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arXiv:math-ph/0201005

Stochastic Dilation of Symmetric Completely Positive Semigroups

Debashish Goswami, Kalyan B. Sinha

Published Jan 3, 2002• 3 citations
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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.

DOI
10.48550/arxiv.math-ph/0201005

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This is a continuation of the study of the theory of quantum stochastic dilation of completely positive semigroups on a von Neumann or C∗C^*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.
Hudson-Parthasarathy (H-P) type quantum stochastic dilation of a class of C 0 semigroups of completely positive maps ( quantum dynamical or Markov semigroups) on a von Neumann or C * algebra, with unbounded generators, is constructed under some assumptions on the semigroup and its generator.The assumption of symmetry with respect to a semifinite trace allows the use of Hilbert space techniques,while that of covariance with respect to an action of a Lie group on the algebra gives a better control on the domain of the generator.A dilation of the dynamical semigroup is obtained, under some further assumptions on the domain of the generator, with the help of a conjugation by a unitary quantum stochastic process satisfying Hudson-Parthasarathy equation in Fock space.
In Chapter 6 we built a theory of quantum stochastic dilation ‘naturally’ associated with an arbitrary Q.D.S. on a von Neumann or C * -algebra with bounded generator. There the computations involved C * or von Neumann Hilbert modules, using the results of [24], map-valued quantum stochastic processes on modules and quantum stochastic integration with respect to them, developed in Chapter 5. It is now natural to consider the case of a Q.D.S. with unbounded generator and ask the same questions about the possibility of dilation. As one would expect, the problem is too intractable in this generality and we need to impose some further structures on it. In this chapter we shall consider a few classes of such Q.D.S. and try to construct H–P and E–H dilation for them. At first, we shall work under the framework of a Lie group action on the underlying algebra, and consider covariant Q.D.S. For H–P dilation, symmetry with respect to a trace is also assumed, whereas a general theory for E–H dilation has been built under the assumption of covariance under the action of a compact group, but without it being symmetric. Then, in the last section, we deal with a class of Q.D.S. on the U.H.F. algebra, described in Chapter 3. In this case, E–H dilation is constructed by a direct iteration using some natural estimates. However, what is common to the methods used in constructing dilation for the different kinds of Q.D.S. mentioned above is the use of a natural locally convex topology, in which the generator (unbounded in the norm topology) is continuous.
Abstract. We consider normal Markovian cocycles on a von Neumann al-gebra 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 generalisa-tion 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 coef-ficient 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.
A covariant version of the dilation theory for quantum dynamical semigrops is established, including both Bhat-Parthasarathy type and Evans-Hudson type dilation. It is shown in particular that every uniformly continuous quantum dynamical semigroup on a separable C*-algebra or a von Neumann algebra, which is covariant under a suitable group-action, admits a covariant Evans-Hudson dilation in some appropriate Fock space. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.