Stability of rotation relations in $C^*$-algebras

TL;DR

This paper proves the stability of approximate rotation relations in certain $C^*$-algebras, showing that near-commuting unitaries can be approximated by exactly commuting unitaries with prescribed relations.

Contribution

It establishes a stability result for rotation relations in unital simple separable $C^*$-algebras with tracial rank at most one and nuclear purely infinite simple $C^*$-algebras, extending previous work.

Findings

Approximate rotation relations can be stabilized to exact relations under certain conditions.

The result applies to both rational and non-degenerate skew-symmetric matrices.

Additional conditions are provided for degenerate matrices in specific $C^*$-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 $\varepsilon>0$, 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 $C^*$-algebra $A$ with tracial rank at most one, such that $$\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}$$ 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 $$\tilde{v}_k\tilde{v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.$$ The same conclusion holds if $\Theta$ is rational or non-degenerate and $A$ is a nuclear purely infinite simple $C^*$-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.