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On simplicity of intermediate $C^{\ast }$-algebras

Part of: Ergodic theory Selfadjoint operator algebras

Published online by Cambridge University Press:  06 June 2019

TATTWAMASI AMRUTAM  and
MEHRDAD KALANTAR
TATTWAMASI AMRUTAM
Affiliation:
Department of Mathematics, 3551 Cullen Blvd, Room 641, Philip Guthrie Hoffman Hall, Houston, TX77204, USA email tamrutam@math.uh.edu, kalantar@math.uh.edu
MEHRDAD KALANTAR
Affiliation:
Department of Mathematics, 3551 Cullen Blvd, Room 641, Philip Guthrie Hoffman Hall, Houston, TX77204, USA email tamrutam@math.uh.edu, kalantar@math.uh.edu
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Abstract

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 $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$.

Keywords

group actionsoperator algebrascrossed product C∗ -algebrasstationary group actionsC∗ -simplicity

MSC classification

Primary: 46L10: General theory of von Neumann algebras 46L55: Noncommutative dynamical systems
Secondary: 37A55: Relations with the theory of $C^*$-algebras 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3181 - 3187
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Copyright
© Cambridge University Press, 2019

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