this section it is not required that j be a homeomorphic embedding.

Remark 3.3.1. As in section 3.1, one still can associate the compactiﬁcation (K, j)

∗ to a C -algebra AK of functions, the compactiﬁcation K is the maximal ideal space

∗ ∗ of AK , moreover AK is isomorphic as a C -algebra to C (MAK ). The less restrictive condition that j is not necessarily a homeomorphic embedding reﬂects in

AK by AK not having the regular separation property. It is clear that the partial order relations and order isomorphisms in section 3.1 can be extended to the compactiﬁcations and C∗-algebras discussed in this section.

Example 3.3.2. Consider the C∗-algebra of constant functions deﬁned on G, the

maximal ideal the space of such algebra is the one-point space {∗}, and the canonical map j ∶ G → ∗ is constant. ({∗}, j) is a compactiﬁcation of G.

Example 3.3.3. A bounded complex-valued function f deﬁned on G is called almost periodic if the G-orbit of f is relatively compact with respect of the norm topology of Cb(G), the bounded complex-valued functions deﬁned on G. That is, YY YY b {Lgf ∶ g ∈ G} is compact in C (G). Equivalently {Rgf ∶ g ∈ G} is compact in Cb(G), thus there is no need to specify weather the left or right action of G is being used. The collection of all almost periodic function is denoted by AP (G). This is a

∗ unital C -algebra and its maximal ideal space MAP (G) is known as the Bohr 75 compactification of G and is denoted by bG. The Bohr compactification of a topological group is itself a topological group and it is maximal among all compact topological groups in the following sense: If K is a compact group and ρ ∶ G → K is a continuous homomorphism, then ρ extends to a continuous homomorphism from bG to K. A topological group G is maximally almost periodic (MAP) if the canonical homomorphism G → bG is injective and G is minimally almost periodic if bG = e. Equivalentely, G is MAP if and only if continuous unitary representations in finite-dimensional Hilbert spaces separate point in G and G is minimally almost periodic if and only if G has no non-trivial continuous finite-dimensional unitary representations. In the case that G is abelian being minimally almost periodic (by 2.1.28) means to have no non-trivial characters as stated in definition 2.6.1.

3.3.1 WAP compactiﬁcation

A function f ∈ Cb(G) is called weakly almost periodic (w.a.p) if the G-orbit of f is relatively weakly compact, that is, its closure in the Banach space

Cb(G) with the weak topology is compact. The collection of all w.a.p functions on G is denoted by W AP (G). The following result asserts that one may consider either the right or left G-orbit and obtain the same collection of w.a.p functions.

Proposition 3.3.4. ([4] corollary 8.2) Let f ∈ Cb(G). The following are equivalent

1. {Rgf ∶ g ∈ G} is relatively weakly compact.

2. {Lgf ∶ g ∈ G} is relatively weakly compact.

3. co({Rgf ∶ g ∈ G}) is relatively weakly compact, where co(A) is the convex hull of A ⊂ Cb(G).

4. co({Lgf ∶ g ∈ G}) is relatively weakly compact. 76

5. {Rgf ∶ g ∈ G} is compact with respect of the weak topology induced by the elements (means) in maximal ideal space of Cb(G).

6. {Lgf ∶ g ∈ G} is compact with respect of the weak topology induced by the elements (means) in maximal ideal space of Cb(G).

Deﬁnition 3.3.5. The maximal ideal space of W AP (G) is called the the weakly almost periodic w.a.p compactiﬁcation and is denoted by W (G).

Deﬁnition 3.3.6. Let S be a semigroup and a topological space. S is called

semitopological semigroup, if the multiplication (x, y) → xy is separately continuous, that is x → ax and x → xa are continuous for every a ∈ S.

Theorem 3.3.7. ([4], Theorem 8.4, Corollary 8.5) Let G be a topological group and

let j ∶ G → W (G) be the canonical map from G to W (G). W (G) is the compact topological semigroup and it is universal in the following sense: for every continuous

homomorphism f ∶ G → S from G to a compact semitopological semigroup S, there exists a unique homomorphism h ∶ W (G) → S such that f = h ○ j.

Every reﬂexive Banach space X has an associated compact semitopological

semigroup Θ(X). Let Θ(X) be the semigroup of all linear operators A ∶ X → X such that YAY ≤ 1, equipped with the weak operator topology. Recall that for a reﬂexive Banach space X all norm bounded sets are relatively weakly compact, in

particular the unit ball BX in X is weakly compact. Notice that Θ(X) is a closed subset of the compact set BB (B endowed with the weak topology) thus Θ(X) is compact with the weak operator topology. It is clear that Θ(X) is semitopological. It happens that every compact semitopological semigroup embeds into a Θ(X) for some reﬂexive X: 77

Theorem 3.3.8. (Shtern [27], Megrelishvili [15]) Every compact semitopological

semigroup is isomorphic to a closed subsemigroup of Θ(X) for some reﬂexive Banach space X.

Θ(X) for X reﬂexive has a subgroup, namely the group of isometries Isow(X) of X with the weak operator topology. For the case of X a reﬂexive space the weak operator topology and the strong operator topology agree:

Theorem 3.3.9. (Megrelishvili [16]) For every reﬂexive Banach space X the weak and strong operator topologies on the group Iso(X) coincide

Remark 3.3.10. The result proved in [16] is stronger than the one stated in theorem 3.3.9. Megrelishvili proved that the weak and strong operator topologies coincide in a group G ⊂ GL(X) for X a Banach space with the point continuity property (which reﬂexive spaces have) and G norm bounded.

Proposition 3.3.11. Let G = Iso(X) be the group of invertible elements of Θ(X), for X a reﬂexive Banach space. The natural action G × Θ(X) → Θ(X) of G on Θ(X) is continuous.

Proof. By what was mentioned in theorem 2.2 in [18], since X is reﬂexive, in order

to show that the action of G on Θ(X) is jointly continuous it is suﬃcient to show

∗ that the action of Isos(X) on the dual space X is continuous. Since

∗ ∗ G = Isos(X) = Isow(X) then Isos(X ) = Isow(X ) are canonically isomorphic. Thus one gets the desired result.

By proposition 3.3.9 this assertion remains true for every compact semitopological semigroup. the group of invertible elements in S is a topological

group and the map (x, y) → xy is jointly continuous on G × S. 78

This means that for every topological group G, the compactiﬁcation W (G) is a G-space that a compact semitopological semigroup, hence by universality of SG and

σLG, there exist G-map SG → W (G) extending the canonical map G → W (G). In its algebra counterpart this means that W AP (G) ⊂ RUCb(G). Since the algebra W AP (G) can be deﬁned using left or right translations it follows that also

b b W AP (G) ⊂ LUC (G), which means that W AP (G) ⊂ UC∧(G), thus one concludes the following:

b Corollary 3.3.12. W AP (G) ⊂ UC∧(G) and W (G) ≤ R(G).

A question about how do the compactiﬁcations W (G) and AP (G) relate to each other is anwered in the following theorem

Theorem 3.3.13. (theorem 14.12 [4])

1. The following conditions on a topological group are equivalent

• AP (G) = Cb(G).

• W AP (G) = Cb(G).

• G is pseudocompact.

2. If G is precompact, then AP (G) = W AP (G)

3.3.2 Eberlein groups

The following deﬁnition introduces the language needed to discuss the natural question on when W AP (G) has the regular separation property and W (G) belongs in the collection C deﬁned in 3.1.

Deﬁnition 3.3.14. Let G be a topological group, G is called an Eberlein group if one of the following equivalent conditions holds: 79

1. W AP (G) has the regular separation property.

2. The canonical homomorphism j ∶ G → W AP (G) is a homeomorphic embedding of topological groups.

3. G is a topological subgroup of a compact semitopological semigroup.

Theorem 3.3.15. (Megrelishvili [16]) For every topological G, the following are equivalent

1. G is an Eberlein group

2. G is a topological subgroup of Isos(X) for a certain reﬂexive Banach space X.

Proof. By proposition 3.3.9 Isos(X) = Isow(X), and by 3.3.8, a topological group G can be embedded into Isow(X) if and only if is a topological subgroup of some compact semitopological semigroup, then the result follows.

In view of corollary 3.3.12, for Eberlein groups the partial order diagram in A and C is given by: b σG∨ TUC (G)

b b σGL S(G) LUC (G) RUC (G)

b R(G) UC∧(G)

W (G) W AP (G)

It is clear that for balanced Eberlein groups (in particular abelian) the diagram becomes two points: 80

b b b b σG∨ = σGL = S(G) = R(G) TUC (G) = LUC (G) = RUC (G) = UC∧(G)

W (G) W AP (G)

The question about whether all topological groups are Eberlein was answered by Megrelishvili the canonical map from R(G) → W (G) and the discoveries by

Uspenskij about R(G) where G = Homeo +(T)

Theorem 3.3.16. [18] Let G = Hom(I) be the group of all orientation-preserving homeomorphisms of I = [0, 1]. Then W (G) is a singleton. Equivalently, every w.a.p function on G is constant.

Nevertheless the following question remains open:

Question 10. (Megrelishvili): Does there exist a non-trivial abelian topological group G for which W (G) is a singleton?

3.3.3 When is S(G) = W (G)?

The question on when for topological groups S(G) = W (G) was answered by A. Megrelishvili, V. Pestov, V. Uspenskij in [18]: Let X be a a compact G-space. The action is called weakly almost periodic if every continuous function on X is weakly almost periodic (or w.a.p), if every continuous function on X is weakly almost periodic

Theorem 3.3.17. For an arbitrary topological group G the following conditions are equivalent

1. Every continuous action of G on a compact space is weakly almost periodic 81

2. Every bounded right uniformly continuous function on G is weakly almost periodic

3. S(G) = W (G)

4. S(G) is a semitopological semigroup with its natural multiplication.

5. G is precompact.

Below the arguments from [18] are presented. An interesting result about about the W AP (G) is the following

Theorem 3.3.18. (Ryll-Nardzewski) For every topological group G, there is a

unique bi-invariant mean on the algebra W AP (G). Moreover, such a mean is the unique left-invariant mean on W (G) as well.

