A topological group $G$ is {\em extremely amenable} if every compact $G$-space has a $G$-fixed point. Let $X$ be compact and $G\subset{\mathrm{Homeo}} (X)$. We prove that the following are equivalent: (1) $G$ is extremely amenable; (2) every minimal closed $G$-invariant subset of $\exp R$ is a singleton, where $R$ is the closure of the set of all graphs of $g\in G$ in the space $\exp (X^2)$ ($\exp$ stands for the space of closed subsets); (3) for each $n=1,2,...$ there is a closed $G$-invariant subset $Y_n$ of $(\exp X)^n$ such that $\cup_{n=1}^\infty Y_n$ contains arbitrarily fine covers of $X$ and for every $n\ge 1$ every minimal closed $G$-invariant subset of $\exp Y_n$ is a singleton. This yields an alternative proof of Pestov's theorem that the group of all order-preserving self-homeomorphisms of the Cantor middle-third set (or of the interval $[0,1]$) is extremely amenable.
Deep Dive into On extremely amenable groups of homeomorphisms.
A topological group $G$ is {\em extremely amenable} if every compact $G$-space has a $G$-fixed point. Let $X$ be compact and $G\subset{\mathrm{Homeo}} (X)$. We prove that the following are equivalent: (1) $G$ is extremely amenable; (2) every minimal closed $G$-invariant subset of $\exp R$ is a singleton, where $R$ is the closure of the set of all graphs of $g\in G$ in the space $\exp (X^2)$ ($\exp$ stands for the space of closed subsets); (3) for each $n=1,2,...$ there is a closed $G$-invariant subset $Y_n$ of $(\exp X)^n$ such that $\cup_{n=1}^\infty Y_n$ contains arbitrarily fine covers of $X$ and for every $n\ge 1$ every minimal closed $G$-invariant subset of $\exp Y_n$ is a singleton. This yields an alternative proof of Pestov’s theorem that the group of all order-preserving self-homeomorphisms of the Cantor middle-third set (or of the interval $[0,1]$) is extremely amenable.
With every 1 topological group G one can associate the greatest ambit S(G) and the universal minimal compact G-space M(G). To define these objects, recall some definitions. A G-space is a topological space X with a continuous action of G, that is, a map G ร X โ X satisfying g(hx) = (gh)x and 1x = x (g, h โ G, x โ X). A map f : X โ Y between two G-spaces is G-equivariant, or a G-map for short, if f (gx) = gf (x) for every g โ G and x โ X.
A semigroup is a set with an associative multiplication. A semigroup X is right topological if it is a topological space and for every y โ X the self-map x โ xy of X is continuous. (Sometimes the term left topological is used for the same thing.) A subset I โ X is a left ideal if XI โ I. If G is a topological group, a right topological semigroup compactification of G is a right topological compact semigroup X together with a continuous semigroup morphism f : G โ X with a dense range such that the map (g, x) โ f (g)x from GรX to X is jointly continuous (and hence X is a G-space).
The greatest ambit S(G) for G is a right topological semigroup compactification which is universal in the usual sense: for any right topological semigroup compactification X of G there is a unique morphism S(G) โ X of right topological semigroups such that the obvious diagram commutes. Considered as a G-space, S(G) is characterized by the following property: there is a distinguished point e โ S(G) such that for every compact G-space Y and every a โ Y there exists a unique G-map f : S(G) โ Y such that f (e) = a.
We can take for S(G) the compactification of G corresponding to the C * -algebra RUCB(G) of all bounded right uniformly continuous functions on G, that is, the maximal ideal space of that algebra.
where N (G) is the filter of neighbourhoods of unity.) The G-space structure on S(G) comes from the natural continuous action of G by automorphims on RUCB(G) defined by gf
. We shall identify G with a subspace of S(G). Closed G-subspaces of S(G) are the same as closed left ideals of S(G).
A G-space X is minimal if it has no proper G-invariant closed subsets or, equivalently, if the orbit Gx is dense in X for every x โ X. The universal minimal compact G-space M(G) is characterized by the following property: M(G) is a minimal compact G-space, and for every compact minimal G-space X there exists a G-map of M(G) onto X. Since Zorn’s lemma implies that every compact G-space has a minimal compact G-subspace, it follows that for every compact G-space X, minimal or not, there exist a G-map of M(G) to X. The space M(G) is unique up to a G-space isomorphism and is isomorphic to any minimal closed left ideal of S(G), see e.g. [1], [9, Section 4.1], [11,Appendix], [10,Theorem 3.5].
A topological group G is extremely amenable if M(G) is a singleton or, equivalently, if G has the fixed point on compacta property: every compact G-space X has a G-fixed point, that is, a point p โ X such that gp = p for every g โ G. Examples of extremely amenable groups include Homeo + [0, 1] = the group of all orientation-preserving selfhomeomorphisms of [0, 1]; U s (H) = the unitary group of a Hilbert space H, with the topology inherited from the product H H ; Iso (U) = the group of isometries of the Urysohn universal metric space U. See Pestov’s book [9] for the proof. Note that a locally compact group = {1} cannot be extremely amenable, since every locally compact group admits a free action on a compact space [12], [9,Theorem 3.3.2].
We refer the reader to Pestov’s book [9] for various intrinsic characterizations of extremely amenable groups. These characterizations reveal a close connection between Ramsey theory and the notion of extreme amenability. The aim of the present paper is to give another characterization of extremely amenable groups, based on a different approach. For a compact space X let H(X) be the group of all self-homeomorphisms of X, equipped with the compact-open topology. Let G be a topological subgroup of H(X). There is an obvious necessary condition for G to be extremely amenable: every minimal closed G-subset of X must be a singleton. However, this condition is not sufficient. For example, let X be the Hilbert cube, and let G โ H(X) be the stabilizer of a given point p โ X. Then the only minimal closed G-subset of X is the singleton {p}, but G is not extremely amenable [11], since G acts without fixed points on the compact space ฮฆ p of all maximal chains of closed subsets of X starting at p. The space ฮฆ p is a subspace of the compact G-space Exp Exp X, where for a compact space K we denote by Exp K the compact space of all closed non-empty subsets of K, equipped with the Vietoris topology2 . It was indeed necessary to use the second exponent in this example, the first exponent would not work. One can ask whether in general for every group G โ H(X) which is not extremely amenable there exists a compact G-space X โฒ derived from X by applying a small number of simple functors, like powers, probability
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