Representing a product system representation as a contractive semigroup and applications to regular isometric dilations

TL;DR

This paper introduces a new analytical tool for Hilbert $C^*$-product system representations, providing a novel proof for dilation existence in doubly commuting cases and extending results to more general semigroups.

Contribution

It presents a new technical approach using product system representations as contractive semigroups, advancing dilation theory for Hilbert $C^*$-product systems.

Findings

Every doubly commuting representation over $ abla^k$ has a regular isometric dilation

Provided sufficient conditions for dilations over general subsemigroups of $ _+^k$

Introduced a new method for analyzing product system representations

Abstract

In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb{R}_+^k$.