Automorphisms of Tensor Products of Irrational Rotation $C^*$-Algebras and the $C^*$-Algebra of Compact Operators

Abstract

Let $\theta$ be an irrational number and $A_\theta$ be the corresponding irrational rotation $C^*$-algebra. Let $\mathbf{K}$ be the $C^*$-algebra of all compact operators on a countably infinite dimensional Hilbert space $H$. Let $\alpha$ be an automorphism of $A_{\theta}\otimes\mathbf{K}$ with $\alpha_*=\mathrm{id}$ on $K_0(A_{\theta}\otimes\mathbf{K})$. If the set of invertible elements in $A_\theta$ is dense in $A_\theta$ or $\alpha$ preserves the canonical dense $*$-subalgebra $F^{\infty}(A_{\theta}\otimes\mathbf{K})$ of $A_{\theta}\otimes\mathbf{K}$, then there are an automorphism $\beta$ of $A_\theta$ and unitary elements $w$ in the double centralizer $M(A_{\theta}\otimes\mathbf{K})$ of $A_{\theta}\otimes\mathbf{K}$ and $W$ in $\mathbf{B}(H)$ such that $\alpha=\mathrm{Ad}(w)\circ(\beta\otimes\mathrm{Ad}(W))$.