Banach algebras with unitary norms
type: Item
instance of: scholarly article
title: Banach algebras with unitary norms
zbMATH Open document ID: 0865.46040
author: Richard V. Kadison
author: Mogens Lemvig Hansen
published in: Pacific Journal of Mathematics
publication date: 1996-12-15T00:00:00Z
review text: The metric nature of the unitary group of a (unital) \(C^*\)-algebra is studied in the Banach algebra framework. Using the Russo-Dye theorem and its recent refinements as a point of departure, a unitary norm on a (complex, unital) Banach algebra is defined to be a norm such that the unit ball is the norm closure of the convex hull of a group of invertible elements in the algebra. If the group is maximal as a bounded subgroup of the group of invertible elements of the algebra, the norm is said to be maximal unitary. It is shown that the unitary group of a \(C^*\)-algebra is a maximal bounded subgroup of the group of invertible elements, that a norm-closed algebra of bounded operators on a Hilbert space whose norm is unitary is a \(C^*\)-algebra, that each finite-dimensional Banach algebra with a maximal unitary norm is isometrically isomorphic to a \(C^*\)-algebra (this fails without the assumption of maximality). The unitary norms on the algebra of continuous complex-valued functions on a compact Hausdorff space are studied. The \(L^1\) norm is shown to be unitary on a Wiener algebra. Using Effros-Ruan and Blecher-Ruan-Sinclair, it is shown that a (unital) \(L^\infty\)-matricially normed Banach algebra with unitary norm is (completely) isometric and isomorphic to a \(C^*\)-algebra.
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 950195
zbMATH Keywords: maximal bounded subgroups
zbMATH Keywords: metric nature of the unitary group
zbMATH Keywords: Russo-Dye theorem
zbMATH Keywords: unitary norm
zbMATH Keywords: convex hull of a group of invertible elements
zbMATH Keywords: maximal unitary
zbMATH Keywords: unitary group of a \(C^*\)-algebra
MaRDI profile type: Publication
full work available at URL: pjm.1996.175.535
OpenAlex ID: W1483296562
DOI: 10.2140/PJM.1996.175.535
description: scientific article; zbMATH DE number 950195