# Doubly Commuting Semigroups of Isometries

## TL;DR

This paper explores the structure and topology of doubly commuting semigroups of isometries, providing new proofs and analyzing differences between finite and infinite dimensions, including the failure of Wold decomposition.

## Contribution

It offers a new proof of Cooper's theorem in the Hilbert module setting and examines the Fell topology on irreducible, doubly commuting isometric representations of , highlighting dimension-dependent topological properties.

## Key findings

- Fell topology is T0 for finite d
- Wold decomposition fails for  with infinite d
- Topology is not T0 in the infinite-dimensional case

## Abstract

In this paper, we discuss the structure of doubly commuting semigroups of isometries. We record a new proof of Cooper's theorem in the Hilbert module setting. We discuss the Fell topology on the set of equivalence classes of irreducible, doubly commuting isometric representations of $\mathbb{R}_{+}^{d}$. We show that if $d$ is finite, the topology is $T_0$. We indicate the pathologies that occur when $d=\infty$. In particular, we show that Wold decomposition fails for isometric representations of $\mathbb{R}_{+}^{\infty}$ and prove that the Fell topology on the set of equivalence classes of irreducible, doubly commuting isometric representations of $\mathbb{R}_{+}^{\infty}$ is not $T_0$