An algebraic construction of quantum flows with unbounded generators

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 satisfies an invariance property for some dense subalgebra A₀ of the C ∗ algebra A 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 verified in four cases: classical random walks on discrete groups, Rebolledo's symmetric quantum exclusion process and flows on the non-commutative torus and the universal rotation algebra.