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SIMPLICITY OF CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS
Part of:
Smooth dynamical systems: general theory
Rings and algebras with additional structure
Topological dynamics
Rings and algebras arising under various constructions
Selfadjoint operator algebras
Published online by Cambridge University Press: 08 April 2019
Abstract
In this article, we consider a twisted partial action 
$\unicode[STIX]{x1D6FC}$ of a group 
$G$ on an associative ring 
$R$ and its associated partial crossed product 
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$. We provide necessary and sufficient conditions for the commutativity of 
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ when the twisted partial action 
$\unicode[STIX]{x1D6FC}$ is unital. Moreover, we study necessary and sufficient conditions for the simplicity of 
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ in the following cases: (i) 
$G$ is abelian; (ii) 
$R$ is maximal commutative in 
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$; (iii) 
$C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$ is simple; (iv) 
$G$ is hypercentral. When 
$R=C_{0}(X)$ is the algebra of continuous functions defined on a locally compact and Hausdorff space 
$X$, with complex values that vanish at infinity, and 
$C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ is the associated partial skew group ring of a partial action 
$\unicode[STIX]{x1D6FC}$ of a topological group 
$G$ on 
$C_{0}(X)$, we study the simplicity of 
$C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ by using topological properties of 
$X$ and the results about the simplicity of 
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © 2019 Australian Mathematical Publishing Association Inc.
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