COVARIANT GNS REPRESENTATION FOR OPEN C*-DYNAMICAL SYSTEMS

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.