Smale space $C^*$-algebras have nonzero projections

Abstract

The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.