An algebraic construction of quantum flows with unbounded generators

Belton, Alexander C. R.; Wills, Stephen J.

Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, pp. 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.

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

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