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 $\alpha$: $A\to A$ such $\alpha\_{\*0}=\mathrm {id}$, $\alpha\_{1}=h$ and the action $\mathbb{Z}\_2$ generated by $\alpha$ has tracial Rokhlin property. Consequently, $C^(A,\mathbb{Z}\_2,\alpha)$ AH-algebra with no dimension growth, Thus our main result can considered as $\mathbb{Z}\_2$-action analogue Lin-Osaka theorem. As consequence, positive answer to lifting problem Blackadar also given for certain split case.