Unbounded quasitraces, stable finiteness and pure infiniteness

Abstract

We prove that if $ A$ is a $ \sigma$ -unital exact $ C^*$ -algebra of real rank zero, then every state on $ K_0(A)$ is induced by a 2-quasitrace on $ A$ . This yields a generalisation of Rainone'\''s work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to $ k$ -graph algebras associated to row-finite $ k$ -graphs with no sources. We show that for any $ k$ -graph whose $ C^*$ -algebra is unital and simple, either every twisted $ C^*$ -algebra associated to that $ k$ -graph is stably finite, or every twisted $ C^*$ -algebra associated to that $ k$ -graph is purely infinite. Finally we provide sufficient and necessary conditions for a unital simple $ k$ -graph algebra to be purely infinite in terms of the underlying $ k$ -graph.