A class of C*-algebras that are prime but not primitive

TL;DR

This paper characterizes when graph C*-algebras are primitive and provides a method to construct large classes of prime but not primitive C*-algebras, expanding understanding of their structure.

Contribution

It offers necessary and sufficient conditions for primitivity in graph C*-algebras and introduces a systematic way to generate prime but nonprimitive C*-algebras.

Findings

Characterization of primitive graph C*-algebras

Method to produce large classes of prime but not primitive C*-algebras

Comparison with Leavitt path algebra results

Abstract

We establish necessary and sufficient conditions on a (not necessarily countable) graph E for the graph C*-algebra C*(E) to be primitive. Along with a known characterization of the graphs E for which C*(E) is prime, our main result provides us with a systematic method for easily producing large classes of (necessarily nonseparable) C*-algebras that are prime but not primitive. We also compare and contrast our results with similar results for Leavitt path algebras.