Hyperfinite Dimensional Representations of Spaces and Algebras of Measures |
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Authors: | Miloš Ziman Pavol Zlatoš |
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Affiliation: | (1) Comenius University, Bratislava, Slovakia |
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Abstract: | Let X be a locally compact topological space and (X, E, Xω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set Xω ⊆ X, such that all internal subsets of Xω are relatively compact in the induced topology and X is homeomorphic to the quotient Xω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient , for certain external subspaces of the hyperfinite dimensional Banach space , with the norm ‖f‖1 = ∑x ∈ X |f(x)|. If additionally X = G is a hyperfinite group, Xω = Gω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of Gω, and G is isomorphic to the locally compact group Gω/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and are isometrically isomorphic as Banach algebras. Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic. |
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Keywords: | 2000 Mathematics Subject Classifications: 28E05, 43A10 03H05, 22D15, 46S20, 54J05 |
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