Elementary operators on prime \(C^*\)-algebras. I
type: Item
instance of: scholarly article
title: Elementary operators on prime \(C^*\)-algebras. I
zbMATH Open document ID: 0648.46052
DOI: 10.1007/BF01442873
author: Martin Mathieu
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164550
review text: The aim of this paper is to study properties of the elementary operator \(Sx=\sum ^{n}_{j=1}a_ jxb_ j\) on a prime \(C^ *\)-algebra A, where \(a_ j,b_ j\in M(A)\), the multiplier algebra of A. In doing this, the primeness of A (in the sense of non-commutative ring theory) is considered to be a fundamental device and exploited in several ways. It is shown that the extended centroid of a prime \(C^ *\)-algebra A is trivial and thus the canonical epimorphism between \(M(A)\otimes M(A)^{op}(M(A)^{op}\) denoting the opposite algebra of M(A)) and El(A), the algebra of all elementary operators on A, is an isomorphism. This extends results of \textit{C. K. Fong} and \textit{A. R. Sourour} [Can. J. Math. 31, 845-857 (1979; Zbl 0408.47028)] for the algebra B(H) and the Calkin algebra C(H); in particular, their use of Voiculescu's non- commutative Weyl-von Neumann theorem is replaced by much more elementary techniques. Applications are characterizations of hermitian-preserving and of completely positive elementary operators on prime \(C^ *\)- algebras as well as a description of their extreme points in the corresponding convex cone [extending results by \textit{M.-D. Choi}, Linear Alg. Appl. 10, 285-290 (1975; Zbl 0327.15018) and others]. Another section of the paper deals with the spectral theory of elementary operators where a complete description is given for the various parts \(\sigma _{\ell}(S)\), AP\(\sigma\) (S), etc. and for \(\sigma\) (S) in terms of the joint spectrum in the sense of R. E. Harte; ultraweak denseness of the range of S (in the case of a \(W^ *\)-algebra) is also discussed. These results generalize many of the results on the spectra of S known so far for B(H), C(H), or other special cases only, mainly established by Fialkow in a series of papers. Using the primeness, however, opens a much more direct and simplified approach. The second part of the paper which will appear in the Glasgow Math. J. 30 (1988) contains characterizations of compact and weakly …
reviewed by: Martin Mathieu
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 4058208
zbMATH Keywords: completely positive elementary operators on prime \(C^*\)-algebras
zbMATH Keywords: spectral theory of elementary operators
zbMATH Keywords: ultraweak denseness
zbMATH Keywords: weakly compact elementary operators
zbMATH Keywords: joint spectrum
zbMATH Keywords: extended centroid
MaRDI profile type: Publication
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OpenAlex ID: W2012155395
description: scientific article; zbMATH DE number 4058208