Tensor products of \(C^*\)-algebras with the ideal property
type: Item
instance of: scholarly article
title: Tensor products of \(C^*\)-algebras with the ideal property
zbMATH Open document ID: 0989.46032
author: Cornel Pasnicu
author: Mikael Rørdam
published in: Journal of Functional Analysis
publication date: 2001-03-19T00:00:00Z
review text: Denote the spatial tensor product of \(C^*\)-algebras \(A\) and \(B\) by \(A \otimes B\). For a state \(\varphi\) of \(A\) the right slice-map \(R_{\varphi} : A \otimes B \to B\) has been defined using \(R_{\varphi}(a \otimes b) = \varphi(a).b\) [\textit{J. Tomiyama}, Tohoku Math. J., II. Ser. 19, 213-226 (1967; Zbl 0166.11401)]. A \(C^*\)-algebra is said to be (tensor product) exact if for \(R_{\varphi}(x) \in D\), \(x \in A \otimes B\) implies \(x \in A \otimes D\), where \(\varphi\) denotes an arbitrary state and \(D\) an arbitrary closed two-sided ideal of \(B\) [cf. \textit{E. Kirchberg}, J. Oper. Theory 10, 3-8 (1983; Zbl 0543.46035)]. A \(C^*\)-algebra is said to have the ideal property if every closed two-sided ideal is generated by its projections. The paper is concerned as to when the ideal property is conserved under spatial tensor products. The authors say this is easily shown to be true if at least one of the \(C^*\)-algebras is exact by using a theorem of \textit{E. Kirchberg} [``The classification of purely infinite \(C^*\)-algebras using Kasparov's theory''. Fields Institute Comunications, in preparation]. The authors show that the ideal property is not necesssarily conserved if neither of the algebras is exact. Let \(B(H)\) denote the \(C^*\)-algebra of all bounded linear operators on a separable infinite dimensional Hilbert space \(H\). This is not exact nor does it have the ideal property. Using the non-exact \(C^*\)-algebra constructed by \textit{M. Dadarlat} [Am. J. Math 122, No. 3, 581-597 (2000; Zbl 0964.46034)], denoted here by \(C\), the authors show that \(B(H) \otimes C\) does not have the ideal property. The authors go further to show that the ideal property need not hold if one replaces the non-separable \(B(H)\) by a separable sub-\(C^*\)-algebra of \(B(H)\).
reviewed by: Aubrey Wulfsohn
Mathematics Subject Classification ID: 46L06
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1542417
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: state
zbMATH Keywords: exact sequences of spatial tensor products
zbMATH Keywords: projection
zbMATH Keywords: slice-map
zbMATH Keywords: ideal property
MaRDI profile type: Publication
full work available at URL: jfan.2000.3630
OpenAlex ID: W1992526876
cites work: Q4176022
cites work: The full C*-algebra of the free group on two generators
cites work: Nonnuclear subalgebras of AF algebras
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DOI: 10.1006/JFAN.2000.3630
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description: scientific article; zbMATH DE number 1542417