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_{*0}=\mathrm {id} , \alpha_{*1}=h and the action of \mathbb{Z}_2 generated by \alpha has the tracial Rokhlin property. Consequently, C^*(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.