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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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