Isomorphisms of subspaces of vector-valued continuous functions
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ISOMORPHISMS OF SUBSPACES OF VECTOR-VALUED CONTINUOUS FUNCTIONS ˇ ∗ and J. SPURNY ´ J. RONDOS Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovsk´ a 83, 18675 Praha 8, Czech Republic e-mails: [email protected], [email protected]ff.cuni.cz (Received July 22, 2020; revised August 3, 2020; accepted August 7, 2020)
Abstract. We deal with isomorphic Banach–Stone type theorems for closed subspaces of vector-valued continuous functions. Let F = R or C. For i = 1, 2, let Ei be a reflexive Banach space over F with a certain parameter λ(Ei ) > 1, which in the real case coincides with the Schaffer constant of Ei , let Ki be a locally compact (Hausdorff) topological space and let Hi be a closed subspace of C0 (Ki , Ei ) such that each point of the Choquet boundary ChHi Ki of Hi is a weak peak point. We show that if there exists an isomorphism T : H1 → H2 with T · T −1 < min{λ(E1 ), λ(E2 )}, then ChH1 K1 is homeomorphic to ChH2 K2 . Next we provide an analogous version of the weak vector-valued Banach–Stone theorem for subspaces, where the target spaces do not contain an isomorphic copy of c0 .
1. Introduction We work within the framework of real or complex vector spaces and write F for the respective field R or C. If E is a Banach space then E ∗ stands for its dual space. We denote by BE and SE the unit ball and sphere in E, respectively, and we write ·, · : E ∗ × E → F for the duality mapping. For a locally compact (Hausdorff) space K, let C0 (K, E) denote the space of all continuous E-valued functions vanishing at infinity. We consider this space endowed with the sup-norm f sup = sup f (x) , x∈K
f ∈ C0 (K, E).
If K is compact, then this space will be denoted by C(K, E). For a compact space K, we identify the dual space (C(K, E))∗ with the space M(K, E ∗ ) of ∗ Corresponding
author. ˇ 17-00941S. The research was supported by the Research grant GACR Key words and phrases: function space, vector-valued Banach–Stone theorem, Amir– Cambern theorem. Mathematics Subject Classification: 47B38, 46A55.
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
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´ J. RONDOS RONDOˇ S and J. SPURNÝ SPURNY
all E ∗ -valued Radon measures on K endowed with the variation norm via Singer’s theorem (see [41, p. 192]). Thus M(K, F) is the usual set of Radon measures on K. Unless otherwise stated, we consider M(K, E ∗ ) endowed with the weak∗ topology given by this duality. Our starting point is the classical Banach–Stone theorem which asserts that, given a pair of compact spaces K and L, they are homeomorphic provided C(K, F) is isometrically isomorphic to C(L, F) (see [21, Theorem 3.117]). The first direction of our research are the so called isomorphic Banach– Stone type theorems, where the assumption of the isometry between the spaces of continuous functions is replaced by an isomorphism T : C(K, F) → C(L, F) with T · T −1 being small. A remarkable generalization of the Banach–Stone theorem in this way was given independently by Amir [3] and Cambern
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