Subsystems of Fock Need Not Be Fock: Spatial CP-Semigroups

TL;DR

This paper demonstrates that subsystems of Fock-based product systems can differ from Fock systems, addressing an open classification problem for CP-semigroups and characterizing spatial CP-semigroups via Christensen-Evans generators.

Contribution

It shows that subsystems of Fock systems need not be isomorphic to Fock systems and characterizes spatial CP-semigroups with Christensen-Evans generators.

Findings

Subsystems of Fock systems can differ from Fock systems.

Provides a classification answer to an open problem in CP-semigroups.

Characterizes spatial CP-semigroups via Christensen-Evans generators.

Abstract

We show that a product subsystem of a time ordered system (that is, a product system of time ordered Fock modules), though type I, need not be isomorphic to a time ordered product system. In that way, we answer an open problem in the classification of CP-semigroups by product systems. We define spatial strongly continuous CP-semigroups on a unital C*-algebra and characterize them as those that have a Christensen-Evans generator.