Stochastic Dilation of Symmetric Completely Positive Semigroups

TL;DR

This paper advances the theory of quantum stochastic dilation for symmetric completely positive semigroups with unbounded generators, utilizing Hilbert space techniques to construct Evans-Hudson flows.

Contribution

It extends previous work by incorporating symmetry and unbounded generators, providing a new approach to dilate semigroups via Evans-Hudson flows.

Findings

Established a dilation framework for symmetric semigroups with unbounded generators

Utilized Hilbert space techniques to handle domain issues

Constructed Evans-Hudson flows for the given semigroups

Abstract

This is a continuation of the study of the theory of quantum stochastic dilation of completely positive semigroups on a von Neumann or $C^*$ algebra, here with unbounded generators. The additional assumption of symmetry with respect to a semifinite trace allows the use of the Hilbert space techniques, while the covariance gives rise to better handle on domains. An Evans-Hudson flow is obtained, dilating the given semigroup.