Recall that if G is an group and F is a C∗-subalgebra that contains the function 1, a mean of F is a positive functional (that is m(f) ≥ 0whenf ≥ 0) µ(1) = 1 = YµY.

A mean is left-invariant if m(gm) = m(f).A right-invariant mean is deﬁned in a similar fashion. A mean that is both left and right invariant is called bi-invariant. As deﬁned in section 2.2 an amenable group G is the one for which there is left-invariant mean on the algebra RUC(G), the collection of such means will be

∗ denoted by LIMG and equipped with the weak topology forms a compact space.

Palch (see [18]) proved that if for a separable metrizable group G, LIMG

contains a Gδ point, then G is precompact. It follows that when G is separable and

LIMG has only one element then, then G is precompact. It is unknown if the same conclusion can be stablished for arbitrary topological groups groups:

Question 11. (Megrelishvili, Pestov, Uspenskij) Is it true that whenever LIMG has a unique element, then G is precompact? 82

The authors provide a positive answer in the case that G is a ω − bounded, where that means that for every neighborhood U of the identity in G there exists a

countable set A ⊂ G such that AU = G. It is clear that in theorem 3.3.17, (2), (3) and (4) are equivalent. (1) implies (2) if one thinks about the action of G on SG. Also it is clear that (5) implies the rest.

Lemma 3.3.19. ([18]) The class of w.a.p groups is closed under forming continuous homomorphic images and topological subgroups.

Lemma 3.3.20. ([18]) Let P be a class of topological groups closed under subgroups and homomorphic images. Unless all groups in P are precompact, there exists a countable metrizable group in P which is not precompact.

Proof. of 3.3.17 Only (1) implies (5) is needed to complete the proof. By lemma 3.3.19 the class of w.a.p. groups satisﬁes the hypothesis or lemma 3.3.20, which means that if not all of them are precompact then there must exist a countable metrizable w.a.p group G that is not precompact, since (1) implies (2) then G has

only one element in LIMG and G must be precompact.

3.4 Free actions on compact spaces

In [24], Pestov introduces the notion of essential set and uses it to give a characterization of when a topological group G has free action on a compact space

X. Recall that an action of G on a compact space X is called free if for every g ∈ G g =~ e, gx =~ x for all x ∈ X, that is every non-identity element of G has no ﬁxed point. If X ⊂ (G, R) is uniformly discrete subspace, then the closure clXof X in the greatest ambit S(G) is canonically homeomorphic to the Stone-Cech compactiﬁcation βX of the space X with the discrete topology X ⊂ G is R-uniformly discrete if for some neighborhood U of the identity in G, Ux ∩ Uy = ∅ 83

for all x, y ∈ X, x =~ y. If V is an open neighborhood of the identity in G such that clV ⊂ U then it follows that the set V ⋅ clX in S(G) is canonically homeomorphic to the product V × βX.

Theorem 3.4.1 (Veech, see [23]). Every locally compact group G admits a free action on a compact space

Proof. Let U ⊂ G be a symmetric compact neighborhood of eG. Using Zorn’s lemma one can construct a maximal uniformly discrete subspace X using such U. Then the

sets U 2x, x ∈ X form a cover of G which implies that clU 2 ⋅ clX matches with all S(G). If x∗ ∈ S(G) then x∗ = u ⋅ x for some u ∈ clU 2 and x ∈ clX, thus x∗ is in the closure of the set uX, which is a maximal uniformly discrete set with respect to the neighborhood of the identity uUu−1. Then one can assume without loss of generality

that x∗ is in the closure of X if one chooses U appropriately. In particular if g ∈ G, choose U such that g ∈~ U 2.

2 Let eG ∈ V ⊂ U. it can be showed that X can be partitioned into ﬁnitely many

pieces X1, ..., Xn such that V ⋅ Xi and g ⋅ Xi are disjoint in G. Then from the discussion above, the topological structure of U ⋅ clX in c(G) it follows that the set

∗ clXi is disjoint from its translations by g. Since x ∈ clXi for some i, one concludes that g ⋅ x∗ = x∗.

A generalization for this proof and when a topological group has a free action on a compact space is given by Pestov in [24] using the following notions.

Deﬁnition 3.4.2. (Pestov, [24]) Let (X, U) be a uniform space, and let F be a family of uniform isomorphism of X. A subset A ⊂ X is called essential with regard of F , or F −essential, if for every entourage of the diagonal V ∈ U and every

ﬁnite collection of transformations f1, f2, ..., fn ∈ F , one has 84

n VA ∩ fiVA =~ ∅ (3.4.1) i=1

Lemma 3.4.3. (V. Pestov [24]) Let g be a uniform isomorphism of a uniform space

(X, U). the following are equivalent:

1. The extension of g over the Samuel compactiﬁcation σX is ﬁxed point-free.

2. X can be covered with ﬁnitely many g−inessential sets.

Theorem 3.4.4. (V. Pestov [24]) For a topological group G the folowing are equivalent:

1. G admits a free action on a compact space

2. G acts freely on the greatet ambit S(G)

3. For every g ∈ G, g =~ e the group G, equipped with the right uniformity and the action on itself by left translations, admits a ﬁnite cover by g−inessential sets.

Remark 3.4.5. Lemma 3.4.3 and theorem 3.4.4 include an extra statement to which the rest are equivalent, such statement includes conditions of large oscillations. For more on this see the reference cited above.

3.5 Applications to the Glasner-Pestov problem

A corollary of theorem 3.3.17 is the following:

Corollary 3.5.1. For non-precompact topological groups W (G)= ~ S(G).

Since minimally almost periodic abelian groups have no non-trivial continuous characters then they can not be precompact, this implies that if G is a minimally

almost periodic abelian group, then W (G)= ~ S(G). In particular, since G is abelian, 85

W (G) is strictly contained in RG. If W (G) is a singleton then one has answer in the positive question 10. In contrast, if W (G) is not a singleton, one can ask if G acts freely or without ﬁxed points on W (G)? More generally one can ask:

Question 12. Can one characterize the class of (abelian) topological groups that act

freely on W (G)?

If any abelian minimally almost periodic group (examples in section 2.7) falls

into this class of groups that act freely on W (G), then one has answered question8 in the negative. Consider the additive topological group introduced in section 2.7. It was shown

that if X is a locally bounded F -space with no convex neighborhood other than ∅ and X, then X is minimally almost periodic. In [4] it is shown that for a complete

metric topological group G, all f ∈ W AP (G) also satisfy the condition that

co{Rgf ∶ g ∈ G}, is relatively compact with respect to the compact-open topology of

b C (G), and Rg is the right multiplication operator and coA denotes the convex hull

of A. By the Arzela-Ascoli theorem this implies that co{Rgf ∶ g ∈ G} is equicontinuous at each point x ∈ X. This induces the following question

Question 13. Let X be a locally bounded F -space with no convex neighborhood

b other than ∅ and X. If f ∈ C (X) is such that co{Rgf ∶ g ∈ G} is equicontinuous at

b each point x ∈ X and {Rgf ∶ g ∈ G} is relatively weakly compact in C (X), does it imply that f is constant?

A positive answer implies that W (X) is a singleton and would solve in the positive question 10. In [24], Pestov proposes the following open problem: 86

Question 14. (Pestov, 3.4.19 [24]) A topological space X is called a kw-space (or: hemicompact space) if it admits a countable cover Kn, n ∈ N by compact subsets in such a way that an S ⊂ X is closed if an only if A ∩ Kn is closed for all n. Is it true that every topological group G that is a kw-space admits a free action on a compact space?

One may ask if W (G) can be the compact space on which a topological group

G that is a Kw-space acts freely. 87 4 Non-weakly mixing topological dynamical

systems

4.1 Topological dynamical systems

In this section, advances in an approach to the Glasner-Pestov problem using tools from the theory of topological dynamics are presented, more speciﬁcally the existence of equicontinuous factors for a topological dynamical system and how this relates to the property of weakly mixing are used to understand when a continuous character may exist for monothetic groups.

4.1.1 Basic deﬁnitions and examples

A dynamical system consist of a non-empty set X together with a transformation T ∶ X → X. In this dissertation, only the case when T is invertible is considered. In such case (X,T ) is normally called a cyclic dynamical systems.

Deﬁnition 4.1.1. A topological dynamical system (X, τ, T ) (sometimes (X,T ) for short) consist of a compact metrizable topological space (X, τ), and a homeomorphism T ∶ X → X.

Remark 4.1.2. Recall that by Urysohn’s metrization theorem a compact space is metrizable if and only if it is Hausdorﬀ and second countable. This means that for a topological dynamical system all spaces are second countable.

Remark 4.1.3. Recall that all metrics in a topological dynamical system (X,T ) are equivalent, since there is a unique uniform structure in a compact space X.

Example 4.1.4. 1. If X is a ﬁnite set and T ∶ X → X is a permutation, (X,T ) is known as a ﬁnite system. 88

2.A Bernoulli system is induced by a non-empty set Ω and it consist of the

Z dynamical system (Ω ,T ) where T is the left shift T (xn)n∈Z = (xn+1)n∈Z.A special case happens when Ω = {0, 1}, so that X = {0, 1}Z = {A ⊂ Z} = 2Z and T is the left shift TA = A − 1 for A ⊂ Z. In this case the system (2Z,T ) is known as the boolean Bernoulli system.

Notice that the maps T n, n ∈ Z can be interpreted as an isomorphisms in several categories:

• As a set isomorphism: T n ∶ X → X from points x ∈ X to points T x ∈ X. If T is a homeomorphism of X then T n is a topological isomorphism.

• As a Boolean algebra isomorphism: T n ∶ 2X → 2X from sets E ⊂ X to subsets T nE ⊂ X. If T is a homeomorphism, thenT n maps open (closed) sets to open (closed) sets.

