Minimal E0–semigroups
نویسنده
چکیده
It is known that every semigroup of normal completely positive maps of a von Neumann can be “dilated” in a particular way to an E0-semigroup acting on a larger von Neumann algebra. The E0-semigroup is not uniquely determined by the completely positive semigroup; however, it is unique (up to conjugacy) provided that certain conditions of minimality are met. Minimality is a subtle property, and it is often not obvious if it is satisfied for specific examples even in the simplest case where the von Neumann algebra is B(H). In this paper we clarify these issues by giving a new characterization of minimality in terms projective cocycles and their limits. Our results are valid for semigroups of endomorphisms acting on arbitrary von Neumann algebras with separable predual. 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05.
منابع مشابه
E0-dilation of strongly commuting CP0-semigroups
We prove that every strongly commuting pair of CP0-semigroups has a minimal E0-dilation. This is achieved in two major steps, interesting in themselves: 1: we show that a strongly commuting pair of CP0semigroups can be represented via a two parameter product system representation; 2: we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation....
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 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 کاملWhat type of dynamics arise in E0-dilations of commuting quantum Markov processes?
Let H be a separable Hilbert space. Given two strongly commuting CP0-semigroups φ and θ on B(H), there is a Hilbert space K ⊇ H and two (strongly) commuting E0-semigroups α and β such that φs ◦ θt(PHAPH) = PHαs ◦ βt(A)PH for all s, t ≥ 0 and all A ∈ B(K). In this note we prove that if φ is not an automorphism semigroup then α is cocycle conjugate to the minimal ∗-endomorphic dilation of φ, and ...
متن کامل2 3 M ay 2 00 7 GENERALIZED CCR FLOWS
We introduce a new construction of E0-semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of C0-semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III E0-semigroups, which can...
متن کامل