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, Tome 51 (2015) no. 1, pp. 349-375.

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.

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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