Stochastic Dilation of Symmetric Completely Positive Semigroups
نویسندگان
چکیده
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.
منابع مشابه
Quantum Stochastic Semigroups
A rigged Hilbert space characterisation of the unbounded generators of quantum completely positive (CP) stochastic semigroups is given. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L (H) is described, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators is constructed.
متن کاملThe Index of a Quantum Dynamical Semigroup
A numerical index is introduced for semigroups of completely positive maps of B(H) which generalizes the index of E0-semigroups. It is shown that the index of a unital completely positive semigroup agrees with the index of its dilation to an E0-semigroup, provided that the dilation is minimal. Introduction. We introduce a numerical index for semigroups P = {Pt : t ≥ 0} of normal completely posi...
متن کاملOn the Index and Dilations of Completely Positive Semigroups
It is known that every semigroup of normal completely positive maps P = {Pt : t ≥ 0} of B(H), satisfying Pt(1) = 1 for every t ≥ 0, has a minimal dilation to an E0-semigroup acting on B(K) for some Hilbert space K ⊇ H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the ind...
متن کاملQuantized Convolution Semigroups
We describe our construction of a continuous tensor product system in the sense of Arveson for a general W ∗-continuous completely positive semigroup of B(H) (H separable). This product system is canonically isomorphic to the product system of the minimal dilation E0-semigroup. We use this construction to show that contrary to previous speculation the minimal dilations of all quantized convolut...
متن کاملGENERALIZED POSITIVE DEFINITE FUNCTIONS AND COMPLETELY MONOTONE FUNCTIONS ON FOUNDATION SEMIGROUPS
A general notion of completely monotone functionals on an ordered Banach algebra B into a proper H*-algebra A with an integral representation for such functionals is given. As an application of this result we have obtained a characterization for the generalized completely continuous monotone functions on weighted foundation semigroups. A generalized version of Bochner’s theorem on foundation se...
متن کامل