# Topological boundaries of unitary representations

## TL;DR

This paper generalizes the Furstenberg boundary concept to unitary representations, introducing the Furstenberg-Hamana boundary as a $ ext{C}^*$-algebraic invariant that captures boundary-like properties of the representation.

## Contribution

It defines the Furstenberg-Hamana boundary for unitary representations, extending boundary theory beyond groups to operator algebraic settings.

## Key findings

- Furstenberg-Hamana boundary is often commutative for quasi-regular representations.
- The boundary can be non-commutative in general cases.
- The boundary has applications in understanding representation properties.

## Abstract

We 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" of the pair $(\Gamma, \pi)$, is a $\Gamma$-invariant subspace of $B(\mathcal H_\pi)$ that carries a canonical $C^*$-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 non-commutative in general. We study various properties of this boundary, and give some applications.