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Equivariant classification of continuous functions on G-spaces. (English. Russian original) Zbl 0558.54030
Russ. Math. Surv. 39, No. 4, 111-112 (1984); translation from Usp. Mat. Nauk 39, No. 4(238), 149-150 (1984).
A compact Hausdorff space X is called an almost Milyutin space if there are a continuous map \(\phi\) : \(D^ m\to X\) of a generalized Cantor discontinuum of the weight m on X and a linear operator u: C(D\({}^ m)\to C(X)\) such that \(\phi^*\circ u\circ \phi^*=\phi^*\), where \(\phi^*: C(T)\to C(D^ m)\) is defined by \(\phi^*(f)=f\circ \phi\); \(f\in C(T)\) and C(X) is the Banach space of all real-valued functions on X endowed with the supremum norm. If X is a compact G-space (for some topological group G), then the formula \((gf)(x)=f(g^{-1}x)\); \(g\in G\), \(f\in C(X)\), \(x\in X\), evidently defines a continuous and linear action of G on C(X) i.e., C(X) becomes a Banach G-space. Theorem 1. Let G be a compact Hausdorff group and let the G-space S be an almost Milyutin space such that the weight \(w(S)=w(S/_ G)=\tau\) and there exists an equivariant embedding \(i: D^{\tau}\times G\to S\). Then the Banach G- space C(S) is equivariantly and linearly homeomorphic to the Banach G- space \(C(D^{\tau}\times G)\). Then the author defines \(G^*\)-extremely disconnectedness and gives equivalent conditions for a compact Hausdorff G-space to be \(G^*\)-extremely disconnected.
Reviewer: S.A.Antonyan
MSC:
54H15 Transformation groups and semigroups (topological aspects)
54C35 Function spaces in general topology
46E15 Banach spaces of continuous, differentiable or analytic functions
57S99 Topological transformation groups
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