• As algebra isomorphism: T n ∶ RX → RX (or T n ∶ CX → CX ), taking real-valued (complex-valued) functions f ∶ X → R (f ∶ X → C) to real-valued (complex-valued) functions T n(f), where T n(f) = f ○ T −n. If (X,T ) is a topological dynamical system and f ∈ C∗(X) the map T n ∶ C∗(X) → C∗(X) where f → f ○ T n is a C∗-algebra isomorphism.

Deﬁnition 4.1.5. A morphism between (topological) dynamical system

(X,T ), (Y,S) is a (continuous) map φ ∶ X → Y such that φ intertwines with T and S, that is φ ○ T = S ○ φ. A morphism φ that has an inverse φ−1 which also is a morphism is called an isomorphism and X and Y are called isomorphic or conjugate.

Example 4.1.6. Let G be a group and let X be a homogeneous space for G, that is, the action of G on X is transitive. Every group element g ∈ G deﬁnes a group 89

action dynamical system (X,Tg) where Tgx = gx. Consider the subgroup

Γx = Stab(x) = {g ∈ G ∶ gx = x} for x ∈ X. Consider the map φ ∶ G~Γx → X deﬁned by −1 −1 φ(gΓx) = gx. If g1, g2 ∈ G and g1Γx = g2Γx then g2 g1 ∈ Γx thus g2 g1x = x, which

means that g1x = g2x, hence φ is well deﬁned. Let y ∈ X, since G acts transitively on

X, then there exists g ∈ G such that gx = y, which implies that φ(gΓx) = y and φ is −1 onto. If g1, g2 ∈ G such that g1x = g2x then g2 g1x = x thus g1Γx = g2Γx and φ is

injective. Notice that φ is an isomorphism of dynamical systems from (X,Tg) to

(G~Γx, z → gz). Indeed, if y ∈ X then φ(g(g0Γx)) = gg0x = Tg(φ(g0Γx)) and if g0x = y

−1 −1 −1 then φ (Tgy) = φ (gg0x) = gg0Γx = gφ (y). This means that the dynamical

system (X,Tg) is isomorphic to (G~Γx, z → gz). A particular instance of a group action system is the circle rotation: Every

α ∈ R induces the topological dynamical system (R~Z,Tα) where Tαx = x + α.

Example 4.1.7. 1. T n, n ∈ Z is an isomorphism. Indeed, T n is a homeomorphism and commutes with T , then T n is a isomorphism of topological dynamical systems.

2. Let (X,T ) be a (topological) dynamical system and suppose that E is a

(closed) subset of X. Then i ∶ (E,TSE) → (X,T ) is a system embedding.

Indeed, notice that T ○ i = i ○ TSE.

Deﬁnition 4.1.8. A factor map or a factor, is a surjective morphism π ∶ X → Y between dynamical systems (X,T ) and (Y,S).

Example 4.1.9. Let α ∈ R and deﬁne (X,T ) as follows: X = (R~Z)2 and

T ∶ (x1, x2) → (x1 + α, x2 + x1). Let (Y,S) be as follows: Y = (R~Z) and S ∶ y → y + α.

It is clear that, the projection map π ∶ X → Y , π(x1, x2) = x1 is a surjective morphism between topological dynamical systems. Thus π is a factor map. 90

If one pictures X as the torus obtained from the product of a horizontal circle and a vertical circle, one can picture the sets π−1({y}), y ∈ Y as a ‘vertical circle’. The shift y → y + α together with the factor map π determine how the vertical circles rotate and angle α over the horizontal circle in X, but S and π do not describe the

movement within the vertical circle that π−1({y}) represents. (See picture below).

In the same way as described in the above example a factor map from (X,T ) to (Y,S) foliates X into ‘vertical fibers’ π−1({y}), y ∈ Y (the fibers indexed by the elements in the factor space Y ). Again the shift S and the factor map π determine how the fibers move in X but they do not govern the dynamics within the vertical

ﬁber π−1({y}).

Deﬁnition 4.1.10. A minimal (topological) dynamical sytems (X,T ) is one that

has no proper (closed) subsystems (Y,TSY ).

Example 4.1.11. It is clear that for a dynamical system (X,T ) the orbit T Zx = {T nx ∶ n ∈ Z} is minimal. Conversely all minimal system arise as an orbit of any of its elements. Thus any dynamical system can be uniquely decomposed as disjoint unions of minimal dynamical systems. It is clear that each of the orbits is

isomorphic to Z~Stab(x). Since all subgroups of Z are of the form NZ, one can conclude that every minimal dynamical system is either (Z~NZ, x → x + 1) or the integer shift (Z, x → x + 1). This shows that all dynamical systems are disjoint unions of these two kinds of minimal systems. In the case that X is ﬁnite one has recovered the fact that each permutation is uniquely decomposable as a product of disjoint cycles.

Remark 4.1.12. For topological dynamical systems: 91

1. It is also true that two minimal subsystems are either disjoint or coincident, otherwise their intersection would be a compact and invariant topological subsystem.

2. Orbits need not to be closed so one look at their closures T Zx which is clearly a subsystem. Also any minimal subsystem is the closure of the orbit of any of its elements.

Example 4.1.13. Not every closure of an element’s orbit is necessarily minimal.

For example, consider the boolean Bernoulli dynamical system, (2Z,A → A − 1), 2Z with the product topology induced by the discrete topology on {0, 1}. Let x = N = {0, 1, 2, ...} ∈ 2Z, so T ZN = {{a, a + 1, a + 2, ...} ∶ a ∈ Z}. It is clear that Z and ∅ are in the closure of T ZN, and T (Z) = Z and T (∅) = ∅. Thus T ZN is not minimal. Moreover, x = N is not contained in any minimal system.

From this example one can conclude that even if a topological dynamical

system (X,T ) is decomposed in its minimal systems there might be points that will not be in any of such minimal subsystems.

Lemma 4.1.14. Every topological dynamical system (X,T ) contains a minimal dynamical system.

Proof. Notice that the intersection of any chain of subsystems of X is again a subsystem, indeed, one uses the ﬁnite intersection property of the chain and the fact that X is compact to assure that the chain has non-empty intersection. Such intersection is T -invariant since each of the subsystems is T -invariant. Thus by Zonr’s lemma there is a minimal dynamical subsystem.

Deﬁnition 4.1.15. Let (X, τ1,T ) and (Y, τ2,S) be topological dynamical systems,

and let (X × Y, τ1 × τ2,T × S) be the product topological dynamical system of 92

(X, τ1,T ) and (Y, τ2,S), where τ1 × τ2 is the product topology and (T × S)(x, y) = (T x, Sy).

Notice that the product dynamical system is a product in the category of topological dynamical systems. Indeed, let (Z, τ3,U) be such that there are morphisms f1 ∶ Z → X, f2 ∶ Z → Y , this means that f1 ○ U = T ○ f1 and f2 ○ U = S ○ f2.

Let f ∶ Z → X × Y such that f(z) = (f1(z), f2(x)) thus one gets:

(T × S) ○ f(z) = (T f1(z), Sf2(z)) = (f1(Uz), f2(Uz)) = f ○ U(z), which means that (T × S) ○ f = f ○ U and that f is a morphism from Z to X × Y . This map f satisﬁes the conditions for making (X × Y, τ1 × τ2,T × S) into a product in the category of topological dynamical systems.

Example 4.1.16. Consider ((R~Z)d, x → x + α), where α ∈ Rd is ﬁxed. This topological dynamical system is minimal if and only if m ⋅ α is not an integer for any m ∈ Zd ∖ {0}.

Deﬁnition 4.1.17. Let (X1, τ1,T1) and (X2, τ2,T2) be topological dynamical systems, and let (X1 ⊍ X2, τ1 ⊍ ×τ2,T1 ⊍ T2) be the disjoint union of topological dynamical system of (X1, τ1,T1) and (X2, τ2,T2), where X1 ⊍ X2 is the disjoint union of X1 and X2, τ1 ⊍ τ2 is the ﬁnest topology that makes all injection maps

φi ∶ Xi → X1 ⊍ X2 continuous, and (T1 ⊍ T2)SXi = Ti.

It is easy to prove that the disjoint union of topological dynamical system is a coproduct inthe category of topological dynamical system.

4.1.2 Recurrence

The study of recurrence in topological dynamical systems consists in studying how often the map T makes a point or an open set return near its original position. The following is the a theorem regarding recurrence of open sets: 93

Theorem 4.1.18. Let (X, τ, T ) be a topological dynamical system, and let

(Uα)α ∈ A be a open cover of X. Then there exists an open set Uα0 in this cover

n such that Uα0 ∩ T Uα0 =~ ∅ for inﬁnitely many n.

Proof. Recall the inﬁnite pigeonhole principle: Whenever Z is colored into ﬁnitely

many colors, at least one of the color classes is infinite. Let (Uα)α∈A be a open cover k of X, by compactness of X it has a finite subcover {Ui}i=1. Consider x ∈ X and the orbit T Zx = {T nx ∶ n ∈}. Each of the elements of T Zx is in one or more of the elements of the finite subcover. Without lost of generality by the pigeonhole

n principle U1 has inﬁnitely many elements for T Zx. Let y = T 0 x for some n0 ∈ Z such

n that y ∈ U1. Then T Zx = T Zy. Thus T U1 ∩ U1 =~ ∅ for inﬁnitely many n’s.

Example 4.1.19. Consider A a compact metric space and the Bernoulli system

Z (A ,S), where S is the left shift such that S{xi}i∈Z = {xi+1}i∈Z. Consider the metric 1 d({xi}, {yi}) = ∑n∈Z 2SnS dA(xn, yn). Notice that d is not shift invariant. If A contains at least two points, then the Bernoulli system (AZ,S) can not be endowed with a shift-invariant metric. Indeed, let d be a compatible metric with

Z A , consider x = (xi)i∈Z = N, xi = 0 if i < 0 and xi = 1 if 0 ≤ i, and let y = Sx. It is clear that x =~ y thus d(x, y)= ~ 0. On the other hand, if d is shift invariant then d(Snx, Sn+1x) = d(x, Sx) = d(x, y) for all n ∈ Z. Since the product topology is the same as point-wise convergence one obtains that Snx → Z and Sny → Z, thus d(Snx, Sny) → 0 which means that d(x, y) = 0, which is a contradiction.

