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.
Keywords
Cite
@article{arxiv.1209.3639,
title = {An algebraic construction of quantum flows with unbounded generators},
author = {Alexander C. R. Belton and Stephen J. Wills},
journal= {arXiv preprint arXiv:1209.3639},
year = {2013}
}
Comments
34 pages. This version contains an expanded introduction and a new section on constructing random walks on discrete groups