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On universal minimal compact \(G\)-spaces. (English) Zbl 0996.22003

Let \(G\) be a topological group. A \(G\)-space \(X\) is minimal if it has no proper \(G\)-invariant closed subsets. The universal minimal compact \(G\)-space \(M_G\) is characterized by the following properties: \(M_G\) has no proper closed \(G\)-invariant subsets, and for any compact \(G\)-space \(Y\) there exists a \(G\)-map \(M_G\rightarrow Y\). V. Pestov in [Trans. Am. Math. Soc. 350, No. 10, 4149-4165 (1998; Zbl 0911.54034)] proved that the universal minimal compact \(G\)-space \(M_G\) of the group \(G=H_+(S^1)\) of all orientation preserving self-homeomorphisms of the circle \(S^1\) can be identified with \(S^1\) (Theorem 6.6). In this paper he asked whether a similar assertion holds for the Hilbert cube \(Q\): if \(G=H(Q)\), are \(M_G\) and \(Q\) isomorphic as \(G\)-spaces? The main result of the paper is
Theorem 1.1. For every topological group \(G\) the action of \(G\) on the universal minimal compact \(G\)-space \(M_G\) is not \(3\)-transitive. Theorem 1.1 implies the negative answer on Pestov’s question. In the final part the author introduces the notion of the greatest ambit \(\mathcal S(G)\) of a topological group \(G\) for the demonstration of the uniqueness of \(M_G\). The greatest ambit \(\mathcal S(G)\) of a topological group \(G\) is by definition the compactification corresponding to the algebra of all right uniformly continuous bounded functions on \(G\). In this part the author proves that for every topological group \(G\) the greatest ambit \(\mathcal S(G)\) has the natural structure of a left-topological semigroup with identity such that the semigroup operation \(\mathcal S(G)\times\mathcal S(G)\rightarrow\mathcal S(G)\) extends the action \(G\times\mathcal S(G)\rightarrow\mathcal S(G)\), and shows that every universal compact minimal \(G\)-space of a topological group \(G\) is isomorphic to a minimal closed left ideal of the greatest ambit \(\mathcal S(G)\).

MSC:

22A05 Structure of general topological groups
22F05 General theory of group and pseudogroup actions
54D30 Compactness
54H15 Transformation groups and semigroups (topological aspects)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
57S25 Groups acting on specific manifolds
22A15 Structure of topological semigroups

Citations:

Zbl 0911.54034
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