Existence of Feller Cocycles on a C * ‐Algebra

The quantum stochastic differential equation d k t = k t ^ θ rβ α d Λ α rβ ( t )is considered on a unital C * ‐algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non‐commutative topological space. 2000 Mathematics Subject Classification 81S25, 46L07, 46L53, 47D06.