Topological freeness for $C^{\ast }$-correspondences

Abstract

We study conditions that ensure uniqueness theorems of Cuntz-Krieger type for relative Cuntz-Pimsner algebras $\mathcal{O}(J,X)$ associated to a $C^{\ast }$-correspondence $X$ over a $C^{\ast }$-algebra $A$. We give general sufficient conditions phrased in terms of a multivalued map $\hat{X}$ acting on the spectrum $\hat{A}$ of $A$. When $X(J)$ is of Type I we construct a directed graph dual to $X$ and prove a uniqueness theorem using this graph. When $X(J)$ is liminal, we show that topological freeness of this graph is equivalent to the uniqueness property for $\mathcal{O}(J,X)$, as well as to an algebraic condition, which we call $J$-acyclicity of $X$. As an application we improve the Fowler-Raeburn uniqueness theorem for the Toeplitz algebra ${\mathcal{T}}_X$. We give new simplicity criteria for ${\mathcal{O}}_X$. We generalize and enhance uniqueness results for relative quiver $C^{\ast }$-algebras of Muhly and Tomforde. We also discuss applications to crossed products by endomorphisms.

arXiv commentWe have updated the list of references, fixed some typos and made other minor improvements. This is the version that will be published

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Cited by in corpus (1)

• Stone duality and quasi-orbit spaces for generalised C*-inclusions