Recall from section 2.6.1 that a syndetic set S ⊂ Z is such that it has bounded gaps, that is S + {1, 2, ..., N} = Z for some N ∈ Z+.

Deﬁnition 4.1.20. A point x in a topological dynamical system (X,T ) can be classiﬁed based on the dynamics of its orbit:

1. x is invariant if T x = x. 94

2. x is periodic if T nx = x for some non-zero n.

3. x is almost periodic if for every > 0 the set S = {n ∈ Z ∶ d(T nx, x) < } is syndetic.

4. x is recurrent if for every > 0, the set {n ∈ Z ∶ d(T nx, x) < } is inﬁnite.

Equivalently there exists a sequence nj of integers such that SnjS → ∞ and

n limj→∞T j x = x.

Since all metrics are equivalent in a compact space the deﬁnition for almost periodic and recurrent are independent of the chosen metric.

It is clear that in the above deﬁnition

(invariant) Ô⇒ (periodic) Ô⇒ (almostperiodic) Ô⇒ (recurrent). These implications are strict, which means that no two of those properties are equivalent.

Example 4.1.21. In the rotation system (R~Z, x → x + α) where α is irrational each x is almost periodic but no point is periodic.

In general it is possible to have a non-recurrent point in a topological dynamical system. Nonetheless, if one studies a minimal dynamical system one gets the following:

Lemma 4.1.22. If (X,T ) is a minimal topological dynamical system then every element of X is almost periodic (thus recurrent).

Proof. Suppose that x ∈ X is not almost periodic, then for some > 0 the set

n + {n ∈ Z ∶ d(T x, x) < } is not syndetic, thus for all m ∈ Z there exists nm such that

n nm if n ∈ [nm − m, nm + m] then d(T x, x) ≥ . Since X is compact, {T x}m∈Z+ has a

h n +h cluster point y. Notice that d(T y, x) = limm→∞d(T m x, x) ≥ , thus x ∈~ T Zy, which contradicts the minimality of X. 95

It turns out that if x is a point in a topological dynamical system, then x is almost periodic if and only if x lies on a minimal system.

Theorem 4.1.23 (Birkhoﬀ recurrence). Every topological dynamical system contains at least one point which is almost periodic (hence recurrent).

Proof. From theorem 4.1.14 every topological dynamical system has a minimal topological dynamical subsystem, then by lemma 4.1.22 all its points are almost periodic.

Remark 4.1.24. Birkhoﬀ recurrence theorem is stronger than theorem 4.1.18, since

n instead of showing that {n ∈ Z ∶ T Uα ∩ Uα =~ ∅} is inﬁnite, one has shown that it is syndetic.

Let G be a topological group, if Γx is co-compact, which means that is the

quotient space G~Γx is compact, then one may consider the topological dynamical

system induced by the group action dynamical system (G~Γx, z → gz) from example

4.1.6. It is clear that G~Γx is a G-space and that the action of G is homogeneous.

One could expect two points in G~Γx to behave similarly, nevertheless this is not the

case in general. Let y, z ∈ G~Γx and let h ∈ G such that hz = y, h. The action of h needs not to preserve the shift z → gz, that is h needs not to be a morphism of topological dynamical systems. In the case that h is a morphism one gets

h ○ g = g ○ h which is the same as hg = gh, meaning that h and g commute. This

means that in order for the points y and z in G~Γx to behave similarly for any pair of points, g needs to be central. In view of this and theorem 4.1.23 one can immediately conclude the following:

Theorem 4.1.25. Let (G~Γx, z → gz) be a topological group quotient dynamical system such that g lies in the center Z(G) of G. Then every point in this system is almost periodic (hence recurrent). 96

4.1.3 Kronecker factor

Deﬁnition 4.1.26. Let (X,T ) be a topological dynamical system:

1. the system is isometric if there exist a metric d on X such that the shift

maps T n ∶ X → X are all isometries.

2. the system is equicontinuous if there exist a metric d on X such that the

shifts T n ∶ X → X form a uniformly equicontinuous family, that is if for all > 0 there exists δ > 0 such that if d(x, y) < δ then d(T nx, T ny) < for all n ∈ Z

These deﬁnitions suggest a more global examination of the group {T n ∶ n ∈ Z}. In particular recall a version of the classical Arzela-Ascoli theorem:

Theorem 4.1.27. (Arzela-Ascoli) Let X be a compact Hausdorﬀ space and Y a metric space. Then F ⊂ C(X,Y ) is compact in the compact-open topology if and only if it is equicontinuous, point-wise relatively compact and closed. C(X,Y ) denotes the continuous functions from X into Y and point-wise relatively compact means that the set {f(x) ∶ f ∈ F } is relatively compact in Y for any x ∈ X.

In the case of the group F = {T n ∶ n ∈ Z} induced by (X,T ) the condition of point-wise relatively compact is satisﬁed trivially since X is compact. This means that (X,T ) is equicontinuous if and only if F = {T n ∶ n ∈ Z} is relatively compact with respect of the compact-open topology. A particular consequence of this is that if (X,T ) is equicontinuous then for any metric d compatible with X the family of shifts F = {T n ∶ n ∈ Z} is equicontinuous with respect of d.

Example 4.1.28. 1. The circle shift x → x + α is both isometric and equicontinuous.

2. On the other hand the boolean Bernoulli shift on {0, 1}Z is neither isometric nor equicontinuous. Since in a compact space all compatible metrics are 97

equivalent, it is enough, from the discussion above, to show that the boolean Bernoulli system is not equicontinuous for one metric, namely:

1 d(x, y) = ∑i∈Z 2SiS Sxi − yiS. This means that:

1 d T nx, T ny x y ( ) = Q SiS S i+n − i+nS i∈Z 2

Let 1 > > 0 and consider x = N and y = N − 1 = T x, one gets

N N 1 d(T x, T y) = 2N+1 but,

−N−1 N −N−1 N −1 −1 d(T (T x),T (T y)) = d(T x, T y) = 1 >

this means that regardless of how small one makes δ > 0, one can always ﬁnd z = T N x and w = T N y such that d(z, w) < δ but d(T kz, T kw)) > . Thus the Bernoulli system is not equicontinuous nor isometric.

Proposition 4.1.29. A topological dynamical system (X,T ) is isometric if and only if it is equicontinuous.

Proof. One direction is immediate. To prove the other direction, let (X,T ) be

n n equicontinuous with respect of the metric d on X. Deﬁne d(x, y) = supnd(T x, T y). It is clear that d is a metric and compatible with X due to properties of supremum.

Also, it makes (X,T ) into an isometric topological system.

Deﬁnition 4.1.30. A topological dynamical system (X,T ) is said to be a Kronecker system if it is isomorphic to a system of the form (K, τ, S), where (K, +, τ) is a compact abelian metrizable topological group, and S ∶ x → x + α is a group rotation for some α ∈ K.

Example 4.1.31. 1. The circle rotation system is a Kronecker system. 98

2. It is easy to show that the product of Kronecker systems is a Kronecker system.

Proposition 4.1.32. Every Kronecker system is equicontinuous (hence isometric).

Proof. Let (K, τ, S) be a Kronecker system. Since (K, +, τ) is a compact topological group the action K × K → K is continuous and K ⊂ Homeo(K) is compact with respect of the compact-open topology of Homeo(K). Since S is a group rotation then G = {Sn ∶ n ∈ Z} ⊂ K ⊂ Homeo(X) is relatively compact with respect of the open-compact topology. By the Arzela-Ascoli theorem then G is equicontinuous

thus (K, τ, S) is equicontinuous.

Not every equicontinuous system is Kronecker. For example consider the

disjoint union of two finite cyclic shift of different order, Z~nZ ⊍ Z~mZ. This is an equicontinuous topological dynamical system that is not a Kronecker system. Nonetheless, when one considers minimal equicontinuous systems the situation is different:

Proposition 4.1.33. Every minimal equicontinuous system (X,T ) is a Kronecker system, i.e. isomorphic to an abelian group rotation (K, τ, x → x + α).

Proof. Let G = {T n ∶ n ∈ Z} the closure with respect of the compact-open topology in Homeo(X). Since (X,T ) is equicontinuous by the Arzela-Ascoli theorem G is compact in such topology. Notice that G is abelian. Recall that the compact-open

topology in Homeo(X) is second countable, since X is a metrizable compact space, thus by Urysohn’s metrization theorem G is metrizable compact topological group.

Let x ∈ X be an arbitraty point. Notice that since X is a G-space, {f(x) ∶ f ∈ G} the G-orbit of x is a non-empty compact subset of X and by minimality of X,

{f(x) ∶ f ∈ G} = X. 99

Let Γx = {f ∈ G ∶ f(x) = x}, it is clear that Γx is a closed (thus compact) subgroup of G. Since {f(x) ∶ f ∈ G} = X then X is G-homogeneous (since x is arbitrary). Then K = G~Γx is homeomorphic to X, indeed, since the map f → f(x) is continuous and K is compact, then the map φ ∶ K → X, φ(fΓx) = f(x) is a homeomorphism. Moreover, φ(T (fΓx)) = T (f(x)) = T φ(fΓx), which means that φ is a morphism, similarly φ−1 is a morphism and (X,T ) is isomorphic to the

Kronecker factor (K = G~Γx, f → T ○ f)

Definition 4.1.34. A factor π ∶ X → Y refines or is finer than another factor π′ ∶ X → Y ′ if it can be factorized as π′ = f ○ π for some continuous map f ∶ Y → Y ′

π X / Y

f ′ π Y ′ Two factors are equivalent if they reﬁne each other. Reﬁnement is a partial order mod equivalences of topological dynamical systems.

Example 4.1.35. The identity factor id ∶ X → X is ﬁner than any other factor X, in particular it is ﬁner than the trivial factor pt ∶ X → {∗}.

