"Type III E_0 semigroups and type III factors"
Let H be a separable infinite dimensional Hilbert space and B(H) be the set of bounded operators on H.
An E_0-semigroup is a 1-parameter semigroup of unit preserving endomorphisms of B(H) satisfying a certain continuity condition. Thanks to Arveson's fundamental work, classification of E_0-semigroups up to cocycle conjugacy is equivalent to that of so called product systems, which are continuous tensor product factorizations of a Hilbert space.
E_0-semigroups are classified into 3 categories, type I, type II, and type III, which perhaps came from a vague analogy to Murray-von Neumann's classification of factors, von Neumann algebras with trivial center. Similar to the case of factors, type I E_0-semigroups are tame and completely classified by Arveson. Though Powers and Tsirelson constructed plenty of type II and type III examples, they are very difficult to analyze.
Although the definition of type III E_0-semigroups is nothing to do with that of type III factors, R. Srinivasan and I recently constructed a family of examples of type III E_0-semigroups that actually produce type III factors.
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