Topological boundaries of unitary representations

2019/01/30 by Bearden, Alex, Kalantar, Mehrdad
#FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1901.10937

Abstract

We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group Γ to the setting of a general unitary representation π: Γ→ B(\mathcal H_π). This space, which we call the "Furstenberg-Hamana boundary" of the pair (Γ, π), is a Γ-invariant subspace of B(\mathcal H_π) that carries a canonical C^*-algebra structure. In many natural cases, including when π is a quasi-regular representation, the Furstenberg-Hamana boundary of π is commutative, but can be non-commutative in general. We study various properties of this boundary, and give some applications.

Related

• Boundary maps, germs and quasi-regular representations
• Spaces invariant under unitary representations of discrete groups
• Uniqueness of trace and C*-simplicity beyond regular representation
• Boundary maps and covariant representations
• Furstenberg boundary of minimal actions