HAAGERUP PROPERTY FOR -CROSSED PRODUCTS

Abstract

Let $\unicode[STIX]{x1D6E4}$ be a countable discrete group that acts on a unital $C^{\ast }$-algebra $A$ through an action $\unicode[STIX]{x1D6FC}$. If $A$ has a faithful $\unicode[STIX]{x1D6FC}$-invariant tracial state $\unicode[STIX]{x1D70F}$, then $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}\circ {\mathcal{E}}$ is a faithful tracial state of $A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}$ where ${\mathcal{E}}:A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}\rightarrow A$ is the canonical faithful conditional expectation. We show that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property if and only if both $(A,\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6E4}$ have the Haagerup property. As a consequence, suppose that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property where $\unicode[STIX]{x1D6E4}$ has property $T$ and $A$ has strong property $T$. Then $\unicode[STIX]{x1D6E4}$ is finite and $A$ is residually finite-dimensional.