E-Semigroups

Abstract

A factor is a von Neumann algebra M ⊆ B(H) having trivial center $$M \cap M' = \mathbb{C} \cdot 1$$, and a factor is said to be of type I if it contains a nonzero minimal projection. The central objects of study in this book are semigroups of endomorphisms of infinite-dimensional type I factors. While it is usually convenient to coordinatize a type I factor M as the algebra ß(H) of all bounded operators on a complex infinite-dimensional Hilbert space H, we will often be led to consider type I subfactors of ß(H) and their commutants inside ß(H). We will also have occasion to deal with semigroups of endomorphisms of more general von Neumann algebras; for example, Chapter 8 concerns dilation theory in that more general setting. But the theory of E0-semigroups acting on ß(H) is still quite far from being well understood, and this will be our focus. For technical reasons it is important that all Hilbert spaces H should be separable. Correspondingly, all von Neumann algebras M must have separable predual M * .KeywordsHilbert SpaceGauge GroupUnitary OperatorVacuum VectorClosed Linear SpanThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.