Article contents
Stability of rotation relations in
$C^*$-algebras
Published online by Cambridge University Press: 21 May 2020
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
$\tau \in T(A)$
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
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.
MSC classification
- Type
- Article
- Information
- Copyright
- © Canadian Mathematical Society 2020
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline17.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline18.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline19.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline20.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline21.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline22.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline23.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline24.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline25.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline26.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline27.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline28.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline29.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline30.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline31.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline32.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210824001229726-0165:S0008414X20000371:S0008414X20000371_inline33.png?pub-status=live)
- 4
- Cited by