Dilation of Markovian cocycles on a von Neumann algebra

Goswami, Debashish; Lindsay, J. M.; Sinha, Kalyan B.; Wills, Stephen J.

Dilation of Markovian cocycles on a von Neumann algebra

Files

pjm-v211-n2-p02-p.pdf(313.51 KB)

Published version

Date

2003-10-01

Authors

Goswami, Debashish

Lindsay, J. M.

Sinha, Kalyan B.

Wills, Stephen J.

Publisher

Mathematical Sciences Publishers (MSP)

Published Version

10.2140/pjm.2003.211.221

Abstract

We consider normal Markovian cocycles on a von Neumann algebra which are adapted to a Fock filtration. Every such cocycle k which is Markov-regular and consists of completely positive contractions is realised as a conditioned ∗-homomorphic cocycle. This amounts to a stochastic generalisation of a recent dilation result for norm-continuous normal completely positive contraction semigroups. To achieve this stochastic dilation we use the fact that k is governed by a quantum stochastic differential equation whose coefficient matrix has a specific structure, and extend a technique for obtaining stochastic flow generators from Markov semigroup generators, to the context of cocycles. Number/exchange-free dilatability is seen to be related to locality in the case where the cocycle is a Markovian semigroup. In the same spirit unitary dilations of Markov-regular contraction cocycles on a Hilbert space are also described. The paper ends with a discussion of connections with measure-valued diffusion.

Keywords

Markovian cocycles , Fock filtration , Von Neumann algebra

Citation

Goswami, D., Lindsay, J.M., Sinha, K.B. and Wills, S.J. (2003) ‘Dilation of Markovian cocycles on a von Neumann algebra’, Pacific Journal of Mathematics, 211(2), pp. 221–247. https://doi.org/10.2140/pjm.2003.211.221