Rotation algebras and Exel trace formula

TL;DR

This paper investigates the structure of certain unitaries in unital C*-algebras, establishing conditions under which they generate algebras isomorphic to rotation algebras and proving the Exel trace formula in this context.

Contribution

It demonstrates that under specific commutation and trace conditions, generated C*-algebras are quotients of rotation algebras, and extends the Exel trace formula to all unital C*-algebras.

Findings

Generated algebras are quotients of rotation algebras under certain conditions.

Exel trace formula holds in any unital C*-algebra.

Approximation of unitaries by rotation algebra unitaries in finite dimensions.

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.