Aligned CP-semigroups

TL;DR

This paper studies aligned CP-semigroups and proves that aligned spatial E_0-semigroups are prime, leading to the existence of many non-cocycle conjugate prime E_0-semigroups of type II_0.

Contribution

It introduces the concept of alignment in CP-semigroups and proves the primeness of aligned spatial E_0-semigroups, establishing new examples of prime E_0-semigroups.

Findings

Aligned spatial E_0-semigroups are prime.

Existence of uncountably many non-cocycle conjugate prime E_0-semigroups of type II_0.

Abstract

A CP-semigroup is aligned if its set of trivially maximal subordinates is totally ordered by subordination. We prove that aligned spatial E_0-semigroups are prime: they have no non-trivial tensor product decompositions up to cocycle conjugacy. As a consequence, we establish the existence of uncountably many non-cocycle conjugate E_0-semigroups of type II_0 which are prime.