Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

In a series of papers [66, 65, 62, 63, 64] TSIRELSON constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by ARVESON [4] for classifying E0-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So we connect each continuous tensor product systems of Hilbert spaces with measure types of distributions of random (closed) sets in [0;1℄ or R+ . These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the TSIRELSON examples, that the classification scheme for product systems into types In, IIn and III is not complete. Moreover, based on a detailed study of this kind of measure types, we construct for each stationary factorizing measure type a continuous tensor product systems of Hilbert spaces such that this measure type arises as the before mentioned invariant. These results are a further step in the classification of all (separable) continuous tensor product systems of Hilbert spaces of type II in completion to the classification of type I done by [4] and combine well with other invariants like the lattice of product subsystems of a given product system. Although these invariants relate to type II product systems mainly, they are of general importance. Namely, the measure types of the above described kind are connected with representations of the corresponding L∞-spaces. This leads to direct integral representations of the elements of a given product system which combine well under tensor products. Using this structure in a constructive way, we can relate to any (type III) product system a product system of type II0 preserving isomorphy classes. Thus, the classification of type III product systems reduces to that of type II (and even type II0) ones. In this circle of ideas it proves useful that we reduce the problem of finding a compatible measurable structure for product systems to prove continuity of one periodic unitary group on a single Hilbert space. As a consequence, all admissible measurable structures (if there are any) on an algebraic continuous tensor product systems of Hilbert spaces yield isomorphic product systems. Thus the measurable structure of a continuous tensor product systems of Hilbert spaces is essentially determined by its algebraic one. GSF — National Research Centre for Environment and Health, Institute of Biomathematics and Biometry, Ingolstädter Landstr.1, D–85758 Neuherberg, Germany, email:[email protected]