An algebraic construction of quantum flows with unbounded generators

Abstract

It is shown how to construct -homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. Our construction is possible provided the generator satises an invariance property for some dense subalgebra A0 of the C algebraA and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A0, must satisfy a growth condition. Furthermore, it is assumed that either the subalgebra A0 is generated by isometries and A is universal, or A0 contains its square roots. These conditions are veried in four cases: classical random walks on discrete groups, Rebolledo’s symmetric quantum exclusion process and ows on the non-commutative torus and the universal rotation algebra. Des cocycles de Feller stochastiques quantiques -homomorphes sont construits pour certains g en erateurs non born es, et ainsi nous obtenons des dilatations pour des semigroupes