Perturbations of Banach algebras
type: Item
instance of: scholarly article
title: Perturbations of Banach algebras
zbMATH Open document ID: 0557.46029
published in: Lecture Notes in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: There are several sources of the considered theory. One is the classical Banach-Stone theorem stating that the spaces C(S) and C(S') are isometric if and only if the compacts S and S' are homeomorphic. Important are also the following generalizations of this result: Uniform algebras A and B are isomorphic if and only if they are isometric (Nagasawa); If the Banach-Mazur distance between the spaces C(S) and C(S') is smaller then two, then these spaces are isometric (Amir, Cambern); If S and S' are compact metric spaces then any isometric embedding of C(S) into C(S') is induced by a continuous map from a subset of S' onto S (Holsztyński). The ''perturbation problems'' mentioned in the introduction are (i) Let A and B be Banach algebras, both contained in a Banach algebra C. Assume that they are close in a geometric sense (distance between their unit balls is small). Does it follow that A and B are isomorphic as algebras, or at least have some common algebraic properties? (ii) Assume that we have defined two multiplications xy and \(x\times y\) on a Banach space A, which are close as bilinear maps. What common algebraic properties of (A,\(\cdot)\) and (A,\(\times)\) must appear? The book gives an account on these and related problems. It is based upon the work of several writers (one should mention B. E. Johnson and R. Rochberg) and a substantial part of its material is due to the author. It is rather a technical book intended for specialists. The basic concept of the book is the concept of an \(\epsilon\)-perturbation of a Banach algebra A, i.e. another continuous multiplication \(\epsilon\)-close in a prescribed sense to the original multiplication of A. In the first chapter the author presents several concepts of an \(\epsilon\)-perturbation and studies their properties. We quote two corollaries to the main theorem. If the Shilov and Choquet boundaries coincide for a uniform algebra A, then for a sufficiently small perturbation \(\times\) of A these boundaries for (…
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 3888342
zbMATH Keywords: Banach-Stone theorem
zbMATH Keywords: Uniform algebras
zbMATH Keywords: Banach-Mazur distance
zbMATH Keywords: isometric embedding
zbMATH Keywords: perturbation problems
zbMATH Keywords: \(\epsilon \) -perturbation
zbMATH Keywords: Shilov and Choquet boundaries
zbMATH Keywords: natural algebras
zbMATH Keywords: uniform algebras
zbMATH Keywords: stability of Shilov-Bishop decomposition of a uniform algebra
zbMATH Keywords: Deformations of algebras of functions on riemann surfaces
zbMATH Keywords: Hochschild cohomology group
zbMATH Keywords: multiplicatively-convex algebra
author: Krzysztof Jarosz
MaRDI profile type: Publication
DOI: 10.1007/BFB0076885
description: scientific article; zbMATH DE number 3888342