Automorphisms, Diffeomorphisms and Strong Morita Equivalence of Irrational Rotation $C^*$-Algebras

Abstract

Let $A_\theta$ be an irrational rotation $C^*$-algebra by $\theta$ and $\mathrm{Aut}(A_\theta)$ (resp. $\mathrm{Diff}(A_\theta)$) be the group of all automorphisms (resp. diffeomorphisms) of $A_\theta$. Let $\mathrm{Int}(A_\theta)$ be the normal subgroup of $\mathrm{Aut}(A_\theta)$ of inner automorphisms of $A_\theta$ and let $\mathrm{Int}^\infty(A_\theta)=\mathrm{Int}(A_\theta)\cap\mathrm{Diff}(A_\theta)$. Let $A_\eta$ be an irrational rotation $C^*$-algebra by $\eta$ which is strongly Morita equivalent to $A_\theta$. In the present paper we will show that $\mathrm{Aut}(A_\theta)/\mathrm{Int}(A_\theta)$ (resp. $\mathrm{Diff}(A_\theta)/\mathrm{Int}^\infty(A_\theta)$) is isomorphic to $\mathrm{Aut}(A_\eta)/\mathrm{Int}(A_\eta)$ (resp. $\mathrm{Diff}(A_\eta)/\mathrm{Int}^\infty(A_\eta)$) and that if $A_\eta$ has a diffeomorphism of non Elliott type, so does $A_\theta$.