A Cuntz-Krieger uniqueness theorem for Cuntz-Pimsner algebras
DOI:
https://doi.org/10.7146/math.scand.a-156724Abstract
We introduce a property of $C^*$-correspondences, which we call Condition (S), to serve as an analogue of Condition (L) of graphs. We use Condition (S) to prove a Cuntz-Krieger uniqueness theorem for Cuntz-Pimsner algebras and obtain sufficient conditions for simplicity of Cuntz-Pimsner algebras. We also prove that if $\mathcal {Q}$ is a topological quiver with no sinks and $X(\mathcal {Q})$ is the associated $C^*$-correspondence, then $X(\mathcal {Q})$ satisfies Condition (S) if and only if $\mathcal {Q}$ satisfies Condition (L). Finally, we consider several examples to compare and contrast Condition (S) with Schweizer's nonperiodic condition.
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