A Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner Algebras

TL;DR

This paper introduces a new property called Condition (S) for C*-correspondences, proving a Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner algebras and exploring conditions for their simplicity.

Contribution

It defines Condition (S) as an analogue of graph Condition (L) and establishes its role in uniqueness theorems and simplicity criteria for Cuntz-Pimsner algebras.

Findings

Condition (S) is equivalent to Condition (L) for topological quivers with no sinks.

Theorem proving a Cuntz-Krieger Uniqueness result using Condition (S).

Examples illustrating the relationship between Condition (S) and Schweizer's nonperiodic condition.

Abstract

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 Q is a topological quiver with no sinks and X(Q) is the associated C*-correspondence, then X(Q) satisfies Condition (S) if and only if Q satisfies Condition(L). Finally, we consider several examples to compare and contrast Condition (S) with Schweizer's nonperiodic condition.