Banach-stone theorems and separating maps |
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Authors: | S Hernandez E Beckenstein L Narici |
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Institution: | (1) Departamento de Matemáticas, Universitat Jaume I, Campus de Penyeta Roja, 12071 Castellón, Spain;(2) Department of Mathematics, St. John's University, 10301 Staten Island, NY, USA;(3) Department of Mathematics, St. John's University, 11439 Jamaica, NY, USA |
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Abstract: | LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain
form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces
of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH
−1 are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries
4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex.
Research supported by the Fundació Caixa Castelló, MI/25.043/92 |
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Keywords: | 46B04 46E40 46E15 |
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