A REMARK ON THE TRACIAL ROKHLIN PROPERTY

Abstract

To explore the difficulties of classifying actions with the tracial Rokhlin property using K-theoretic data, we construct two $\mathbb{Z}_{2}$ actions $\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}$ on a simple unital AF algebra $A$ such that $\unicode[STIX]{x1D6FC}_{1}$ has the tracial Rokhlin property and $\unicode[STIX]{x1D6FC}_{2}$ does not, while $(\unicode[STIX]{x1D6FC}_{1})_{\ast }=(\unicode[STIX]{x1D6FC}_{2})_{\ast }$, where $(\unicode[STIX]{x1D6FC}_{i})_{\ast }$ is the induced map by $\unicode[STIX]{x1D6FC}_{i}$ acting on $K_{0}(A)$ for $i=1,2$.