Topological freeness for $C^*$-correspondences

TL;DR

This paper establishes new conditions for the uniqueness of Cuntz-Krieger type algebras associated with $C^*$-correspondences, using topological and algebraic criteria, and applies these to improve existing theorems and criteria.

Contribution

It introduces general sufficient conditions for uniqueness theorems based on multivalued maps and graph structures, extending previous results to broader classes of $C^*$-algebras.

Findings

Topological freeness of associated graphs characterizes uniqueness.

New simplicity criteria for Cuntz-Pimsner algebras.

Enhanced and generalized uniqueness theorems for relative quiver $C^*$-algebras.

Abstract

We study conditions that ensure uniqueness theorems of Cuntz-Krieger type for relative Cuntz-Pimsner algebras $\mathcal{O}(J,X)$ associated to a $C^*$-correspondence $X$ over a $C^*$-algebra $A$. We give general sufficient conditions phrased in terms of a multivalued map $\widehat{X}$ acting on the spectrum $\widehat{A}$ 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 $C^*$-algebras of Muhly and Tomforde. We also discuss applications to crossed products by endomorphisms.