Quantum stochastic processes for maps on Hilbert C*-modules

Abstract

We discuss pairs (φ, ) of maps, where φ is a map between C * -algebras and is a φ-module map between Hilbert C * -modules, which are generalization of representations of Hilbert C * -modules. A covariant version of Stinespring's theorem for such a pair (φ, ) is established, and quantum stochastic processes constructed from pairs ({φ t }, { t }) of families of such maps are studied. We prove that the quantum stochastic process J = {J t } constructed from a φ-quantum dynamical semigroup = { t } is a j-map for the quantum stochastic process j = { j t } constructed from the given quantum dynamical semigroup φ = {φ t }, and that J is covariant if the φquantum dynamical semigroup is covariant. C 2011 American Institute of Physics.Let ϕ : A → B be a * -homomorphism between two C * -algebras and let X and Y be Hilbert C * -modules over A and B, respectively. A map from X to Y is said to be a ϕ-representation if (x), (y) = ϕ( x, y ) for all x, y in X. Then any ϕ-representation is linear and satisfies the module property (xa) = (x)ϕ(a) for x ∈ X and a ∈ A. Such maps are naturally obtained from representations of the linking algebras of Hilbert C * -modules. It is known that the quotient map from X to X I is also a φ-morphism, where I is an ideal of A, φ : A → A/I is a quotient map and X I is an ideal submodule of X. In general, for any linear map φ we can also define a φ-map between Hilbert C * -modules in the same way. Asadi 2 constructed a Stinespring-type representation associated with (φ, ), where φ is completely positive, is a φ-map, and Bhat, Ramesh, and Sumesh 7 modified Asadi's construction.The main purpose of this paper is, motivated by Refs. 2, 7, and 20, to study a covariant version of Stinespring's theorem for a pair (φ, ) of maps on Hilbert C * -modules, and motivated by Refs. 1,5,6,and 19, to study the reconstruction theorem for quantum stochastic process from the pair ({φ t }, { t }) of families of such maps. Such a pair (φ, ) with a * -homomorphism φ is useful for the study of morphisms for the category of Hilbert C * -modules over possibly different C * -algebras. Furthermore, for such a pair ({φ t }, { t }) we observe that {φ t } is an asymptotic homomorphism if and only if { t } is an asymptotic module (see Example 4.2). This paper is organized as follows: In Sec. II we recall some basic notions and the concrete realization for Hilbert C * -modules. We discuss about typical examples of φ-representations of Hilbert C * -modules. In Sec. III we study a covariant version of Stinespring's theorem for a pair of two maps on C * -algebras and Hilbert C * -modules, respectively. In Sec. IV we introduce a notion of φ-quantum dynamical semigroups on Hilbert C * -modules and give an example related to an asymptotic homomorphism. We study the reconstruction theorem for quantum stochastic processes ({ j t }, {J t }) from a pair ({φ t }, { t }) of families of maps. We prove that if each t is a φ t -map (t ≥ 0), then the quantum stochastic process J = {J t } constructed from the φ-map becomes a j-map (Theorem 3.2). Her...

CP-H-extendable maps between Hilbert modules and CPH-semigroups

“…It is unclear if E ∞ is full, even if E is full and E ⊙ is full, or if E ∞ may be possibly {0}.But in any case, T (co)restricts to a minimal strictly CPH 0 -semigroup on E ∞ associated with τ. Necessarily, τ (co)restricts to a CP-semigroup on B E ∞ .It might be worth to compare the results in this section with Heo and Ji[HJ11], who investigated semigroups that, in our terminology, are CP-H-extendable, but who call them CPsemigroups.…”

“…Unfortunately, this is not so: There are more CP-extendable maps than τ-maps; see Section 3. We, therefore, strongly object to use the name CP-maps between Hilbert modules as meaning τ-maps, which was proposed recently by several authors; see, for instance, Heo and Ji [HJ11], or Joita [Joi12].…”

Covariant representations on KreinC∗-modules associated to pairs of two maps

“…In this section, we construct a covariant representation on a Krein C * -module associated to a pair of two covariant maps on Krein C * -modules, which may be regarded as a generalization of Theorem 3.2 in [12].…”

“…More generally, Heo-Hong-Ji [11] provided such a KSGNS type representation on a Krein C * -module for an α-completely positive map on a C * -algebra or a * -algebra. Moreover, Heo-Ji [12] constructed a Stinespring type covariant representation for a pair of a covariant completely positive map ρ and a covariant ρ-map. In this paper, motivated by the results in [3,13,11,12,14], we construct a KSGNS type covariant representation for a pair of a covariant α-completely positive map ρ on a C * -algebra and a covariant ρ-map on a Krein C * -module, using the KSGNS type representations on Krein C * -modules associated to α-completely positive maps.…”

“…Moreover, Heo-Ji [12] constructed a Stinespring type covariant representation for a pair of a covariant completely positive map ρ and a covariant ρ-map. In this paper, motivated by the results in [3,13,11,12,14], we construct a KSGNS type covariant representation for a pair of a covariant α-completely positive map ρ on a C * -algebra and a covariant ρ-map on a Krein C * -module, using the KSGNS type representations on Krein C * -modules associated to α-completely positive maps. Furthermore, we give a new covariant J-representation of a crossed product of a C * -algebra by a locally compact group and a new covariant map on the crossed product of a Hilbert C * -module by a locally compact group.…”