Proposition 4.1.36. Every factor of a minimal topological dynamical system is also minimal

Proof. Recall that since π is a morphism then π ○ T = S ○ π, thus π ○ T 2 = π ○ T ○ T = S ○ π ○ T = S ○ S ○ π = S2 ○ π Thus by induction π ○ T n = Sn ○ π. Let π ∶ (X,T ) → (Y,S) be a factor map and (X,T ) be a minimal topological dynamical system. Let w, z ∈ Y and let x, y ∈ X such that π(x) = w and π(y) = z. By minimality of X one gets T Zx = T Zy = X, this means that there exists a

n sequence {nj ∈ Z ∶ j ∈ Z} such that T j x → y as j → ∞. Thus by continuity of π, 100

π(T nj x) → π(y) = z rewriting this one gets: Snj (πx) → z which means that Snj w → z, thus since z is arbitrary then SZw = Y and since w is arbitrary, one obtains that (Y,S) is minimal.

Definition 4.1.37. Let π ∶ X → Y and π′ ∶ X → Y ′ be two factors, one defines the join π ∨ π′ ∶ X → Y ∨ Y ′, where Y ∨ Y ′ = {(π(x), π′(x))} is a compact subspace of the product system Y × Y ′. π ∨ π′ is onto and a morphism, thus a factor. It follows from the properties of the product that π ∨ π′ is the least common refinement π and π′.

Proposition 4.1.38. Let π ∶ X → Y and π′ ∶ X → Y ′ be two factors such that Y and Y ′ are isometric. Then π ∨ π′ ∶ X → Y ∨ Y ′ is also isometric.

Definition 4.1.39. Given a chain (πα)α∈A of factors πa ∶ X → Yα, i.e. πα refines πβ for all α > β, one can define the inverse limit

π = lim←(πα)α∈A ∶ X → Y = lim←(Yα)α ∈ A by letting fαβ ∶ Yα → Yβ be the factoring

maps for all α > β. Observe that fβγ ○ fαβ = fαγ for all α > β > γ. Y is the compact space deﬁned as a subset of ∏α Yα as follows

Y = {(Yα)α∈A ∶ fαβ(yα) = yb whenever α > β} and setting π(x) = (πα(x))α∈A. π is a factor of X.

Proposition 4.1.40. ([30]) Let (πα)α∈A be a chain of factors πa ∶ X → Yα with

Yα = (Yα, τα,Sα) isometric, then the inverse limit π ∶ X → Y of the πα is such that Y is also isometric.

The following theorem is a consequence of the previous proposition, Zorn’s lemma, and the fact that ({∗}, id∗) is an equicontinuous factor for any topological dynamical system.

Theorem 4.1.41. For every topological dynamical system (X,T ) there is a factor π ∶ X → Y with dynamical (Y,S) isometric, and which is maximal with respect to 101

reﬁnement among all such factors with this property. This factor is unique up to equivalences.

Deﬁnition 4.1.42. If (X,T ) is minimal in the previuos theorem, by propositions 4.1.33 and 4.1.36 this maximal isometric factor is a Kronecker system and thus is known as the Kronecker factor of the minimal topological dynamical system

(X,T ).

4.1.4 Characterizations of existence of non-trivial equicontinuous factors

For minimal topological dynamical system (X,T ), its Kronecker factor is a maximal equicontinuous factor, nonetheless the Kronecker factor might be the trivial factor. In this section theorems 4.1.46, 4.1.54 and corollary 4.1.47 characterize weather the Kronecker factor is trivial or not.

Deﬁnition 4.1.43. Let (X,T ) be a topological dynamical system, λ ∈ C is called an eigenvalue if there is f ∈ C∗(C), f ≡~ 0 such that T f = λf, in such case f is called and eigenfunction. λ is called trivial eigenvalue if λ = 1.

Proposition 4.1.44. If λ is an eigenvalue for T then λ ∈ T, moreover there is a unimodular function g ∶ X → T which is an eigenfunction of λ.

Proof. Consider A = {a ∈ X ∶ f(a) = 0} where f is an eigenfunction of λ, it is clear that A is a closed set of X. Notice that if a ∈ A, f(T (a)) = (T −1f)(a) = λ−1f(a) = 0, thus A is invariant under T . By minimality of (X,T ) and since f ≡~ 0 then A = ∅. On the other hand if SλS= ~ 1 then, without lost of generality assume SλS > 1 (otherwise look at the system (X,T −1)), let b ∈ X such that f(b)= ~ 0 and using the fact that X

ni is compact let (ni)i∈Z an increasing sequence of positive integers such that T b is a convergent subsequence T ni b → y. Notice that λni f(b) = f(T ni b) this implies that

ni f f(T b) → f(y) = 0 thus y ∈ A. Which is a contradiction. Then SλS = 1. Let g = SfS , which is well deﬁned since A = ∅. 102

Proposition 4.1.45. If (X,T ) is a minimal topological dynamical system and λ an eigenvalue, then the eigenspace {f ∈ C∗(X) ∶ T f = λf} is one-dimensional.

Proof. In proposition 4.1.44, it was proven that if f is an eigenfunction of λ then

f(x)= ~ 0 for all x ∈ X. Let f and f ′ be eigenfuctions of λ, since (X,T ) is minimal,

n for any y ∈ X there exists a sequence ni such that T i x → y. Notice that

f(x) λ−ni f(x) f(T ni x) f(y) = = → f ′(x) λ−ni f ′(x) f ′(T ni x) f ′(y) ′ but the left hand side is constant, thus f(y) = c0f (y) for all y ∈ X, where c0 is constant.

Theorem 4.1.46. Let g ∶ X → T a unimodular eigenfunction with non-trivial eigenvalue, then g ∶ X → g(X) ⊂ T is an isometric factor of X, where the factor dynamical system is (g(X), z → λz¯ ). Thus g = χ ○ π where π ∶ X → K is the Kronecker factor and χ ∶ K → T is a character of K. Conversely, all functions of the form c0χ ○ π where c0 is a constant are eigenfunctions.

Proof. Let g ∶ X → T be an eigenfunction with non-trivial λ =~ 1, then g(X) is compact and T is not the identity on X. Since λ =~ 1 then g(X)= ~ {1}, indeed suppose that g(x) = 1 for some x ∈ X, then g(T −1x) = λg(x) = λ =~ 1. Notice that λg¯ (x) = (T −1g)(x) = g(T x) ∈ g(X), thus z ∈ λz¯ is a homeomorphism of g(X). Recall that g is an eigenfunction, i.e. λg = T g, which implies that g ○ T = λg¯ , thus g is a morphism of topological dynamical systems. Thus g ∶ (X,T ) → (g(X), z → λz¯ ) is a factor. Recall that the Kronecker factor (Y,S) (where T ○ π = π ○ S) can be seen as

n (K = {S ∶ n ∈ Z}~Γx, z → Sz) as in proposition 4.1.33, and the following commutative diagram is satisﬁed: 103

π (X,T ) / (K, z → Sz) χ g ( (g(X) ⊂ T, z → λz¯ )

Thus for z ∈ K, χ(Sz) = λχ¯ (z) thus χ(Snz) = (λ¯)nχ(z) = χ(Sn)χ(z) taking limits to generate arbitrary elements of K, one gets χ(yz) = χ(y)χ(z), which means that χ is a character and g = χ ○ π.

Conversely, if f = c0χ ○ π where c0 is a constant and χ a character of K then since π is a morphism and χ a character one gets:

−1 T f(x) = c0χ(π(T x)) = c0χ(S○π(x)) = c0χ(S)χ(π(x)) = χ(S)(c0χ○π(x)) = χ(S)f(x)

thus f is an eigenfunction.

An immediate consequence of this theorem is the following corollary:

Corollary 4.1.47. Let (X,T ) be a minimal topological dynamical system then the following are equivalent:

1. (X,T ) has a non-trivial eigenvalue.

2. The Kronecker factor of (X,T ) is not trivial.

3. (X,T ) has a non-trivial equicontinuous factors.

Deﬁnition 4.1.48. A topological dynamical system (X,T ) is topologically transitive if for every pair U, V ⊂ X non-empty open sets, there exists n ∈ Z such that T nU ∩ V =~ ∅. 104

Proposition 4.1.49. A topological dynamical system (X,T ) is topologically transitive if and only if it is the closure of the orbit of one of its elements.

Proof. (⇐) Let x0 such that T Zx0 is dense in X, then there exists n0 ∈ Z such that

n n −n T 0 x0 ∈ V and there exists n1 ∈ Z such that T 1 x0 ∈ U. This means that x0 ∈ T 1

n ( N and it follows that T 0 x0 ∈ T n0 − n1)U. Let N = n0 − n1 then T U ∩ V =~ ∅. (⇒) By contradiction, suppose that T Zx is not dense in X for all x ∈ X. Recall ∞ that X is second countable by Urysohn’s metrization theorem. Let {Ui}i=1 be a

Z Z countable base. For x ∈ X let Uix be such that T x ∩ Uix = ∅ (it exists since T x is

n not dense in X). By hypothesis there is an nj ∈ Z such that T j Uix ∩ Uj =~ ∅ for any

+ k j ∈ Z , then the set ⋃k∈Z T (Uix ) is a open dense in X. Nevertheless

k Z k x ∈~ ⋃k∈Z T (Uix ) since T ∩ Uix ∅. Consider now A = ⋂x∈X ⋃k∈Z T Uix . On one hand k side, A = ∅ since x ∈~ ⋃k∈Z T (Uix ), on the other hand by the Baire category theorem, the intersection of dense open sets is dense in X thus A =~ ∅, which is a contradiction. Then T Zx is dense in X for some x ∈ X.

Remark 4.1.50. If one considers G = {T n ∶ n ∈ Z} with the compact-open topology, then X becomes a G-space. The space X together with a point for which T Zx is dense in X is an ambit, as deﬁned in section 3.2.1. Another way of expressing the property of topologically transitive is that the G-space X is an ambit.

