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 $thetain (-1/2, 1/2)$ 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 $$ |uv-e^{2pi itheta}vu|<dtandeqn {1over{2pi i}}tau(log(uvu^*v^*))=theta, $$ then there exists a pair of unitary matrices $tilde{u}$ and $tilde{v}$ such that $$ tilde{u}tilde{v}=e^{2pi itheta} tilde{v}tilde{u},,, |u-tilde{u}|<epandeqn |v-tilde{v}|<ep. $$ Furthermore, a generalization of this for all real $theta$ is obtained for unitaries in unital infinite dimensional simple $C^*$-algebras of tracial rank zero.
Download