Topological freeness for C*-correspondences

Abstract

We study conditions that ensure uniqueness theorems of Cuntz–Krieger type for relative Cuntz–Pimsner algebras associated to a ⁎-correspondence X over a ⁎-algebra A. We give general sufficient conditions phrased in terms of a multivalued map acting on the spectrum of A. When is of Type I we construct a directed graph dual to X and prove a uniqueness theorem using this graph. When is liminal, we show that topological freeness of this graph is equivalent to the uniqueness property for , 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 . We give new simplicity criteria for . We generalize and enhance uniqueness results for relative quiver ⁎-algebras of Muhly and Tomforde. We also discuss applications to crossed products by endomorphisms.