Rotation algebras and Exel trace formula

Abstract

We found that if $u$ and $v$ are any two unitaries in a unital $C^{\ast }$-algebra with $\Vert uv-vu\Vert <2$ such that $uv{u}^{\ast }{v}^{\ast }$ commutes with $u$ and $v,$ then the \SCA\, $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of the rotation algebra $A_{\theta }$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^{\ast }$-algebra. Let $\theta \in (-1/2,1/2)$ be a rational number. We prove the following: For any $\ep>0,$ there exists $\dt>0$ satisfying the following: if $u$ and $v$ are two unitary matrices such that $$ \|uv-e^{2πiθ}vu\|<\dt\andeqn {1\over{2πi}}τ(\log(uvu^*v^*))=θ, $$ then there exists a pair of unitary matrices $\tilde{u}$ and $\tilde{v}$ such that $$ \tilde{u}\tilde{v}=e^{2πiθ} \tilde{v}\tilde{u},\,\, \|u-\tilde{u}\|<\ep\andeqn \|v-\tilde{v}\|<\ep. $$ Furthermore, a generalization of this for all real $\theta$ is obtained for unitaries in unital infinite dimensional simple $C^{\ast }$-algebras of tracial rank zero.