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

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  21 May 2020

Jiajie Hua  and
Qingyun Wang
Jiajie Hua*
Affiliation:
College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314000, China
Qingyun Wang
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA e-mail: qingyunw@uoregon.edu
*
e-mail: huajiajie2006@hotmail.com
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Abstract

Let $\Theta =(\theta _{j,k})_{3\times 3}$ be a nondegenerate 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

$$\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}\end{align*}$$
for all $\tau \in T(A)$ and $j,k=1,2,3,$ where $\log _{\theta }$ is a continuous branch of logarithm (see Definition 4.13) 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
$$\begin{align*}\tilde{v}_k\tilde{v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.\end{align*}$$

The same conclusion holds if $\Theta $ is rational or nondegenerate 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 conditions to get the above conclusion.

Keywords

rotation relationsC*-algebrasstabilitynoncommutative toriunitary

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L35: Classifications of $C^*$-algebras

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Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 1171 - 1203
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© Canadian Mathematical Society 2020

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