An essential representation for a product system over a finitely generated subsemigroup of $\mathbb{Z}^{d}$

TL;DR

This paper proves that every product system over a finitely generated subsemigroup of b^d can be represented via a semigroup of unital normal *-endomorphisms on an infinite-dimensional separable Hilbert space, linking algebraic and operator-theoretic structures.

Contribution

It establishes a universal representation theorem for product systems over finitely generated subsemigroups of b^d using operator algebra techniques.

Findings

Existence of a Hilbert space b that models the product system.

Construction of a semigroup of unital normal *-endomorphisms representing the product system.

Isomorphism between the given product system and one derived from the endomorphism semigroup.

Abstract

Let $S \subset \mathbb{Z}^{d}$ be a finitely generated subsemigroup. Let $E$ be a product system over $S$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $\alpha:=\{\alpha_x\}_{x \in S}$ of unital normal $*$-endomorphisms of $B(\mathcal{H})$ such that $E$ is isomorphic to the product system associated to $\alpha$.