On simplicity of intermediate C*-algebras

Abstract

We prove simplicity of all intermediate $C^{\ast }$-algebras $C_r^{\ast }(\Gamma )\subseteq \mathcal{B}\subseteq \Gamma {⋉}_{r}C(X)$ in the case of minimal actions of $C^{\ast }$-simple groups $\Gamma$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar (arXiv:1712.10133v2). We show that the Powers' averaging property holds for the reduced crossed product $\Gamma {⋉}_r\mathcal{A}$ for any action $\Gamma \curvearrowright \mathcal{A}$ of a $C^{\ast }$-simple group $\Gamma$ 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 $\Gamma {⋉}_r\mathcal{A}$.

References in corpus (3)

• The ideal structure of reduced crossed products
• Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems
• Stationary C*-dynamical systems

Cited by in corpus (1)

• A generalized Powers averaging property for commutative crossed products