Deﬁnition 4.1.51. A topological dynamical system (X,T ) is topologically weakly mixing if the product system (X × X,T × T ) is topologically transitive.

Lemma 4.1.52. (Krylov-Bogolubov lemma, see [30]) Let (X,T ) be a topological dynamical system. Then there exists a T -invariant probability measure µ on X.

Lemma 4.1.53. Let (X,T ) be a minimal system and f ∶ X → R a T -invariant function with at least one point of continuity, then f is constant. 105

Proof. Let x0 be a continuity point and x an arbitrary point in X. Since T Zx is

dense in X and since the value f(T nx) does not depend on n it follows that

f(x) = f(x0)

Theorem 4.1.54. ([10] and [30]) Let (X,T ) be a minimal topological dynamical system then (X,T ) is topologically weakly mixing if and only if (X,T ) has no non-trivial eigenvalues.

Proof. (⇒)Let (X,T ) be minimal and topologically weakly mixing, let π ∶ (X,T ) → (Y,T ) be an equicontinuous factor. If (x, x′) ∈ X × X is a point such that its T × T -orbit is dense in X × X. Let (y, y′) = (π(x), π(x′)), since π is onto, then the T × T -orbit of (y, y′) is dense in Y × Y . Since (Y,T ) is equicontinuous let d

n n ′ ′ be a metric for which T is an isometry, then d(z, w) = limid(T i y, T i y ) = d(y, y ) for any z, w ∈ Y , since (T × T )Z(y, y′) is dense in Y × Y . Hence Y is a trivial point space.

(⇐) Assume that (X × X,T × T ) is not topologically transitive then one can construct and equicontinuous factor (Z,T ) of (X,T ). By the Krylov-Bogolubov lemma X admits an invariant Borel measure µ. The support of µ is a non-empty closed invariant subset of X, thus it must be all of X, again by minimality of X. By

deﬁnition there are U, V ⊂ X × X non-empty open sets such that (T × T )nU ∩ V = ∅

n for all n. Let K = ⋃n(T × T ) U, K is compact, proper and T × T -invariant subset of X × X, also K has non-empty interior (and exterior). On the other hand, the projection of K to either factor of X × X is a non-empty compact invariant subset of X thus it must be all of X by minimality of X. Let K(x) = {y ∈ X ∶ (x, y) ∈ K} and

let fx = 1K(x) be the indicator function for the set K(x), which is an element of

1 1 L (X, µ). The space L (X, µ) is a metric space, and the shift map UT f = f ○ T is

1 isometric since µ is T invariant. Deﬁne π ∶ X → L (X, µ) by the formula π(x) = fx. By the T -invariance of K, π preserves the shifts, that is, π is a morphism, now one 106

1 only has to prove that π is continuous to see that π from X to π(X) ⊂ L (X, µ, UT ) is a non-trivial factor. f x π x y dµ y Consider the scalar function ( ) = ∫X ( )( ) ( ). From the dominated convergence theorem and the fact that K is closed, one can see that f is upper semi-continuous, and then continuous in at least one point (lemma 2..4.13 [30]),

then by lemma 4.1.53 f must be constant. Observe that since K is T × T -invariant and µ is T -invariant, then f is also T-invariant. Notice that since K is closed then

1 one has that lim supx→x0 fx(y) ≤ fx0 (y) for any x0 ∈ X, thus fx converges in L (X, µ)

to 0 outside of the support of fx0 by the dominated convergence theorem. This,

1 together with the fact that f is constant imply that fx converges to fx0 in L (X, µ) on all of X, thus π is continuous.

4.2 Non-trivial equicontinuous factors and existence of non-trivial characters

Let (X,T ) be a topological dynamical system and π ∶ (X,Y ) → (Y,S) a factor.

n n Let G1 = {T ∶ n ∈ Z} and G2 = {S ∶ n ∈ Z}. Recall that the basic neighborhoods of the identity in the topology of uniform convergence (compact-open topology) for the

˜ i groups G1 and G2, are of the form V = {S ∈ G2 ∶ ∀y ∈ Y, (y, gy) ∈ V } and ˜ i U = {T ∈ G1 ∶ ∀x ∈ X, (x, gx) ∈ U} respectively, where U is an entourage of the

diagonal in UX and V is an entourage of the diagonal in UY . Deﬁne the

n n homomorphism α ∶ G1 → G2 such that α(T ) = S for all n ∈ Z. α is continuous with respect to the topologies of uniform convergence. Indeed, since π is a continuous map between compact spaces, π is uniformly continuous. This means that for every

m ˜ V ∈ UY there exists U ∈ UX , such that π × π(U) ⊂ V . If T ∈ U then for all x ∈ X, (x, T mx) ∈ U which implies by uniform continuity that (πx, π(T nx)) ∈ V , since π is 107

a morphism it follows that (πx, π(T nx)) = (πx, Snπx) = (y, α(T n)y) ∈ V for all y ∈ Y . This means that α(U˜) ⊂ V˜ , thus α is a continuous homomorphism. Now, consider the Kronecker factor K as in the proof of theorem 4.1.46, and

observe that the projection p ∶ G → K where G = {Sn ∶ n ∈ Z}, is a continuous homomorphism, thus the map χ ○ p is a character for G as well. This means that χ ○ p ○ α is continuous character for {T n ∶ n ∈ Z} with the compact-open topology. Theorem 4.1.46 states that a non-trivial eigenvalue of a minimal system (X,T ) induces a non-trivial character χ of the Kronecker factor of (X,T ) and vice versa. This means that if (X,T ) has a non-trivial eigenvalue then the group {T n ∶ n ∈ Z} with the compact-open topology has a non-trivial character. On the other hand, if

G = {T n ∶ n ∈ Z} with the compact-open topology has a non-trivial character then G acts non-trivially on the circle group by rotations, which implies that due to the universal property of the universal minimal compact G-space MG, the topological dynamical system (G, MG) has a non-trivial equicontinuous factor. The discussion above together with theorem 4.1.54 gives us the following theorem:

Theorem 4.2.1. Let (X,T ) be a minimal topological dynamical system, then the following are equivalent:

1. (X,T ) has a non-trivial eigenvalue.

2. The Kronecker factor of (X,T ) is not trivial.

3. (X,T ) has a non-trivial equicontinuous factors.

4. The Kronecker factor has a non-trivial character.

5. (X,T ) is not topologically weakly mixing. 108

And all of them imply the following two equivalent statements:

• The group {T n ∶ n ∈ Z} with the compact-open topology has a non-trivial character.

• (MG,T ) has an equicontinuous factor, where MG is the universal minimal compact G-space and G = {T n ∶ n ∈ Z}

4.3 Vietoris topology and the space of maximal chains

4.3.1 Vietoris topology

In this section, K is assumed to be a compact space unless otherwise stated.

Deﬁnition 4.3.1. Let X be a topological space. Let K(X) be the collection of all compact subsets of X. The Vietoris topology on K(X) is the topology generated by sets of the form:

1. {A ∈ K(X) ∶ A ⊂ U} where U is a given open set of X

2. {A ∈ K(X) ∶ A ∩ U ≠ ∅} where U is a given open set of X

From the deﬁnition one can see that a basic open set in the Vietoris topology on K(X) is of the form:

{A ∈ K(X) ∶ A ⊂ U0 & A ∩ U1 ≠ ∅ & ... & A ∩ Un ≠ ∅}

where U0,U1, ..., Un are given open sets of X.

Consider the open set in the Vietoris topology {A ∈ K(X) ∶ A ⊂ ∅} = {∅} which means that ∅ is an isolated point of K(X). The subspace OF K(X) that does not contain this isolated point. is denoted by Exp(X) = K(X) ∖ {∅}.

Proposition 4.3.2. (Theorem 4.2, Michael [19]) Exp (K) is compact. 109

Deﬁnition 4.3.3. C ⊂ Exp (K) is a chain if for every F,E ∈ C either F ⊂ E or E ⊂ F .

Lemma 4.3.4. Let C ⊂ Exp (K) be a chain and let B ∈ Exp (K). If there exist F ∈ C such that F ⊂~ B and B ⊂~ F then B ∈~ C.

Proof. Deﬁne an open neighborhood O of B whose intersection with C is empty.

Let x ∈ F ∖ B and y ∈ B ∖ F . Since F is closed and y ∈~ F there is U1 open in K such that y ∈ U1 and F ∩ U1 = ∅. Let K ∖ {x} = U0 since x ∈~ B then B ⊂ U0. Now consider

O = {A ∈ Exp (K) ∶ A ⊂ U0 & A ∩ U1 ≠ ∅}

Clearly B ∈ O. Now, let E ∈ C: Case 1: E ⊂ F then E ∩ U1 = ∅ which implies that E ∈~ O. Case 2: F ⊂ E. Since x ∈ F then x ∈ E thus E ⊂~ U0, thus E ∈~ O. This implies that B ∈~ C.

Proposition 4.3.5. If C ⊂ Exp (K) is a chain so is its closure.

Proof. Let B1,B2 ∈ C. By lemma 4.3.4, C ∪ {B1} is also a chain and by the same

argument (C ∪ {B1}) ∪ {B2} is also a chain. This implies that either B1 ⊂ B2 or

B2 ⊂ B1.

Corollary 4.3.6. A maximal chain is closed in Exp K.

Since a maximal chain C is a closed set in Exp (K) (thus compact in Exp (K) by proposition 4.3.2) then C is an element of Exp Exp (K).

Deﬁnition 4.3.7. Let Φ(K) ⊂ Exp Exp (K) (or Φ for short) be the collection of all maximal chains of Exp K. 110

In order to understand Φ as a topological space one has to understand the topology of Exp Exp (K), in the following lines a discussion about the open sets of this topology is presented.

Remark 4.3.8. Each basic open set R of Exp Exp (K) is determined by open sets

O0, O1, ..., On of Exp (K). At the same time each of these Oi is determined by open sets Uij ⊂ K. In order to gain more clarity on the topology of Exp Exp (K) consider its sub-basic open sets.

