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

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.