Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations

TL;DR

This paper proves that spatial Markov semigroups on the algebra of bounded operators on a Hilbert space admit Hudson-Parthasarathy dilations if and only if they are spatial, using abstract classification methods rather than stochastic differential equations.

Contribution

It provides a new proof that spatial Markov semigroups admit Hudson-Parthasarathy dilations, avoiding Hilbert module techniques and relying on Arveson system classification.

Findings

Spatial Markov semigroups admit Hudson-Parthasarathy dilations if and only if they are spatial.

The proof uses abstract classification methods instead of stochastic differential equations.

Results extend to more general C*-algebras with suitable adaptations.

Abstract

We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on $\mathscr{B}(G)$ ($G$ a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of $E_0$-semigroups on $\mathscr{B}(H)$ by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general $C^*$-algebras) have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here in order to make the result available without any appeal to Hilbert modules.)