Multiparameter Decomposable Product Systems

TL;DR

This paper generalizes the structure of decomposable product systems from one-parameter cases to higher-dimensional cones, characterizing them via isometric representations and 2-cocycles, and computing cocycles for shift semigroups.

Contribution

It extends Arveson's classification to multiparameter settings, describing the structure using isometric representations and cocycles, and computes cocycles for specific semigroups.

Findings

Structure of higher-dimensional decomposable product systems characterized by isometric representations and 2-cocycles.

Computed the space of 2-cocycles for shift semigroups from transitive cone actions.

Generalized the classification of product systems beyond the one-parameter case.

Abstract

In [8], Arveson proved that a $1$-parameter decomposable product system is isomorphic to the product system of a CCR flow. We show that the structure of a generic decomposable product system, over higher dimensional cones, modulo twists by multipliers, is given by an isometric representation $V$ of the cone and a certain $2$-cocycle for $V$. Moreover, we compute the space of $2$-cocycles for shift semigroups associated to transitive actions of a higher dimensional cone.