Approximately Inner $*$-Derivations of Irrational Rotation $C^*$-Algebras

Abstract

Let $\theta$ be an irrational number and $A_\theta$ be the corresponding irrational rotation $C^*$-algebra. For any $k\in N\cup\{\infty\}$ let $A_\theta^k$ be the dense $*$-subalgebra of $k$-times continuously differentiable elements in $A_\theta$ with respect to the canonical action of the two dimensional torus and let $A_\theta^0=A_\theta$. In the present paper we will show that there is an approximately inner $*$-derivation of $A_\theta^\infty$ to $A_\theta^\infty$ which is not inner if and only if $\theta$ is a non-generic irrational number.