Equivalent conditions for the general Stone-Weierstrass problem
type: Item
instance of: scholarly article
title: Equivalent conditions for the general Stone-Weierstrass problem
zbMATH Open document ID: 0579.46040
DOI: 10.1007/BF01626397
author: Ichiro Fujimoto
author: S.-E. Takahasi
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155096
review text: The setting of the Stone-Weierstraß\ problem is generalized to the cone \(CP(A,H)\) of all completely positive linear maps of a unital \(C^*\)-algebras \(A\) into the \(C^*\)-algebra \(B(H)\) of all bounded linear operators on a Hilbert space \(H\). It is shown that some conditions are equivalent to that of Stone-Weierstraß\ conjecture, i.e., a \(C^*\)-subalgebra \(B\) of \(A\) which contains the identity of \(A\) separates the pure states of \(A\). One of such conditions is that \(B\) separates the pure completely positive linear maps \(P_ A(H)\) of \(A\) into \(B(H)\). As a consequence, a new generalized spectrum ``\(\oplus_{\dim H\leq \alpha_ A}P_ A(H)\)'' is introduced, where \(\alpha_ A=\sup_{\pi \in \hat A}\dim \pi\) and \(\hat A\) is the set of all unitary equivalence classes of the set of irreducible representations. This generalized spectrum contains pure states and irreducible representations as proper subsets.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L30
zbMATH DE Number: 3927669
zbMATH Keywords: Stone-Weierstrass problem
zbMATH Keywords: completely positive linear maps of a unital \(C^*\)-algebras
zbMATH Keywords: pure states
zbMATH Keywords: generalized spectrum
zbMATH Keywords: irreducible representations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3927669