Simplicity of Partial Crossed Products

TL;DR

This paper investigates the conditions under which partial crossed products of rings, especially continuous function algebras, are commutative or simple, linking algebraic properties with topological characteristics.

Contribution

It provides necessary and sufficient conditions for the commutativity and simplicity of partial crossed products, particularly for algebras of continuous functions under partial group actions.

Findings

Criteria for commutativity of partial crossed products.

Conditions for simplicity involving topological properties.

Extension of simplicity results to partial skew group rings.

Abstract

In this article, we consider a twisted partial action $\alpha$ of a group $G$ on a ring $R$ and it is associated partial crossed product $R*_{\alpha}^wG$. We study necessary and sufficient conditions for the commutativity and simplicity of $R*_{\alpha}^wG$. Let $R=C(X)$ the algebra of continuous functions of a topological space $X$ on the complex numbers and $C(X)*_{\alpha}G$ the partial skew group ring, where $\alpha$ is a partial action of a topological group $G$ on $C(X)$. We study some topological properties to obtain results on the algebra $C(X)$. Also, we study the simplicity of $C(X)*_{\alpha}G$ using topological properties and the results about the simplicity of partial crossed product obtained for $R*_{\alpha}^wG$.