2012/09/17 by Belton, Alexander C. R., Wills, Stephen J.
#46N50 #47D06 #60J27 (Secondary) #81S25 (Primary) 46L53 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1209.3639
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 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 processes and flows on the non-commutative torus and the universal rotation algebra.