Affiliation:
1. Dipartimento di Ingegneria, Universitá di Roma "Tor Vergata", via del Politecnico 00133 Roma, Italy
Abstract
An open C*-dynamical systems is a triple {𝔄, Φ, φ} where 𝔄 is a C* algebra with unit, φ as its state, Φ : 𝔄 → 𝔄 a unital completely positive map with φ ◦ Φ = φ. Such system is called purely implementable if there exists a representation π of 𝔄 in the bounded operators on Hilbert spaces [Formula: see text], an isometry [Formula: see text] and a V-invariant vector [Formula: see text] such that π(Φ(a)) = V*π(a)V and φ(a) = 〈Ω, π(a)Ω〉 for all a ∈ 𝔄, with Ω cyclic vector for the algebra generated by π(𝔄) and V. The quadruple [Formula: see text] is said be a covariant GNS representation of the open C*-dynamical system. We prove that every open C*-dynamical system is purely implementable. In the case when the dynamics Φ is a *-homomorphism this result was obtained by Niculescu Ströh and Zsidó. Moreover we prove that, in the case of *-homomorphisms, the above-mentioned construction provides an alternative construction of a minimal dilation in the sense of Kümmerer in which the dilation of the dynamics Φ has the same ergodic properties as Φ.
Publisher
World Scientific Pub Co Pte Lt
Subject
Applied Mathematics,Mathematical Physics,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
2 articles.
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