Finding dilations of Markov semigroups to cocycle perturbations of noises is a fundamental problem of quantum probablity and quantum dynamics. A noise is hereby a reversible quantum dynamics with a filtration that is independent in some notion of quantum independence. The noise is a dilation of the trivial Markov semigroup; the scope is turning it into a dilation of a nontrivial Markov semigroup by perturbation with a unitary cocycle. In 2004, we pointed out that, in a sense, every noise has a filtration that is conditional (or amalgamated, or operator-valued) monotone independent; it is a monotone noise. But there may be filtrations that are independent in other notions of independences, for instance, in conditional (or amalgamated) free independence. In 2009, we proved that a Markov semigroup admits a dilation to a cocycle perturbation of a noise if (and, obviously, only if) it is spatial. A Markov semigroup is spatial if it "dominates" an "elementary" CP-semigroup. Spatiality of a Markov semigroup is reflected by its GNS product system and, in fact, in the construction of the dilation the GNS-system and the classification of $E_0$-semigroups by their product systems up to "stable" cocycle conjugacy plays a crucial role. But this construction is "general abstract nonsense". In particular, it is unclear if the unitary cocycle is adapted to a suitable independent filtration or if it fulfills a quantum stochastic differential equation. After recalling some details about these facts, we introduce free product systems an illustrate how to contruct a conditional (or amalgamated) free noise from every free product system. We show how every spatial (tensor!) product system generates a free product system. For instance, as we pointed out in 2001, the time ordered product systems (that is, the spatial type I systems) generate the free product systems of full Fock modules with the same index. For uniformly continuous continuous Markov semigroups, we constructed suitable unitary cocycles with our quantum stochastic calculus on the full Fock module from 2000. Here, we ask if it is possible to do the same, purely algebraically without any calculus, starting from the GNS-system of an arbitrary spatial Markov semigroup. This talk continues in a sense the talk from Goa about Spatial Quantum Dynamics two weaks ago, but it is independent it in that we repeat the essential results. But while the talk in Goa concentrates on spatial dynamics in general, this talk will concentrate on the relation of spatial product systems, free product systems, and free noises.
Free Product Systems
SKEIDE, Michael
2013-01-01
Abstract
Finding dilations of Markov semigroups to cocycle perturbations of noises is a fundamental problem of quantum probablity and quantum dynamics. A noise is hereby a reversible quantum dynamics with a filtration that is independent in some notion of quantum independence. The noise is a dilation of the trivial Markov semigroup; the scope is turning it into a dilation of a nontrivial Markov semigroup by perturbation with a unitary cocycle. In 2004, we pointed out that, in a sense, every noise has a filtration that is conditional (or amalgamated, or operator-valued) monotone independent; it is a monotone noise. But there may be filtrations that are independent in other notions of independences, for instance, in conditional (or amalgamated) free independence. In 2009, we proved that a Markov semigroup admits a dilation to a cocycle perturbation of a noise if (and, obviously, only if) it is spatial. A Markov semigroup is spatial if it "dominates" an "elementary" CP-semigroup. Spatiality of a Markov semigroup is reflected by its GNS product system and, in fact, in the construction of the dilation the GNS-system and the classification of $E_0$-semigroups by their product systems up to "stable" cocycle conjugacy plays a crucial role. But this construction is "general abstract nonsense". In particular, it is unclear if the unitary cocycle is adapted to a suitable independent filtration or if it fulfills a quantum stochastic differential equation. After recalling some details about these facts, we introduce free product systems an illustrate how to contruct a conditional (or amalgamated) free noise from every free product system. We show how every spatial (tensor!) product system generates a free product system. For instance, as we pointed out in 2001, the time ordered product systems (that is, the spatial type I systems) generate the free product systems of full Fock modules with the same index. For uniformly continuous continuous Markov semigroups, we constructed suitable unitary cocycles with our quantum stochastic calculus on the full Fock module from 2000. Here, we ask if it is possible to do the same, purely algebraically without any calculus, starting from the GNS-system of an arbitrary spatial Markov semigroup. This talk continues in a sense the talk from Goa about Spatial Quantum Dynamics two weaks ago, but it is independent it in that we repeat the essential results. But while the talk in Goa concentrates on spatial dynamics in general, this talk will concentrate on the relation of spatial product systems, free product systems, and free noises.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
