1 results
On simplicity of intermediate
$C^{\ast }$-algebras
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 06 June 2019, pp. 3181-3187
- Print publication:
- December 2020
-
- Article
- Export citation
-
We prove simplicity of all intermediate
$C^{\ast }$-algebras 
$C_{r}^{\ast }(\unicode[STIX]{x1D6E4})\subseteq {\mathcal{B}}\subseteq \unicode[STIX]{x1D6E4}\ltimes _{r}C(X)$ in the case of minimal actions of 
$C^{\ast }$-simple groups 
$\unicode[STIX]{x1D6E4}$ on compact spaces 
$X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar [Stationary 
$C^{\ast }$-dynamical systems. Preprint, 2017, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product 
$\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ for any action 
$\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{A}}$ of a 
$C^{\ast }$-simple group 
$\unicode[STIX]{x1D6E4}$ on a unital 
$C^{\ast }$-algebra 
${\mathcal{A}}$, and use it to prove a one-to-one correspondence between stationary states on 
${\mathcal{A}}$ and those on 
$\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$.
