Stochastic Dilations of Uniformly Continuous Completely Positive Semigroups

Abstract

For an arbitrary uniformly continuous completely positive semigroup (ℑ _t : t≥ 0) on the space B(ɧ₀) of bounded operators on a Hilbert space ɧ₀, we construct a family (U(t): t ≥ 0) of unitary operators on a Hilbert space ℌ₀ = ɧ₀ ⊗ ℌ and a conditional expectation E₀ from B(ℌ₀) to B(ℌ₀), such that, for arbitrary t ≥0, X ∈ B(ɧ₀) ℑ _t (X) = E₀[U(t)X ⊗ IU(t)^†]. The unitary operators U(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.

Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.