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.