Stability of rotation relations in $C^{\ast }$-algebras

Abstract

Let $\Theta =({\theta }_{j,k}{)}_{3\times 3}$ be a non-degenerate real skew-symmetric $3\times 3$ matrix, where ${\theta }_{j,k}\in [0,1).$ For any $\epsilon >0$, we prove that there exists $\delta >0$ satisfying the following: if $v_1,v_2,v_3$ are three unitaries in any unital simple separable $C^{\ast }$-algebra $A$ with tracial rank at most one, such that $$\|v_kv_j-e^{2πi θ_{j,k}}v_jv_k\|<δ\,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2πi}τ(\log_θ(v_kv_jv_k^*v_j^*))=θ_{j,k}$$ for all $\tau \in T(A)$ and $j,k=1,2,3,$ where ${\log }_{\theta }$ is a continuous branch of logarithm for some real number $\theta \in [0,1)$, then there exists a triple of unitaries ${\tilde{v}}_1,{\tilde{v}}_2,{\tilde{v}}_3\in A$ such that $$\tilde{v}_k\tilde{v}_j=e^{2πiθ_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.$$ The same conclusion holds if $\Theta$ is rational or non-degenerate and $A$ is a nuclear purely infinite simple $C^{\ast }$-algebra (where the trace condition is vacuous). If $\Theta$ is degenerate and $A$ has tracial rank at most one or is nuclear purely infinite simple, we provide some additional injectivity condition to get the above conclusion.

31 pages