Rotation algebras and Exel trace formula

Abstract

We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $\|uv-vu\|<2$ such that $uvu^*v^*$ 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 $C^*$-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 $$ \|uv-e^{2\pi i\theta}vu\|<\dt\andeqn {1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta, $$ then there exists a pair of unitary matrices $\tilde{u}$ and $\tilde{v}$ such that $$ \tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\, \|u-\tilde{u}\|<\ep\andeqn \|v-\tilde{v}\|<\ep. $$ Furthermore, a generalization of this for all real $\theta$ is obtained for unitaries in unital infinite dimensional simple $C^*$-algebras of tracial rank zero.