Twisted partial group algebra and related topological partial dynamical system (2411.09824v1)

Abstract: 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 ).