Topological quivers as multiplicity free relations

Abstract

For a $C^*$-correspondence $\mathcal{E}$ over a $C^*$-algebra $A$ the restricted correspondence $\mathcal{R}(\mathcal{E})$ over the ideal $I=\overline{\langle\mathcal{E},\mathcal{E}\rangle}$ of $A$ is introduced. The Cuntz-Pimsner algebra $\mathcal{O}_{\mathcal{R}(\mathcal{E})}$ is the unaugmented $C^*$-algebra associated with $\mathcal{E}$. For a topological quiver $G$ an associated multiplicity free quiver, or topological relation, $G^{1}$ is introduced. The Cuntz-Pimsner algebra $\mathcal{O}_{\mathcal{R}(\mathcal{E})}$ of the correspondence $\mathcal{E}$ associated with $G$ is contained in the algebra $\mathcal{O}_{\mathcal{R}(\mathcal{E}^{1})}$ for the correspondence $\mathcal{E}^{1}$ associated with $G^{1}$ if the source map for the quiver is proper on an appropriate codomain. The unaugmented Cuntz-Pimsner algebras for $G$ and $G^{1}$ are isomorphic if the left action for the correspondence $\mathcal{R}(\mathcal{E})$ is by compact adjointable maps and if the kernel for the left action is complemented in $I$. There are counter examples if either condition fails.