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Extremely Amenable Groups and Banach Representations
Ronquillo Rivera, Javier Alfredo

2018, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
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 is a subset of 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 in 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 L^p[0,1], p in (0,1), 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.
Vladimir Uspenskiy (Advisor)
126 p.

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Rivera, Javier Ronquillo Accepted Dissertation 3-8-18 Sp18.pdf (749.68 KB) View|Download