Symmetries of simple $A\mathbb{T}$-algebras

Abstract

Let $A$ be a unital simple $A\mathbb{T}$-algebra of real rank zero. Given an order two automorphism $h:K_1(A)\to K_1(A)$, we show that there is an order two automorphism $\alpha$: $A\to A$ such that ${\alpha }_{\ast 0}=\mathrm{id}$, ${\alpha }_{\ast 1}=h$ and the action of ${\mathbb{Z}}_2$ generated by $\alpha$ has the tracial Rokhlin property. Consequently, $C^{\ast }(A,{\mathbb{Z}}_2,\alpha )$ is a simple unital AH-algebra with no dimension growth, and with tracial rank zero. Thus our main result can be considered as the ${\mathbb{Z}}_2$-action analogue of the Lin-Osaka theorem. As a consequence, a positive answer to a lifting problem of Blackadar is also given for certain split case.