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A Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner Algebras

Menevşe Eryüzlü, Mark Tomforde

TL;DR

This work introduces Condition (S), a correspondence-internal analogue of graph Condition (L), and uses it to prove a Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner algebras when the left action is injective. It then derives sufficient conditions for simplicity of 𝒪X, first in the injective-left-action case via (S) and absence of X-invariant ideals, and then in full generality by dilating to a larger correspondence 𝐗̄ over 𝐀̄ and applying Kwaśniewski–Meyer results. The paper also shows that for topological quivers, (S) is equivalent to Condition (L), reinforcing (S) as the natural analogue of (L) beyond graphs. Through an examination of Schweizer's nonperiodic condition, it argues that (S) provides a more robust, generally applicable framework for CK-Un uniqueness and simplicity in Cuntz-Pimsner algebras, while outlining open questions about necessity and noninjective extensions.

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.

A Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner Algebras

TL;DR

This work introduces Condition (S), a correspondence-internal analogue of graph Condition (L), and uses it to prove a Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner algebras when the left action is injective. It then derives sufficient conditions for simplicity of 𝒪X, first in the injective-left-action case via (S) and absence of X-invariant ideals, and then in full generality by dilating to a larger correspondence 𝐗̄ over 𝐀̄ and applying Kwaśniewski–Meyer results. The paper also shows that for topological quivers, (S) is equivalent to Condition (L), reinforcing (S) as the natural analogue of (L) beyond graphs. Through an examination of Schweizer's nonperiodic condition, it argues that (S) provides a more robust, generally applicable framework for CK-Un uniqueness and simplicity in Cuntz-Pimsner algebras, while outlining open questions about necessity and noninjective extensions.

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.
Paper Structure (8 sections, 17 theorems, 75 equations)

This paper contains 8 sections, 17 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner algebras
  4. Simplicity of Cuntz-Pimsner algebras
  5. Simplicity when the left action is not necessarily injective
  6. Topological Quivers and Condition (L)
  7. An Examination of Schweizer's Simplicity Result
  8. Concluding Remarks

Key Result

Lemma 3.3

Let $X$ be a $C^*$-correspondence over $A$ and let $\zeta \in X^{\otimes m}$ be a nonreturning vector. If $(\psi, \pi) : (X,A) \to B$ is any covariant representation, then for any $n \in \{ 1, \ldots, m-1 \}$ and $\xi \in X^{\otimes n}$, we have

Theorems & Definitions (71)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Definition 3.1
  • ...and 61 more