Convergence in the dual of a \(\sigma\)-complete \(C^*\)-algebra
type: Item
instance of: scholarly article
title: Convergence in the dual of a \(\sigma\)-complete \(C^*\)-algebra
zbMATH Open document ID: 1054.46039
author: J. D. Maitland Wright
author: James K. Brooks
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-10T00:00:00Z
review text: Let \(B\) be a monotone \(\sigma\)-complete C*-algebra, that is, a C*-algebra which possesses a unit element and so that each upper bounded, monotone increasing sequence of self-adjoint elements of \(B\) has a least upper bound. In particular, every von Neumann algebra is a monotone \(\sigma\)-complete C*-algebra. It is also known that the class of monotone \(\sigma\)-complete C*-algebras is strictly larger than the class of von Neumann algebras. In the paper under review, the authors provide sufficient conditions to assure that a sequence in the dual of a monotone \(\sigma\)-complete C*-algebra is weakly convergent. Concretely, they prove that if \((\mu_n)\) is a sequence in the dual of a monotone \(\sigma\)-complete C*-algebra \(B\) such that \((\mu_n (p))\) converges for each projection \(p\in B,\) then \((\mu_n)\) converges weakly to a bounded functional on \(B\). This interesting result extends the non-commutative version of the Dieudonné theorem proved by \textit{J. K. Brooks, K. Saitô} and \textit{J. D. M. Wright} [J. Math. Anal. Appl. 276, No.~1, 160-167 (2002; Zbl 1018.46031)], where the hypotheses are essentially stronger because, in the latter, it is assumed that the sequence \((\mu_n)\) converges on each range projection in \(B^{**}\). In a second section, the authors study the absolute continuity of bounded linear functionals on a monotone \(\sigma\)-complete C*-algebra.
reviewed by: Antonio M. Peralta
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 2085916
zbMATH Keywords: monotone \(\sigma\)-complete C*-algebras
zbMATH Keywords: absolute continuity of bounded linear functionals
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.02.004
OpenAlex ID: W2035110200
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DOI: 10.1016/J.JMAA.2004.02.004
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description: scientific article; zbMATH DE number 2085916