1. Consider R = {P ∈ Exp Exp (K) ∶ P ∩ O =~ ∅}. The fact that P ∈ Exp Exp (K) implies that P is a collection of closed sets of K, i.e. P ⊂ Exp (K). Now, the fact that P ∩ O =~ ∅ implies that P and O have a common element (a closed set from K). This means that there is F ∈ P such that satisﬁes with all the

requirements given by the open sets U1, ..., Un ⊂ K that determine O . Another way of looking at it is to say that as long as there is one element in P that

satisﬁes with the conditions coming from the Ui’s (about being a good

approximation for certain closed set) then P is in R.

2. Consider R = {P ∈ Exp Exp (K) ∶ P ⊂ O}. The contrast with the above case is that for P to be in R all F ∈ P have to satisfy the requirements given by the

open sets U1, ..., Un ⊂ K that determine O. In other words all elements of P

have to satisfy the conditions coming from Ui’s (about being a good approximation form certain closed set).

Using these ideas about the sub-basic open sets in Exp Exp (K), now when looking at the set

R = {P ∈ Exp Exp (K) ∶ P ⊂ O0,P ∩ Oi =~ ∅, i = 1, 2, ...n} 111

one knows that P ∈ R if and only if each element of P follows the requirements given by the U0j (approximate a given closed set inside the O0 neighborhood) and that at least one of its elements follows the requirements given by (Uij, i > 0)

(approximate a given closed set inside the (Oi, i > 0) neighborhood).

Example 4.3.9. Let D ∈ Exp Exp (K) with D not a chain. Since D is not a chain then there are closed sets F,E ∈ D such that there exist x ∈ F ∖ E and y ∈ E ∖ F . Using the ideas in remark 4.3.8 one can build a speciﬁc kind of neighborhood R for D. Consider:

O1 = {A ∈ Exp (K) ∶ A ⊂ U10,A ∩ U11 =~ ∅}

where U10 = K ∖ E and x ∈ U11 ⊂ U11 ⊂ F

O2 = {A ∈ Exp (K) ∶ A ⊂ U20,A ∩ U21 =~ ∅}

where U20 = K ∖ F and x ∈ U21 ⊂ U21 ⊂ E

It is clear that F ∈ O1 and E ∈ O2, since O1 and O2 were built to be neighborhoods that ‘approximate’ F and E respectively. Moreover one can see that

O1 ∩ O2 = ∅, indeed w.l.o.g let B ∈ O1 then B ⊂ K ∖ E thus B ∩ U21 = ∅ since

U21 ⊂ E. Notice that the fact that O1 and O2 are disjoint does NOT necessarily mean that an element from O1 are disjoint to an element from O2. Deﬁne:

R = {P ∈ Exp Exp (K) ∶ P ∩ O1 =~ ∅,P ∩ O2 =~ ∅}

′ ′ If P ∈ R then P has an element F that ‘approximates’ F (meaning F ∈ O1)

′ ′ and an element E that ‘approximates’ E (meaning E ∈ O2). Thanks to the way the approximation is deﬁned one has that x ∈ F ′ ∖ E′ and y ∈ E′ ∖ F ′ giving that F ′ ⊂~ E′ 112

and E′ ⊂~ F ′, which implies that P can’t be a chain and that R contains no chain. In particular notice that D ∈ R

An immediate implication of the previous example is the following proposition:

Proposition 4.3.10. Let D ∈ Exp Exp (K) if D is not a chain then D ∈~ Φ

Proof. R from the example above is an open neighborhood that contains D and whose intersection with Φ is empty. Thus D ∈~ Φ.

Proposition 4.3.11. (See [35]) Let C ∈ Exp Exp (K) be a non-maximal closed chain then C ∈~ Φ

Proposition 4.3.12. Φ is a closed set of Exp Exp (K)

Proof. Proposition 4.3.11 together with proposition 4.3.10 imply that φ = φ¯. Thus Φ is closed and thus compact set of Exp Exp (K).

Suppose that G is a topological group and K is a compact G-space. Then the

natural action of G on Exp (K) is continuous, hence Exp (K) is also a G-space, and so is Exp Exp (K). Since the compact set Φ ⊂ Exp Exp (K) is G-invariant, then Φ is also a G-space.

4.3.2 On the action of G on MG

In [35], Uspenskij showed that the action of G on the universal minimal

compact G-space MG is not 3-transitive Below a proof following [35].

Deﬁnition 4.3.13. Let H and G be topological groups with H ⊂ G. Consider the natural action of H on any compact G-space X, if this action always has a ﬁxed point H is called relatively extremely amenable with respect of G. 113

Proposition 4.3.14. Let G be a topological group and let p ∈ MG. If

Sbp = {g ∈ G ∶ gp = p} is the stabilizer of p. Then Sbp is relatively extremely amenable with respect of G.

Proof. Let X be a compact G-space. By deﬁnition there exists a onto G-map f ∶ MG → X. Since f is a G-map gf(p) = f(gp) = f(p) for all g ∈ Sbp. Thus f(p) is a

Sbp-ﬁxed point. Since X is arbitrary then Sbp is relatively extremely amenable with respect of G

Deﬁnition 4.3.15. Let G be a topological group and X be a compact G-space. G

is called shackled to X if the action of G on Φ ⊂ Exp Exp (X) has a ﬁxed point.

Example 4.3.16. Let G be a topological group, p ∈ MG and Sbp = {g ∈ G ∶ gp = p}.

By proposition 4.3.14 Sbp is shackled to any G-space X, in particular it is shackled to MG, that is there exists there exists a maximal chain C of closed subsets of MG

that is Sbp-invariant.

The following result is immediate.

Proposition 4.3.17. Let H,G topological groups with H ⊂ G, let X be a compact G-space:

H is extremely amenable Ô⇒ H is relatively extremely amenable with respect of G Ô⇒ H is shackled to X.

Deﬁnition 4.3.18. The action of G on a G-space X is called n-transitive if

SXS ≥ n and for any triples(a1, a2, ..., an) and (b1, b2, ..., bn) of distinct points in X

there exist g ∈ G such that gai = bi, i = 1, 2, ..., n.

Theorem 4.3.19. (Usepenskij [35]) Let G be a topological group, X a compact

G-space, p ∈ X and H = Sbp = {g ∈ G ∶ gp = p}. If H is shackled to X then the action of G on X is not 3-transitive. 114

Proof. By hypothesis there is a maximal chain C ∈ Φ such that it is invariant under the action of H. This implies that if g ∈ H and F ∈ C then gF ∈ C, which also implies that F and gF need to be comparable (with the ⊂ relation). By contradiction suppose that the action of G on X is 3-transitive.

Since C is maximal then C contains a singleton {∗}. If ∗ =~ p then the 2-transitivity of the action implies that there is g ∈ G (really g ∈ H) such that gp = p and g∗ = ∗′ where ∗′ is any point in X. Since g ∈ H and the sets {∗} and g{∗} = {∗′} are not comparable then this situation is impossible thus ∗ = p and every element of C contains {p} Assuming that the action of G on X is 3-transitive implies that SXS ≥ 3. Since C is a chain there exist F ∈ C and a, b ∈ X such that {p}= ~ F =~ X, a ∈ F ∖ {p} and b ∈ X ∖ F . Again thanks to 3-transitivity there exists g ∈ G (really g ∈ H) such that gp = p, ga = b, gb = a. Since g ∈ H, F and gF need to be comparable. Nevertheless a ∈ F ∖ gF and b ∈ gF ∖ F .(Ð→←Ð) Thus the action of G on X is not 3-transitive

Corollary 4.3.20. Let G be a top group, X a compact G-space, p ∈ X and H = {g ∈ G ∶ gp = p}. If the action of G on X is 3-transitive then H is not shackled to X, in particular H is not extremely amenable.

Corollary 4.3.21. For every topological group G the action of G on the universal

minimal compact G−space MG is not 3−transitive.

Proof. By example 4.3.16 we see that Sbp is shackled to MG thus by theorem 4.3.19

the action of G on MG is not 3-transitive

Remark 4.3.22. Let G = Homeo(Iω), the homeomorphism group of the Hilbert cube Iω endowed with the compact-open topology. In [35], Uspenskij used corollary 115

ω 4.3.21 to show that the universal minimal compact G-space MG is not I , since the action of G on Iω is known to be 3-transitive.

4.4 The space of maximal chains of subcontinua

Another perspective on the methods used by Usepenkij in [35] is the following:

As one studies the universal minimal G-space MG, one can study the action of G through the study of the map f ∶ MG → Φ(MG), which exists due to the universality of MG. It is through the study of f and the action of G on Φ(MG) that Uspenskij established the the non 3-transitivity of the action of G on MG.

Inspired in this idea, Y. Gutman studied the subspace M = M(X) ⊂ Φ(X) of maximal chains of continua of a G-space X. In [12], Gutman proved that for certain

G-spaces, the action of G on M is minimal. Again, by the universality of MG there is an onto G-map f ∶ MG → M. Gutman proved that the action of G on MG is not 1-transitive by showing that the action of M is not 1-transitive. Below definitions and the most important theorems from [12] to establish these results are presented Recall that a continuum is a non-empty compact connected metric space. A Peano continuum is a continuum that admits at every point a neighborhood basis of open connected sets. Denote by c(X) all subcontnua of X, similarly to definition 4.3.7 define M(X) to be the collection of maximal chains in c(X). For X a non-trivial continuum, M(X) can be characterized as the space of connected (in Exp (X)) maximal chains, that is M(X) = Φ ∩ c(Exp (X)) (see Lemma 2.3 [12]). Let X be a G-space, the action of G on X is called locally transitive if for any open set U ⊂ X and x ∈ U the set {gx ∶ g ∈ GU } is a neighborhood of x, where

GU = {g ∈ G ∶ gx = x for x ∈~ U}. Let X be the Hilbert cube or a closed manifold of dimension 2 or higher, then any group containing one of the following groups is locally transitive on X: (1) G = Homeo 0(X), the arcwise connected component of 116

the identity in Homeo (X). (2) If X is an orientable manifold, G = Homeo+(X), the group of orientation preserving homeomorphisms. (3) For X a smooth manifold,

G = Diﬀeo0(X), the arcwise connected component of the identity in the diﬀeomorphism group of X.

