Some Results on Multiparameter Decomposable Product Systems and CAR Flows [HBNI Th260] 

Abstract

The content of this thesis comes from the study of product systems over closed convex cones. Product systems arise in the study of E0 - semigroups. An E0 -semigroup is a σ- weakly continuous semigroup of normal, unital endomorphisms of B(H). Introduced in the 1980’s by Powers, the subject of E0 -semigroups has been an active area of research ever since. Arveson introduced product system associated to one parameter E0 -semigroups as a tool to address their classification problem, which was later extended to E0 -semigroups over higher dimensional cones. In line with these studies, this thesis also explores product systems over a cone. This thesis deals with two classification problems related to multiparameter product systems. The first problem is concerned with the classification of decomposable product systems over closed convex cones. We describe the structure of a decomposable product system over a closed convex cone, and show that such a product system is described by an isometric representation of the cone and a 2-cocycle associated to the isometric representation. As an illustration of these results, we compute the cohomology groups involved in the description of a decomposable product system in a concrete situation. The second problem deals with CAR flows over cones. We first show that the map which sends an isometric representation of a cone to the corresponding CAR flow is in- jective, i.e. for two isometric representations V (1) and V (2) , the product systems of the corresponding CAR flows are isomorphic if and only if V (1) and V (2) are unitary equivalent. Finally, we show that if we restrict attention to isometric representations of the cone with commuting range projections, then we can describe type I CAR flows in explicit terms. We also compute the gauge group and index for this subclass of CAR flows.