On a lifting question of Blackadar

Abstract

Let $A$ be the AF algebra whose scaled ordered group $K_0\left(A\right)$ is $(G\oplus H,(G_{+}\setminus \left\{0\right\})\oplus H\cup \{(0,0)\},\tilde{g}\oplus 0)$, where $(G,G_{+},\tilde{g})$ is the scaled ordered group $K_0\left(B\right)$ of a unital simple AF algebra $B$, and $H$ is a countable torsion-free Abelian group. Let $\sigma$ be an order 2 scaled ordered automorphism of $K_0\left(A\right)$, defined by $\sigma (g,h)=(g,-h)$, where $(g,h)\in G\oplus H$. We show that there is an order $2$ automorphism $\alpha$ of $A$ such that ${\alpha }_{\ast }=\sigma$. This gives a partial answer to a lifting question posed by Blackadar. Incidentally, the lift $\alpha$ we construct has the tracial Rokhlin property. Consequently, the crossed product $C^{\ast }({\mathbb{Z}}_2,A,\alpha )$ is a unital simple AH algebra with no dimension growth.