The extend centroid of C*-algebras
type: Item
instance of: scholarly article
title: The extend centroid of C*-algebras
zbMATH Open document ID: 0669.46026
DOI: 10.1007/BF01189582
author: Pere Ara
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let R be a semiprime ring. Then the extended centroid of R, C(R), can be viewed as the center of the symmetric ring of quotients of R, denoted by S(R). If A is a \(C^*\)-algebra then the involution of A extends in a natural way to a positive definite involution on S(A) and, consequently, a partial order is defined in S(A) by taking as a positive cone the set of elements of the form \(\sum^{n}_{i=1}x^*_ ix_ i\), where \(x_ i\in S(A)\). It is described the bounded subring of S(A) with respect to this partial order in terms of the algebras of multipliers of the closed essential ideals of A. The author uses this description, as well as the Dauns-Hofmann Theorem to obtain a characterization of C(A) in terms of the primitive spectrum of A. By using this characterization it is proved that the extended centroid of a prime \(C^*\)-algebra is trivial and that, if the primitive spectrum of a \(C^*\)-algebra is Hausdorff, then its extended centroid is in fact the maximal quotient ring of the center of the algebra of multipliers.
reviewed by: Pere Ara
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 4096162
zbMATH Keywords: semiprime ring
zbMATH Keywords: C*-algebra
zbMATH Keywords: positive definite involution
zbMATH Keywords: algebras of multipliers
zbMATH Keywords: Dauns-Hofmann theorem
zbMATH Keywords: extended centroid of a prime C*- algebra
zbMATH Keywords: primitive spectrum of a C*-algebra
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01189582
OpenAlex ID: W2039566041
description: scientific article; zbMATH DE number 4096162