Constructing Proper Markov Semigroups for Arveson Systems

TL;DR

This paper demonstrates that a specific Markov semigroup derived from an $E_0$-semigroup does not consist of endomorphisms and cannot be its tail flow, impacting the construction of certain type III Markov semigroups.

Contribution

It proves that Floricel's Markov semigroup is not composed of endomorphisms and cannot be realized as a tail flow, revealing limitations in existing constructions.

Findings

Floricel's Markov semigroup does not consist of endomorphisms.

It cannot be the tail flow of an $E_0$-semigroup.

Provides a pathway to construct proper type III Markov semigroups.

Abstract

We show that the Markov semigroup obtained by Floricel in [Flo08] compressing the $E_0$-semigroup of Skeide [Ske06], does not consist of endomorphisms. It, therefore, cannot be the tail flow of an $E_0$-semigroup. As a corollary of our result, Floricel's construction will allow to get examples of proper type III Markov semigroups that are not tensor products of simpler ones, provided we find type III Arveson systems that do not factor into tensor products.