The isolated ideal of a correspondence associated with a topological quiver (Q851538)

A \(C^{\ast}\)-correspondence over a \(C^{\ast}\)-algebra \(A\) is a right Hilbert \(C^{\ast}\)-module \(\mathcal E\) over \(A\) together with a left action defined by a \(\ast\)-morphism \(\varphi_{A}\). In the first part of this paper, the author shows that for a given \(C^{\ast}\)-correspondence \(\mathcal E\), \(N= \operatorname{ker}\varphi_{A}\cap\langle \mathcal E,\mathcal E\rangle^{\bot}\) is an \(\mathcal E\)-invariant and \(\mathcal E\)-saturated ideal of \(A\), and considers the quotient correspondence \(\mathcal E_{N}\). Also he shows that the Cuntz--Pimsner algebra associated with \(\mathcal E_{N}\) can be identified with the relative Cuntz--Pimsner algebra associated with \(\mathcal E\). In the second part of the paper, he applies these results to the correspondences associated with a topological quiver. Let \(G=(X,E,r,s,\lambda)\) be a topological quiver with vertex space \(X\) and edge space \(E\) and let \(\mathcal E(G)\) be the \(C^{\ast}\)-correspondence over \(C_0 (X)\) associated with \(G\). The author shows that \(N=\{f \in C_0(X);f_{s(E)\cap r(E)}=0\}\), and the quotient \(C^{\ast}\)-correspondence \(\mathcal E(G)_{N}\) over \(A/N\) is the same with the \(C^{\ast}\)-correspondence \(\mathcal E(G^{N})\) over \(C_0(s(E)\cap r(E))\) associated with the restricted topological quiver \(G^{N}=(s(E)\cap r(E),E,r,s,\lambda)\). Finally, the author shows that \(G\) satisfies condition (L) (respectively, condition (K)) if and only if \(G^{N}\) satisfies condition (L) (respectively, condition (K)).