Rotation Algebras and the Exel Trace Formula

Abstract

We show that if $u$ and $v$ are any two unitaries in a unital ${{C}^{*}}$–algebra such that $\left\| uv\,-\,vu \right\|\,<\,2$ and $uv{{u}^{*}}{{v}^{*}}$ commutes with $u$ and $v$, then the ${{C}^{*}}$–subalgebra ${{A}_{u,v}}$ generated by $u$ and $v$ is isomorphic to a quotient of some rotation algebra ${{A}_{\theta }}$, provided that ${{A}_{u,v}}$ has a unique tracial state. We also show that the Exel trace formula holds in any unital ${{C}^{*}}$–algebra. Let $\theta \,\in \,\left( -1/2,\,1/2 \right)$ be a real number. For any $\in \,>\,0$, we prove that there exists $\delta \,>\,0$ satisfying the following: if $u$ and $v$ are two unitaries in any unital simple ${{C}^{*}}$–algebra $A$ with tracial rank zero such that

$$\left\| uv\,-\,{{e}^{2\pi i\theta }}vu \right\|\,<\,\delta \,\,\,\text{and}\,\,\frac{1}{2\pi i}\tau \left( \log \left( uv{{u}^{*}}{{v}^{*}} \right) \right)\,=\,\theta ,$$

for all tracial states $\tau$ of $A$, then there exists a pair of unitaries $\widetilde{u}$ and $\widetilde{v}$ in $A$ such that

$$\widetilde{u}\widetilde{v}\,=\,{{e}^{2\pi i\theta }}\widetilde{v}\widetilde{u},\,\,\,\,\,\,\,\left\| u\,-\,\widetilde{u} \right\|\,<\,\in \,\,\,\text{and}\,\,\left\| v\,-\,\widetilde{v} \right\|\,<\,\in.$$