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.

设 $\Theta =(\theta _{j,k})_{3\times 3}$ 是一个非退化实斜对称 $3\times 3$ 矩阵，其中 $\theta _{j,k}\in [ 0,1).$ 对于任何 $\varepsilon>0$ ，我们证明存在 $\delta>0$ 满足以下条件：如果 $v_1,v_2,v_3$ 是任何单位简单可分离 $C^* $ -algebra A与tracial rank 至多为1，使得 $$\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\, \,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{ j,k}\end{align*}$$ 对于所有 $\tau \in T(A)$ 和 $j,k=1,2,3,$ 其中 $\log _{\theta }$ 是某个实数 $\theta \in [0, 1)$ 的对数的连续分支（见定义 4.13） ，则存在三元组 $\tilde {v}_1,\tilde {v}_2,\tilde {v}_3\in A$ 使得 $$\begin{align*}\tilde{v}_k\tilde{ v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\, \,\|\波浪号{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.\end{align*}$$

如果 $\Theta $ 是有理的或非退化的，并且A是核的纯无限简单 $C^*$ - 代数（其中迹条件是空的），则相同的结论成立 。

如果 $\Theta $ 是退化的并且A 的迹阶至多为 1 或核纯无限简单，我们提供一些额外的注入条件来得到上述结论。