Twisted partial group algebra and related topological partial dynamical system

Abstract

Given a group \( G \), a field \( κ\), and a factor set \( σ\) arising from a partial projective \( κ\)-representation of \( G \). This leads to the construction of a topological partial dynamical system \( (Ω_σ, G, \hatθ) \), where \( Ω_σ\) is a compact, totally disconnected Hausdorff space, and \( σ\) acts as a twist for \( \hatθ \). We show that the twisted partial group algebra \( κ_{par}^σ G \) can be realized as a crossed product \( {\mathscr L}(Ω_σ) \rtimes_{(\hatθ, σ)} G \), with \( {\mathscr L}(Ω_σ) \) denoting the \( κ\)-algebra of locally constant functions \( Ω_σ\to κ\). The space \( Ω_σ\) corresponds to the spectrum of a unital commutative subalgebra in \( κ_{par}^σ G \), generated by idempotents. By describing \( Ω_σ\) as a subspace of the Bernoulli space \( 2^G \), we examine conditions under which the spectral partial action \( \hatθ \) is topologically free, impacting the ideal structure of \( κ_{par}^σ G \). We further explore generating idempotent factor sets of \( G \) and present conditions on them to ensure the topological freeness of \( \hatθ \). Inspired by Exel's semigroup \( \mathcal{S}(G) \), which governs partial actions and representations of \( G \) and relates to \( κ_{par}G \), we characterize the twisted partial group algebra \( κ_{par}^σG \) as generated by a \( κ\)-cancellative inverse semigroup constructed from elements of \( Ω_σ\). When \( Ω_σ\) is discrete, we demonstrate that \( κ_{par}^σ G \) decomposes into a product of matrix algebras over twisted subgroup algebras, generalizing known results for finite \( G \).