Constructing Proper Markov Semigroups for Arveson Systems

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.