Christensen and Evans showed that, in the language of Hilbert modules, a bounded derivation on a von Neumann algebra with values in a two-sided von Neumann module (i.e. a sufficiently closed two-sided Hilbert module) are inner. Then they use this result to show that the generator of a normal uniformly continuous completely positive (CP-) semigroup on a von Neumann algebra decomposes into a (suitably normalized) CP-part and a derivation like part. The backwards implication is left open.In these notes we show that both statements are equivalent among themselves and equivalent to a third one, namely, that any type I tensor product system of von Neumann modules has a unital central unit. From existence of a central unit we deduce that each such product system is isomorphic to a product system of time ordered Fock modules. We, thus, find the analogue of Arveson's result that type I product systems of Hilbert spaces are symmetric Fock spaces.On the way to our results we have to develop a number of tools interesting on their own right. Inspired by a very similar notion due to Accardi and Kozyrev, we introduce the notion of semigroups of completely positive definite kernels (CPD-semigroups), being a generalization of both CP-semigroups and Schur semigroups of positive definite C-valued kernels. The structure of a type I system is determined completely by its associated CPD-semigroup ARTICLE IN PRESS$ This work has been supported by a DAAD-DST Project based Personnel exchange Programme. and the generator of the CPD-semigroup replaces Arveson's covariance function. As another tool we give a complete characterization of morphisms among product systems of time ordered Fock modules. In particular, the concrete form of the projection endomorphisms allows us to show that subsystems of time ordered systems are again time ordered systems and to find a necessary and sufficient criterion when a given set of units generates the whole system. As a byproduct we find a couple of characterizations of other subclasses of morphisms. We show that the set of contractive positive endomorphisms are order isomorphic to the set of CPD-semigroups dominated by the CPD-semigroup associated with type I system. r
Abstract. We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra, some standard results concerning the spectral properties exhibited by temperature states for disordered quantum spin systems. We discuss the Arveson spectrum and its connection with the Connes and Borchers Γ-invariants for such W * -dynamical systems. In the case of KMS states exhibiting a natural property of invariance with respect to the spatial translations, some interesting properties, associated with standard spinglass-like behaviour, emerge naturally. It covers infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice Z d . In particular, we show that a temperature state of the systems under consideration can generate only a type III von Neumann algebra (with the type III 0 component excluded). Moreover, in the case of the pure thermodynamic phase, the associated von Neumann is of type III λ for some λ ∈ (0, 1], independent of the disorder. Such a result is in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The present approach can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.Mathematics Subject Classification: 46L55, 82B44, 46L35.
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