# Smale space C*-algebras have nonzero projections

## TL;DR

This paper proves that stable and unstable C*-algebras from mixing Smale spaces always contain nonzero projections, impacting their structural understanding and aligning with the Elliott classification program.

## Contribution

It establishes the existence of nonzero projections in these algebras, a key structural property previously unresolved.

## Key findings

- Stable and unstable C*-algebras from mixing Smale spaces contain nonzero projections.
- Homoclinic, stable, and unstable algebras have real rank zero.
- Results support the Elliott classification program for these algebras.

## 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.