Peer-Reviewed Journal Details
Mandatory Fields
Belton, Alexander C.R.; Wills, Stephen J.
Annales De LInstitut Henri Poincaré (B) Probabilités Et Statistiques
An algebraic construction of quantum flows with unbounded generators
Optional Fields
quantum dynamical semigroup quantum Markov semigroup CPC semigroup strongly continuous semigroup semigroup dilation Feller cocycle higher-order Itô product formula random walks on discrete groups quantum exclusion process non-commutative torus
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 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 process and flows on the non-commutative torus and the universal rotation algebra.
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