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. The construction is possible provided the generator satisfies an invariance property for some dense subalgebra A_0 of the C* algebra A and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A_0, must satisfy a growth condition. Furthermore, it is assumed that either the subalgebra A_0 is generated by isometries and A is universal, or A_0 contains its square roots. These conditions are verified in four cases: classical random walks on discrete groups, Rebolledo's symmetric quantum exclusion processes and flows on the non-commutative torus and the universal rotation algebra.