A functional calculus for Banach PI-algebras
type: Item
instance of: scholarly article
title: A functional calculus for Banach PI-algebras
zbMATH Open document ID: 0577.46050
author: Denis Luminet
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let A be a unital Banach algebra. Take \(a_ 1,...,a_ n\in A\) and let B be the closed subalgebra of A they generate. The algebras \({\mathcal D}(\Omega)\) of entire matrix-valued functions were introduced by J. L. Taylor, who asked if they led to a functional calculus, generalizing the Shilov-Waelbroeck-Arens-Calderon theorem. We show that a necessary condition for a functional calculus map \({\mathcal D}(\Omega)\to A\) to exist is that B satisfy a polynomial identity; sufficient conditions are that B be a topological subquotient of a Banach Azumaya algebra, or that \(n=2\) and B satisfy all identities of \(2\times 2\) matrices. For closed subalgebras of Banach Azumaya algebras, we obtain a functional calculus on polynomial polyhedra containing the joint spectrum. Various properties of algebras of matrix-valued functions are studied, including domains of holomorphy.
Mathematics Subject Classification ID: 46H30
Mathematics Subject Classification ID: 32D05
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16Rxx
zbMATH DE Number: 3923447
zbMATH Keywords: unital Banach algebra
zbMATH Keywords: entire matrix-valued functions
zbMATH Keywords: functional calculus
zbMATH Keywords: Shilov-Wealbroeck-Arens-Calderon theorem
zbMATH Keywords: polynomial identity
zbMATH Keywords: Banach Azumaya algebra
zbMATH Keywords: algebras of matrix-valued functions
zbMATH Keywords: domains of holomorphy
zbMATH Keywords: PI-algebras
MaRDI profile type: Publication
full work available at URL: pjm.1986.125.127
OpenAlex ID: W1989317117
DOI: 10.2140/PJM.1986.125.127
description: scientific article; zbMATH DE number 3923447