Topological Boundaries of Unitary Representations

Abstract

AbstractWe introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $\Gamma $ to the setting of a general unitary representation $\pi : \Gamma \to B(\mathcal H_\pi )$. This space, which we call the “Furstenberg–Hamana boundary” (or "FH-boundary") of the pair $(\Gamma , \pi )$, is a $\Gamma $-invariant subspace of $B(\mathcal H_\pi )$ that carries a canonical $C^{\ast }$-algebra structure. In many natural cases, including when $\pi $ is a quasi-regular representation, the Furstenberg–Hamana boundary of $\pi $ is commutative but can be noncommutative in general. We study various properties of this boundary and discuss possible applications, for example in uniqueness of certain types of traces.