Pure states and p-commutative Banach *-algebras
type: Item
instance of: scholarly article
title: Pure states and p-commutative Banach *-algebras
zbMATH Open document ID: 0656.46040
DOI: 10.1007/BF01153581
author: Wayne Tiller
author: Robert S. Doran
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155301
review text: Banach *-algebras with bounded approximate identities are considered. If every element x of a Banach *-algebra A from this class satisfies the condition \(\sigma_ A(x)=\{f(x):\) \(f\in P(A)\}\), where \(\sigma_ A(x)\) is the spectrum of x in A and P(A) is the set of all pure states of A, then A is P-commutative (Theorem 1). An isomorphism theorem is proved, which involves P-commutative Banach *-algebras with bounded approximate identities, and pure states. It is shown that the norm of every positive functional f on a symmetric, P-commutative Banach *-algebra with an approximate identity \(\{e_{\alpha}\}\) bounded by one, is computed by the rule \(\| f\| =\lim_{\alpha}f(e_{\alpha})\) (Theorem 3).
Mathematics Subject Classification ID: 46K05
Mathematics Subject Classification ID: 46L30
zbMATH DE Number: 4071674
zbMATH Keywords: Banach *-algebras with bounded approximate identities
zbMATH Keywords: spectrum
zbMATH Keywords: pure states
zbMATH Keywords: P-commutative Banach *-algebras
zbMATH Keywords: positive functional
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
cites work: Q3215824
cites work: Q3730456
cites work: P-commutativity of the Banach algebra \(L_ 1^{(G)}\)
cites work: Extensions of Pure Positive Functionals on Banach ∗ -Algebras
cites work: Approximate identities and factorization in Banach modules
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cites work: P-Commutative Banach ∗ -Algebras
cites work: A characterization of multiplicative linear functionals in complex Banach algebras
description: scientific article; zbMATH DE number 4071674