Complements to various Stone-Weierstrass theorems for \(C^*\)-algebras and a theorem of Shultz
type: Item
instance of: scholarly article
title: Complements to various Stone-Weierstrass theorems for \(C^*\)-algebras and a theorem of Shultz
zbMATH Open document ID: 0778.46035
DOI: 10.1007/BF02099015
author: Lawrence G. Brown
published in: Communications in Mathematical Physics
publication date: 1992-09-26T00:00:00Z
review text: There are various versions (due to Kaplansky, Glimm, Longo, Popa and others) of the Stone-Weierstrass theorem for non-commutative \(C^*\)- algebras. The general setting of the non-commutative Stone-Weierstrass theorem is as follows: Let \(A\) be a \(C^*\)-algebra, \(S\) a set of states of \(A\) and \(B\) a \(C^*\)-subalgebra of \(A\) that separates \(S\cup\{0\}\). Then, under various assumptions on \(A\), \(B\) or \(S\) it can be concluded that \(B=A\). Typically, \(S\) is the set of pure states or the set of factorial states of \(A\). In this article complements of these theorems of the following form are proved: Let \(A\) be a \(C^*\)-algebra, \(S\) a set of states of \(A\) and \(B\) a \(C^*\)-subalgebra of \(A\). Let \(x\in A\) such that any two elements of \(S\cup\{0\}\) that coincide on \(B\), also agree on \(x\). Then, under various assumptions on \(A\), \(B\) or \(S\) it can be concluded that \(x\in B\).
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 54719
zbMATH Keywords: Stone-Weierstrass theorem for non-commutative \(C^*\)-algebras
reviewed by: Joachim Cuntz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02099015
OpenAlex ID: W1976547104
description: scientific article; zbMATH DE number 54719