Edward Beckenstein

Rings of continuous functions with values in a topological field

Let F be a complete topological field. We undertake a study of the ring C(X, F) of all continuous F-valued functions on a topological space X whose topology is determined by C(X, F), in that it is the weakest making each function in C(X, F) continuous, and of the ring C*(X, F) of all continuous F-valued functions with relatively compact range, where the topology of X is similarly determined by C*(X, F). The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of X in the former case and of the Stone-Cech compactification of X in the latter. Most attention is given to the case where F and X are ultraregular; in this case we determine conditions on F that permit a development parallel to the classical theory where F is the real number field. One example of such conditions is that the cardinality of F be nonmeasurable and that the topology of F be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirotas theorem concerning the bornologicity of C(X) is obtained. Our goal is to develop a theory parallel to that of realcompact spaces and realcompactifications by replacing the topological field R of real numbers with a complete topological field F, with emphasis on the case where F is ultraregular. (An ultraregular space is a Hausdorff space for which the clopen sets form a basis, a clopen set being one that is both open and closed. An ultranormal space is a Hausdorff space each closed subset of which has a fundamental system of neighborhoods consisting of clopen sets. There exist ultraregular spaces that are not ultranormal [25], [261.) Our emphasis on ultraregular fields is prompted by the fact that every field whose topology is given by a valuation or an ultrametric is ultraregular (and, in fact, is ultranormal). If X and Y are topological spaces, C(X, Y) denotes the set of all continuous functions from X into Y, and C*(X, Y) denotes the set of all continuous functions f from X into Y such that f(X) is compact. If F is a topological field, C(X, F) and C*(X, F) are both F-algebras; if, moreover, F is locally compact, Received by the editors July 24, 1973. AMS (MOS) subject classifications (1970). Primary 54D35, 54D60, 54E15.

Banach-Stone Theorems and Separating Maps*

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.

The Hahn-Banach Theorem: The Life and Times

Abstract Without the Hahn-Banach theorem, functional analysis would be very different from the structure we know today. Among other things, it has proved to be a very appropriate form of the Axiom of Choice for the analyst. (It is not equivalent to the Axiom of Choice, incidentally; it follows from the ultrafilter theorem which is strictly weaker.) Riesz and Helly obtained forerunners of the theorem in the turbulent mathematical world of the early 1900s. Hahn and Banach independently proved the theorem for the real case in the 1920s. Then there was Murrays extension to the complex case—easy, once you realize that ( χ ) = Re ( χ ) − iRe (i χ ). Can continuous linear maps 06 be extended as easily as linear functionals? Banach and Mazur had already proved that they could not in 1933 but it was not until Nachbins 1950 result that a definitive answer was achieved to this more general question. In this article, we discuss the mathematical world into which the theorem entered, examine its connection to the axiom of choice, look at some ancestors, mention some of its consequences and consider some of its principal variations.

Automatic continuity of linear maps on spaces of continuous functions

Let C(S)and C(T) denote the sup-normed Banach spaces of real- or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map A∶C(T)→C(S) is calledseparating if when two functions x and y from C(T) have disjoint cozero sets then so do Ax and Ay. In the spirit of [3] and [4], we show that separating maps are automatically continuous in some important cases (Theorems 2.4 and 2.5). If a separating map is continuous, then it must be a continuous multiple of a composition map (Theorem 2.2). If A is injective, separating and detaching (Def. 2.4) then S and T are homeomorphic (Theorem 2.1).

Biseparating Maps and Rings of Continuous Functions

ABSTRACT: C(T) and C(S) denote the rings of real‐ or complex‐valued continuous functions on the Tihonov spaces T and S endowed with their respective compact‐open topologies. An additive bijection H:C(T)→C(S) such that xy= 0 ⇔HxHy= 0 is called biseparating. As shown in 4.5, the existence of such an H implies that the repletions (= realcompactifications) rep T and rep S of T and S are homeomorphic, thus generalizing Hewitts well‐known result about isomorphisms of rings of continuous functions. “Separating maps” (defined in the first section) serve as a general utensil for investigating rings of continuous functions and automatic continuity results between them. Their utility stems principally from the fact that a separating map H: C(T)→C(S) induces a continuous “support” map h:S→cT, where cT denotes a compactification of T. This notion of support of a separating map is introduced and discussed in the second section.

On continuous extensions

We consider various possiblities concerning the continuous extension of continuous functions taking values in an ultrametric space. In Section 1 we consider Tietze-type exetension theorems concerning continuous extendibility of continuous functions from compact and closed subsets to the whole space. In Sections 2 and 3 we consider extending “separated” continous functions in such a way that, certain continuous extensions remain separated. Functgions taking values in a complete ultravalued field are dealt with in Section 2, and the real and complex cases in Section 3.

Banach Algebras Over Valued Fields

Abstract By “Gelfand theory” here is meant the study of the consequences of topologizing the maximal ideals of a Banach algebra. The theory is most rich when the underlying field is that of the complex numbers. If the underlying field is 3R or some other valued field, a theory can still be developed however and that is discussed here. First the Gelfand theory for complex Banach algebras is reviewed briefly; then the analogous theory for the case when the field carries a non-archimedean real-valued valuation is presented. In the course of the latter discussion, a Stone-Weierstrass theorem is needed. In the last part of the paper some versions of the Stone-Weierstrass theorem which hold in algebras of continuous functions over fields with nonarchi - medean valuation are discussed.

The Fourier Transform

Note. In this chapter, unless otherwise indicated, all functions are complex-valued functions of a real variable. Given integrable functions of t, f, and k the function

The Discrete and Fast Fourier Transforms

g\left( \omega \right) = \int {_{E}f\left( t \right)k} \left( {t,\omega } \right)dt