A Banach *-algebra associated with the action of a group on a topological space

2025/04/30 by Roland, Tabaré
#46H10 (Secondary) #47L65 (Primary) 37B05 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2505.00108

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 ℓ¹(Σ). We will study how different dynamical properties of Σ are characterized as analytic-algebraic properties of ℓ¹(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 ℓ¹(Σ).

Citations

• Prime and semiprime Lie ideals in C*-algebras
• The ideal separation property for reduced group C^*-algebras
• Semiprime ideals in C*-algebras
• Prime ideals in C*-algebras and applications to Lie theory
• Detecting ideals in reduced crossed product C*-algebras of topological dynamical systems
• Symmetry and Spectral Invariance for Topologically Graded C*-Algebras and Partial Action Systems
• A modern look at algebras of operators on L^p-spaces
• Topologically irreducible representations of the Banach *-algebra associated with a dynamical system
• Continuous orbit equivalence rigidity
• The ideal structure of reduced crossed products

Related

• Hermitian crossed product Banach algebras
• Topologically irreducible representations of the Banach *-algebra associated with a dynamical system
• Dynamical systems of type (m,n) and their C*-algebras
• Amenable crossed product Banach algebras associated with a class of C^∗-dynamical systems. II
• Amenability of group actions on compact spaces and the associated Banach algebras