An algebraic construction of quantum flows with unbounded generators
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, p. 349-375

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.

Des cocycles de Feller stochastiques quantiques *-homomorphes sont construits pour certains générateurs non bornés, et ainsi nous obtenons des dilatations pour des semigroupes dynamiques quantiques fortement continus sur des C * algèbres. Ceci généralise la construction d’un processus de Feller classique et de son semigroupe à partir d’un générateur donné. Notre construction est possible à condition que le générateur satisfasse une propriété d’invariance pour une sous-algèbre dense 𝒜 0 de la C * algèbre 𝒜 et obéisse aux relations de structure nécessaires; les itérations du générateur, lorsqu’elles sont appliquées à une famille génératrice de 𝒜 0 , doivent satisfaire à une condition de croissance. De plus, il est supposé que soit la sous-algèbre 𝒜 0 est engendrée par les isométries et 𝒜 est universelle, ou bien 𝒜 0 contient ses racines carrées. Ces conditions sont vérifiées dans quatre cas: marches aléatoires classiques sur les groupes discrets, le processus d’exclusion quantique symétrique introduit par Rebolledo et des flux sur le tore non commutatif et l’algèbre de rotation universelle.

DOI : https://doi.org/10.1214/13-AIHP578
Classification:  81S25,  46L53,  46N50,  47D06,  60J27
Keywords: 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
@article{AIHPB_2015__51_1_349_0,
     author = {Belton, Alexander C. R. and Wills, Stephen J.},
     title = {An algebraic construction of quantum flows with unbounded generators},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     pages = {349-375},
     doi = {10.1214/13-AIHP578},
     zbl = {06412908},
     mrnumber = {3300974},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_1_349_0}
}
Belton, Alexander C. R.; Wills, Stephen J. An algebraic construction of quantum flows with unbounded generators. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, pp. 349-375. doi : 10.1214/13-AIHP578. http://www.numdam.org/item/AIHPB_2015__51_1_349_0/

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