Type III factors as invariants of type III $E_0$-semigroups
Dr. R. Srinivasan
Chennai Mathematical Institute.
E_0 -semigroups are semigroups of *-endomorphisms on B(H), the algebra of all bounded operators on a separable Hilbert space. They are divided into three types, namely type I,II and III. In the eighties W. Arveson completely classified type I E_0 smeigroups. In 1987 Robert Powers discovered the first example of a type III E_0-semigroup. After 13 years, in 2000 Boris Tsirelson produced an uncountable family of non-isomorphic type III product systems.(Product systems forms a complete invariant for E_0-semigroups). Inspired by Tsirelson's construction me and Bhat gave a purely operator algebraic construction of the same, and on the other hand Masaki Izumi studied the perturbation of the semigroup of Unilateral shift of index 1 by Hilbert-Schmidt operators.
As a consequences of all these developments, a new construction of E_0-semigroups called generalized CCR flows are introduced and studied by me and Masaki Izumi. I will explain this construction and talk about the necessary and sufficient condition for them to be of type III. Finally I will talk about the connection between type III factors and type III E_0-semigroups, and how type III factors distinguish an uncountable family of type III E_0-semigroups, which can not be distinguished by the invariants introduced earlier. This is a joint work with Masaki Izumi.