A Banach *-algebra associated with the action of a group on a topological space
Tabaré Roland
TL;DR
The paper studies the Banach *-algebra ℓ^1(Σ) arising from a countable infinite group G acting by homeomorphisms on a compact space X, and it connects dynamical properties of the system Σ=(G,X,σ) to analytic-algebraic properties of ℓ^1(Σ). It develops representations π_x from trivial isotropy representations and proves a central equivalence (Theorem topFreeThm) linking topological freeness to the ideal structure of ℓ^1(Σ), enabling characterizations of minimality, transitivity, and residual topological freeness. It then analyzes when freeness of the action can be detected via closed non-self-adjoint ideals, showing freeness implies all closed ideals are self-adjoint in general, but the converse fails in general; for abelian torsion-free G the action is free if and only if all closed ideals are self-adjoint. These results illuminate how much dynamical information is preserved in the ℓ^1(Σ) setting versus the C*-crossed product envelope and broaden noncommutative spectral-synthesis ideas beyond Z-actions.
Abstract
Given a discrete and countable infinite group $G$ acting via homeomorphism on a compact and Hausdorff space $X$, we consider $Σ$ the dynamical system associated to the action. One can naturally associate to $Σ$ the crossed product type Banach *-algebra $\ell^1(Σ)$. We will study how different dynamical properties of $Σ$ are characterized as analytic-algebraic properties of $\ell^1(Σ)$. The dynamical properties of $Σ$ that we will study are topological freeness, minimality, topological transitivity and residual topological freeness. We will also see how, under some assumptions on $G$, one can detect if $Σ$ is free by detecting closed non-self-adjoint ideals of $\ell^1(Σ)$.