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 $A_\theta$.
Citation
Kazunori KODAKA. "Automorphisms, Diffeomorphisms and Strong Morita Equivalence of Irrational Rotation $C^*$-Algebras." Tokyo J. Math. 12 (2) 415 - 427, December 1989. https://doi.org/10.3836/tjm/1270133189
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