Representations of the Banach algebra \(\ell ^ 1(S)\)
type: Item
instance of: scholarly article
title: Representations of the Banach algebra \(\ell ^ 1(S)\)
zbMATH Open document ID: 0633.46057
DOI: 10.1007/BF02573128
author: R. G. McLean
author: Hans J. Kummer
published in: Semigroup Forum
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134919
review text: Let S be a monoid equipped with an involution *. Then the Banach space \(\ell^ 1(S)\) becomes a Banach *-algebra when multiplication is defined by convolution. This algebra is *-semisimple if and only if the bounded *-representation of S, on Hilbert spaces, separate points of S. We give a new proof of this result using properties of von Neumann algebras. It is also shown that when S is the free monoid on a countable set of generators then \(\ell^ 1(S)\) has a separating sequence of finite dimensional irreducible *-representations.
Mathematics Subject Classification ID: 46K10
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 4030264
zbMATH Keywords: monoid equipped with an involution *
zbMATH Keywords: Banach *-algebra
zbMATH Keywords: convolution
zbMATH Keywords: *- semisimple
zbMATH Keywords: bounded *-representation
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: free monoid on a countable set of generators
zbMATH Keywords: separating sequence of finite dimensional irreducible *-representations
MaRDI profile type: Publication
cites work: Representations of Foundation Semigroups and their Algebras
cites work: The Banach algebra \(\ell^1(S)\)
cites work: The l 1 -Algebra of a Commutative Semigroup
description: scientific article; zbMATH DE number 4030264