Tensor products of \(C^*\)-algebras with the ideal property
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 \(B(H)\).
exact sequences of spatial tensor products
scientific article; zbMATH DE number 1542417
scientific article