Let I be an interval and let Cs(I,X) denote the collection of all continuous simple (injective) paths p ∶ I → X, these paths are called arcs. A space is called strongly arcwise-inseparable (SAI) if and only if any non-empty open and

connected set U ⊂ X and for any arc p ∈ Cs([a, b],U) the set U ∖ p([a, b]) is connected and non-empty. A space is called strongly R-inseparable (SRI) if and only if any non-empty open and connected set U ⊂ X and for any arc

p ∈ Cs([a, b],X) the set U ∖ p([a, b]) is connected and non-empty. Notice that (SRI) implies the property (SAI). Some examples of spaces with this properties are as follows: Closed manifolds of dimension 2 are SAI, closed manifolds of dimension 3 or higher and the Hilbert cube are SRI. The term closed manifod refers to a compact manifold without boundary.

Definition 4.4.1. (Gutman [12]) The members of Cs([0, ∞],X) are called rays.A ¯ R+-chain is any element of M(X) such that c = (ct)t∈[0,∞] and there exists a ray ρ with ct = ρ([0, t]) for all t < ∞ When the last condition is satisfied, one says that c is induced by the ray ρ. Notice that c∞ = X, since c is maximal thus ρ([0, ∞]) is dense in X. Let R be defined as follows:

R = {c ∈ M(X) ∶ c is R¯ +-chain}

It turns out that for some spaces, the chains in M(X) can be approximated by the elements of R. 117

Theorem 4.4.2. (Gutman [12]) Let X be a Peano continuum which is SAI, then

R¯ = M(X), the closure taken with respect to the Vietoris topology in Exp Exp (X).

Deﬁnition 4.4.3. (Gutman [12]) Let N ∈ N. Equip M(X)N with the product

N N N topology. Deﬁne the subspace R∗ ⊂ M as follows: (c1, ..., cN ) ∈ R∗ if and only if:

(1)ci = {ρ([0, t])}t∈R+ ∪ {X} ∈ R, i = 1, ..., n

(2)ρi(R+) ∩ ρj(R+) = ∅, 1 ≤ i < j ≤ n

Theorem 4.4.4. (Gutman [12]) Let X be a Peano continuum which is SRI, then

¯ N N R∗ = M(X) .

One of the most interesting phenomena found by Gutman is that for a continuum G-space X that satisfy certain hypothesis, the space M(X) ⊂ Φ(X), of maximal chains of continua of X is minimal. This is formalized in the following theorem

Theorem 4.4.5. (Theorem 6.5 Gutman [12]) Let G act locally transitively on a Peano continuum X which is strongly arcwise-inseparable. Then the action of G on

M(X) is minimal

Finally, this allows to improve on Uspenskij’s result, corollary 4.3.21.

Theorem 4.4.6. (Corollary 6.6 Gutman [12]) Let G act locally transitively on a Peano continuum X which is strongly arcwise-inseparable, then the action of G on the universal minimal G-space MG is not 1-transitive.

Proof. It is enough to show that the action of G on M(X) is not 1-transitive, since if the action of G on MG was 1-transitive then the action of G on all minimal G-spaces would be 1-transitive. 118

Let c ∈ R, c induced by the ray ρ. Let r ∈ X. Deﬁne v = {B(r, t)}t∈R. Since no arc is SAI it follows that one cannot map balls B(r, t) (homeomorphically) onto arcs of the form ρ([0, a]). This implies that there does not exist g ∈ G so that g(v) = c, from which one concludes that G on M is not 1-transitive.

This corollary implies that the action of G = Homeo (X) on MG is not 1-transitive, for X a closed manifold of dimension 3 or higher or the Hilbert cube.

Nonetheless it is unknown if M(X) itself plays the roll of MG for this G. The following question remains open:

Question 15. (Gutman [12]) Is the universal minimal space for the group

Homeo (X), X being a closed manifold of dimension 3 or higher or the Hilbert cube, equal to the space M(X)?

4.5 Applications to the Glasner-Pestov problem

Some applications to the theory developed in this chapter is given below. Theorem 4.2.1 can be applied to speciﬁc minimal spaces or operators to show the existence of characters for groups of the form {T n ∶ n ∈ Z} with the compact-open topology.

Theorem 4.4.5, states that for G = {T n ∶ n ∈ Z} acting locally transitively on a Peano continuum X which is arcwise-inseparable (SAI) the action of G on M(X) is minimal, that is (M(X),T ) is a minimal topological dynamical system. It follows by theorem 4.2.1 that if such system has a non-trivial equicontinuous factor then G

has a non-trivial character, or which is the same if (M(X),T ) is not weakly mixing then G has a non-trivial character. The above discussion establishes the following theorem: 119

Theorem 4.5.1. Let G = {T n ∶ n ∈ Z} ⊂ Homeo (X) act locally transitively on a Peano continuum X which is arcwise-inseparable (SAI). If (M(X),T ) is not weakly mixing then G has a non-trivial continuous character.

By corollary 4.4.6 the action of G on M(X) is not 1-transitive, this leaves the question open to weather this action is weakly mixing or not, if the action was 2-transitive, immediately one gets that the system is weakly mixing.

In order to study the weakly mixing property of the system (M(X),T ) one has to ask when (M(X) × M(X),T × T ) is topologically transitive. By proposition 4.1.49, (M(X) × M(X),T × T ) is topologically transitive when there is an element

(a1, a2) ∈ M(X) × M(X) whose T × T orbit is dense in M(X) × M(X). In the case that such element does not exist then the system is not weakly mixing. From

2 theorem 4.4.4 one knows that R∗ = M(X) × M(X). One can study the existence of

2 an element (a1, a2) ∈ M(X) × M(X) whose T × T orbit approximates elements in R∗ to learn about the topologically transitivity of (M(X) × M(X),T × T ).

If there is any case in which M(X) = MG where X satisﬁes the hypothesis in theorem 4.5.1, then again by theorem 4.2.1, (M(X),T ) being weakly mixing implies that G has no non-trivial characters.

Besides the need to investigate if (M(X),T ) is weakly mixing in theorem 4.5.1, the following questions arise:

Question 16. How can one characterize the groups G = {T n ∶ n ∈ Z} ⊂ Homeo (X) that act locally transitively on X a Peano continuum space which is arcwise-inseparable (SAI)?

Question 17. Can the hypothesis of locally transitive be relaxed and still get a

minimal M(X) to broaden the class of groups to which theorem 4.5.1 can be applied? 120

A diﬀerent approach to use theorem 4.2.1 is to look for non-trivial eigenvalues

of (X,T ). In that sense one needs eigenvalues of T acting on C∗(X). One theorem that implies the existence of eigenvalues for Banach algebras under certain hypothesis is 2.1.34, so one can consider T in the Banach algebra of bounded linear

operator on C∗(X).

Theorem 4.5.2. Let (X,T ) be a minimal topological dynamical system, Ω an open set of C, and h a holomorphic function on Ω. If T acting on C∗(X) is such that T = h(U) where U is a linear bounded operator of the Banach algebra C∗(X), and ∅ =~ σ(U) ⊂ Ω with h(σ(U)) ∖ {1}= ~ ∅, where σ(U) is the set of eigenvalues of U, then (X,T ) has a non-trivial equicontinuous factor. Thus {T n ∶ n ∈ Z} has a non-trivial character.

This theorem is abstract and to really understand the algebraic counterpart of the phenomenon in theorem 4.2.1 one has to deepen in the study of linear bounded

operators of C∗(X). Maybe then, one can apply use theorem 2.1.34 in a more concrete fashion. 121 5 Future work

In section 3.5 and 4.5, some questions for future research were raised. Besides these paths of investigation there is another approach to the Glasner-Pestov problem that will be considered for future research.

Recall from section 2.1.5 that an abelian unital Banach algebra A is semisimple if radA = {0}. The set radA is the set of all quasi-nilpotent elements of A. x ∈ A is

n 1~n quasi-nilpotent if ρ(x) = limn→∞ Yx Y = 0. Recall that the Gelfand transform from

∗ ∗ A → C (MA) has kernel radA, this means that A~radA is isomorphic to C (MA) as C∗-algebras. In particular if there is an element x ∈ A that is not quasi-nilpotent

then MA is not empty and A has a multiplicative homomorphism (character) λ, such that λ(x)= ~ 0 . This leads to the following:

Proposition 5.0.1. Let A be a unital commutative Banach algebra. If there exists x ∈ A such that YxnY > Aebn, A, b > 0, n ∈ Z+ then A has a character λ, such that λ(x)= ~ 0.

The condition YxnY > Aebn, A, b > 0 implies that x is not quasi-nilpotent. Let X be a compact Hausdorﬀ (metric) space. Consider T ∈ Homeo(X) such that T has no ﬁxed points, and consider the polynomial algebra A′ generated by T . Let E be a Banach space on which A′ acts upon as a subalgebra of B(E), the space of bounded linear operators of E with the norm topology. One can endow A′ with such norm and take its closure to get the unital commutative Banach algebra

A = A′. If the norm topology on A, restricted to {T n ∶ n ∈ Z} matches with the compact-open topology from Homeo(X) then the existence of a character λ, ∋ λ(T )= ~ 0, for the Banach algebra A implies the existence of a non-trivial continuous character for the topological group {T n ∶ n ∈ Z} (endowed with the compact-open topology). 122

This means that if a Banach space E can be found as above and the norm on A inherited from B(E) is such that YT nY > Aebn, A, b > 0, n ∈ Z+, then {T n ∶ n ∈ Z} has a non-trivial continuous character.

A natural space to make A′ act upon is the free locally convex space generated by X. One of our future research projects is the investigation of this action and whether or not a construction as the one mentioned above is possible. 123 References

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