Automorphisms of Unital $C^*$-Algebras Which are Strongly Morita Equivalent to Irrational Rotation $C^*$-Algebras

Abstract

Let $B$ be a unital $C^*$-algebra which is strongly Morita equivalent to an irrational rotation $C^*$-algebra. Then Rieffel showed that it is isomorphic to $A_\theta\otimes M_n$ where $A_\theta$ is an irrational rotation $C^*$-algebra and $M_n$ is the $n\times n$ matrix algebra over $C$. In the present paper we will show that for any automorphism $\alpha$ of $A_\theta\otimes M_n$ there are unitary elements $w\in A_\theta\otimes M_n$, $W\in M_n$ and an automorphism $\beta$ of $ A_\theta$ such that $\alpha=\mathrm{Ad}(w)\circ(\beta\otimes\mathrm{Ad}(W))$.