The splitting of operator algebras. II (Q795285)

Let \(\{A_{\alpha}:\alpha \in \Lambda \}\) be a family of \(C^*\)- algebras with direct sum A and let B be a \(C^*\)-subalgebra of A. Then B is said to split if B coincides with the direct sum of the family \(\{\pi_{\alpha}B:\alpha \in \Lambda \}\) where \(\pi_{\alpha}\) is the canonical projection of A onto \(A_{\alpha}\). If I denotes the closed two-sided ideal in A which is the restricted direct sum of the family \(\{\pi_{\alpha}B:\alpha \in \Lambda \}\) then the main result gives criteria for splitting in terms of \(B/_{B\cap I},\) and certain subsets of the primitive ideal space of B. The results can be alternatively phrased in terms of irreducible representations. In addition the corresponding result for \(W^*\)-algebras is proved and examples are given to show that in a sense the result is the best possible. [For part I see \textit{S.-K. Tsui} and the author, ibid. 84, 201-215 (1979; Zbl 0424.46042)].