Pseudo-diagonals and uniqueness theorems

Abstract

We examine a certain type of abelian C*-subalgebra that allows one to give a unified treatment of two uniqueness theorems: for graph C*algebras and for certain reduced crossed products. This note is meant to complement the paper [NR], which provides one of the two main examples of our results, by offering a conceptual treatment of the Uniqueness Theorem for the C*-algebras associated with graphs satisfying condition (L) and its generalization found in [Sz]. As pointed out in many places in graph C*-algebra literature (see for instance [Re1]), for graphs satisfying condition (L), a natural abelian C*-subalgebra (which is referred to in [NR] as the “diagonal”) turns out to give substantial information about the ambient (graph) C*-algebra. A similar treatment was proposed by Kumijian in [Ku], where he introduced the notion of C*-diagonals. In this paper we explain how, by considerably weakening several hypotheses in Kumjian’s definitions (in particular by getting rid of normalizers entirely), one can still obtain several key results, the most significant one being an “abstract” uniqueness property (see Theorem 3.1 below). Another illustration of this general approach is given in the context of reduced crossed products of abelian C*-algebras by essentially free actions of discrete groups, where we recover another uniqueness result due to Archbold and Spielberg [AS]. Our treatment focuses on the unique state extension property, as discussed in [KS] and [An], by weakening the global requirement made in the so-called Extension Property discussed in [An]. 1. Notation and preliminaries Notation. For any C*-algebra A, we denote by S(A) its set of states and by P (A) the set of pure states. Given a C*-subalgebra B ⊂ A, using the Hahn-Banach Theorem, it follows that any φ ∈ S(B) has at least one extension to some ψ ∈ S(A), so the set Sφ(A) = {ψ ∈ S(A) : ψ ∣∣ B = φ} is non-empty, convex and weak*-compact. Remark that if φ ∈ P (B), then every extreme point in Sφ(A) is in fact an extreme point in S(A); thus the intersection Sφ(A) ∩ P (A) is non-empty. Received by the editors June 26, 2011 and, in revised form, March 8, 2012. 2010 Mathematics Subject Classification. Primary 46L10; Secondary 46L30.