Extremely Amenable Groups and Banach Representations 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 Javier Ronquillo Rivera May 2018 © 2018 Javier Ronquillo Rivera. All Rights Reserved. 2 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