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.
Regularity of Villadsen algebras and characters on their central sequence algebras
