Twisted partial group algebra and related topological partial dynamical system

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