A Banach-Stone Theorem for completely regular spaces (original) (raw)

Review of the Banach-Stone Theorem

Journal of Development Review

This is a quick overview of the isomorphism between spaces of continuous functions, or C(X) type spaces, that depend on compact Hausdorff spaces outfitted with the uniform norm. When two compact metric spaces, X and Y, are homeomorphic, Banach assumed the problem in 1932. He came to the conclusion that if C(X) and C(Y) are isometric isomorphic, then X and Y are homeomorphic. Stone then generalized this outcome for a general compact Hausdorff space in 1937. Then it is frequently referred to as the Banach-Stone theorem. There are numerous variations of this classic result. We can derive the topological features of X and Y from Gelfand and Kolmogoroff's algebraic version, which was published in 1939.

Banach-stone theorems and separating maps

Manuscripta Mathematica, 1995

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.

A multiplicative Banach-Stone theorem

Contemporary Mathematics, 2015

The Banach-Stone theorem states that any surjective, linear mapping T between spaces of continuous functions that satisfies T (f) − T (g) = f − g , where • denotes the uniform norm, is a weighted composition operator. We study a multiplicative analogue, and demonstrate that a surjective mapping T , not necessarily linear, between algebras of continuous functions with T (f)T (g) = f g must be a composition operator in modulus.

A New Class of Banach Spaces and Its Relation with Some Geometric Properties of Banach Spaces

Abstract and Applied Analysis, 2012

By introducing the concept ofL-limited sets and thenL-limited Banach spaces, we obtain some characterizations of it with respect to some well-known geometric properties of Banach spaces, such as Grothendieck property, Gelfand-Phillips property, and reciprocal Dunford-Pettis property. Some complementability of operators on such Banach spaces are also investigated.

On algebras of Banach algebra-valued bounded continuous functions

Rocky Mountain Journal of Mathematics, 2016

Let X be a completely regular Hausdorff space. We denote by C(X, A) the algebra of all continuous functions on X with values in a complex commutative unital Banach algebra A. Let C b (X, A) be its subalgebra consisting of all bounded continuous functions and endowed with the uniform norm. In this paper, we give conditions equivalent to the density of a natural continuous image of X × M(A) in the maximal ideal space of C b (X, A).

Realcompactness and Banach-Stone theorems

2000

For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.

The Mazur-Ulam property for a Banach space which satisfies a separation condition

arXiv (Cornell University), 2022

We study C-rich spaces, lush spaces, and C-extremely regular spaces concerning with the Mazur-Ulam property. We show that a uniform algebra and the real part of a uniform algebra with the supremum norm are C-rich spaces, hence lush spaces. We prove that a uniformly closed subalgebra of the algebra of complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is C-extremely regular provided that it separates the points of the underlying space and has no common zeros. In section 3 we exhibit descriptions on the Choquet bounday, theŠilov bounday, the strong boundary points. We also recall the definition that a function space strongly separates the points in the underlying space. We need to avoid the confusion which appears because of the variety of names of these concepts; they sometimes differs from authors to authors. After some preparation, we study the Mazur-Ulam property in sections 4 through 6. We exhibit a sufficient condition on a Banach space which has the Mazur-Ulam property and the complex Mazur-Ulam property (Propositions 4.11 and 4.12). In section 5 we consider a Banach space with a separation condition (*) (Definition 5.1). We prove that a real Banach space satisfying (*) has the Mazur-Ulam propety (Theorem 6.1), and a complex Banach space satisfying (*) has the complex Mazur-Ulam property (Theorem 6.3). Applying the results in the previous sections we prove that an extremely Cregular complex linear subspace has the complex Mazur-Ulam property (Corollary 6.4) in section 6. As a consequence we prove that any closed subalgebra of the algebra of all complex-valued continuous functions defined on a locally compact Hausdorff space has the complex Mazur-Ulam property (Corollary 6.5).