Continuous families of E_0-semigroups

Abstract

We consider families of E0-semigroups continuously parametrized by a compact Hausdorff space, which are cocycle-equivalent to a given E0-semigroup β. When the gauge group of β is a Lie group, we establish a correspondence between such families and principal bundles whose structure group is the gauge group of β.Let H be a Hilbert space, which we will always assume to be separable and infinitedimensional, and let B(H) denote the * -algebra of all bounded operators over H. An E 0semigroup acting on B(H) is a point-σ-weakly continuous family β = {β t : B(H) → B(H)} t≥0 of unital * -endomorphisms such that β 0 = id. We direct the reader to [Arv03] for a general reference on the theory of E 0 -semigroups.This paper studies continuous families of E 0 -semigroups parametrized by a compact Hausdorff space, where the E 0 -semigroups are all cocycle equivalent to a given E 0 -semigroup β. Suitable notions of continuity and equivalence are introduced below. If the gauge group G of β is a Lie group, we show that such continuous families are classified by principal G-bundles. Gauge groups of E 0 -semigroups were computed by Arveson in [Arv89] for the type I case. In the type II case, the gauge groups were computed recently for several classes of examples by Alevras, Powers and Price in [APP06] and by Jankowski and the second author in [JM11]. Indeed, in many of the known examples the gauge group is a Lie group. Specializing to the case of E 0 -semigroups of type I, one can recast those principal bundles as vector bundle invariants. The case of continuous families of single endomorphisms of B(H) (of finite index) was studied in [Hir04], where it was shown that such families of endomorphisms are classified by vector bundle invariants of dimension given by the index of the endomorphism. We thus obtain an analogy between the case of continuous families of endomorphisms and continuous families of E 0 -semigroups.In the case of families of one-parameter automorphism groups, the gauge group is R, hence the principal bundle invariants are trivial. We treat this case separately, since we establish this triviality result under (a priori) weaker continuity assumptions, using techniques from [Bar54]. This corresponds to a parametrized version of Wigner's theorem.Given an E 0 -semigroup β acting on B(H) we will say that a strongly continuous family of unitary operators {U t ∈ U (H) : t ≥ 0} is a β-cocycle if U 0 = 1 and U t+s = U t β t (U s ) for all t, s ≥ 0. We emphasize that in this paper all cocycles will be unitary cocycles. We will denote by C β the set of all β-cocycles. An E 0 -semigroup α is cocycle equivalent to β if there exists a β-cocycle U t such that α t (X) = U t β t (X)U * t for all t ≥ 0 and X ∈ B(H). We will denote by E β be the set of all E 0 -semigroups acting on B(H) which are cocycle equivalent to the E 0 -semigroup β.If H is separable, then the unitary group U (H) is a Polish group when endowed with the relative strong operator topology. Recall that a Polish space is a topological space with a separable completely m...