The Index of a Quantum Dynamical Semigroup
نویسندگان
چکیده
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 positive maps on the algebra B(H) of all bounded operators on a separable Hilbert space H. This index is defined in terms of basic structures associated with P , and generalizes the index of E0-semigroups. In the case where Pt(1) = 1, t ≥ 0, we show that the index of P agrees with the index of its minimal dilation to an E0-semigroup. The key ingredients are the existence of the covariance function (Theorem 2.6), the relation between units of P and units of its minimal dilation (Theorem 3.6), and the mapping of covariance functions (Corollary 4.8). No examples are discussed here, but another paper is in preparation [5]. 1. The metric operator space of a completely positive map. We consider the real vector space of all normal linear maps L of B(H) into itself which are symmetric in the sense that L(x) = L(x), x ∈ B(H). For two such maps L1, L2 we write L1 ≤ L2 if the difference L2 − L1 is completely positive. Every operator a ∈ B(H) gives rise to an elementary completely positive map Ωa by way of Ωa(x) = axa , x ∈ B(H). 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05.
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