On algebras of Banach algebra-valued bounded continuous functions

Canadian Journal of Mathematics, 1989

We are concerned in this paper with the study of homomorphisms between different algebras of continuous functions, especially the algebras of real functions which are either weakly continuous on bounded sets or weakly uniformly continuous on bounded sets on a Banach space (see definitions below). These spaces of weakly [uniformly] continuous functions appeared in relation with some questions in Infinite-dimensional Approximation Theory (see [4], [6], [11], [12], [13] and [16]); and since the structure of these function spaces is closely related with properties of different weak topologies (the bounded-weak and bounded-weak* topologies, respectively) and with the structure of Banach spaces on which they are defined, their study also presents interest from the point of view of Banach space theory, as can be seen in [2], [12] or [17].

Integral Equations and Operator Theory, 2000

1998

Let A be a unital Banach function algebra with character space Φ A . For x ∈ Φ A , let M x and J x be the ideals of functions vanishing at x, and in a neighbourhood of x, respectively. It is shown that the hull of J x is connected, and that if x does not belong to the Shilov boundary of A then the set {y ∈ Φ A : M x ⊇ J y } has an infinite, connected subset. Various related results are given.

p-Adic Numbers, Ultrametric Analysis and Applications, 2018

The main goal of this work is to study the Gelfand spaces of some commutative Banach algebras with unit within the space of bounded linear operators. We will also show, under special condition, that this algebra is isometrically isomorphic to some space of continuous functions defined over a compact. Such isometries preserve idempotent elements. This fact will allow us to define the respective associated measure which is known as spectral measure. Let us also notice that this measure is obtained by restriction of the reciprocal of the Gelfand transform to the set of characteristic functions of clopen subsets of the spectrum of above algebra. We will finish this work showing that each element of such algebras described above can be represented as an integral of some continuous function, where the integral has been defined through the spectral measure.

Scientiae mathematicae Japonicae, 2010

Let X be a topological space, Ë a cover of X and C b (X, Ã; Ë) the algebra of all Ã-valued continuous functions on X, which are bounded on every S ∈ Ë. Necessary and sufficient conditions for a subalgebra A of C b (X, Ã; Ë) to be dense in C b (X, Ã ; Ë) in the topology τ Ë of Ë-convergence and in the Ë-strict topology β Ë on C b (X, Ã ; Ë) are given. Also, necessary and sufficient conditions for the completeness of the topological algebras (C b (X, Ã; Ë),τ Ë) and (C b (X, Ã; Ë),β Ë) are given.

Proceedings of the American Mathematical Society

We obtain an explicit description of the spectrum (set of closed maximal ideals) of $H_{b}(X)$ , algebra of analytic functions on a Banach space X which are bounded on bounded subsets. We show that the spectrum of $H_{b}(X)$ admits a natural linear structure. Some applications to the algebra of uniformly continuous and bounded analytic functions on the unit ball $B \subset X$ are indicated.

Journal of Functional Analysis, 1988

We show that a Banach algebra A such that A* has the property (V*) of A. Pelczynski is Arens-regular. A strong uniqueness property of the extension of the product on a unital C*-algebra A to A ** is proved. The algebraic structure of the bidual of a C*-algebra can be obtained through the local reflexivity principle. We give examples which show that the results are sharp. c 1988 Academic PESS, ITIC. Let A be a Banach algebra. There are two natural ways [2,3] of extending the product of A to A ** Namely, if L, (resp. R,) is the left (resp. . right) multiplication by a E A and if x, y E A**, the products x x 1 y and x x z y are defined by their values of fc A * as <XXI xf)= (4 (a-, (Y? C(f)))) (xx2 YJ-) = (YY (a+ (4 Wf)))). Less technically, the point is that one has to "choose one side" for starting the extension of the product by o*-continuity. The products may be distinct even if A is commutative: an algebra A is said to be Arens- regular if the two products coincide (see the very useful survey [ 131). We consider in this paper the relations between Arens-regularity and the infinite-dimensional geometry of Banach spaces. We prove in Section I an easy-to-check condition on the Banach space A which implies that A is Arens-regular for any Banach algebra structure (Proposition 1.1) and give *l?quipe d'Analyse,

Trends in Mathematics, 2019

Let IK be an ultrametric complete field and let E be an open subset of IK of strictly positive codiameter. Let D(E) be the Banach IK-algebra of bounded strictly differentiable functions from E to IK, a notion whose definition is detailed. It is shown that all elements of D(E) have a derivative that is continuous in E. Given a positive number r > 0, all functions that are bounded and are analytic in all open disks of diameter r are strictly differentiable. Maximal ideals and continuous multiplicative semi-norms on D(E) are studied by recalling the relation of contiguity on ultrafilters: an equivalence relation. So, the maximal spectrum of D(E) is in bijection with the set of equivalence classes with respect to contiguity. Every prime ideal of D(E) is included in a unique maximal ideal and every prime closed ideal of D(E) is a maximal ideal, hence every continuous multiplicative semi-norm on D(E) has a kernel that is a maximal ideal. If IK is locally compact, every maximal ideal of D(E) is of codimension 1. Every maximal ideal of D(E) is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. The Shilov boundary of D(E) is equal to the whole set of continuous multiplicative semi-norms. Many results are similar to those concerning algebras of uniformly continuous functions but some specific proofs are required.

Journal of Functional Analysis, 1997

We show among other things that if B is a Banach function space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, with the property that for some odd natural number p>1, b 1Â p # B for all b # B, then B=C 0 (X).

Mediterranean Journal of Mathematics, 2013

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X, A) be the algebra of all continuous A-valued functions on X equipped with the topology of K(X)-convergence. Moreover, let M (A) and M(A) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X, A) and show that there exist bijections from M (C(X, A)) onto X × M (A) and M(C(X, A)) onto X × M(A). We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand-Mazur algebra with jointly continuous multiplication.