Continuous Families of E0-semigroups

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 E0semigroup acting on B(H) is a point-σ-weakly continuous family β = {βt : B(H)→ B(H)}t≥0 of unital ∗-endomorphisms such that β0 = id. We direct the reader to [Arv03] for a general reference on the theory of E0-semigroups. This paper studies continuous families of E0-semigroups parametrized by a compact Hausdorff space, where the E0-semigroups are all cocycle equivalent to a given E0-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 E0-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 E0-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 E0-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 E0-semigroup β acting on B(H) we will say that a strongly continuous family of unitary operators {Ut ∈ U(H) : t ≥ 0} is a β-cocycle if U0 = 1 and Ut+s = Utβt(Us) for all t, s ≥ 0. We emphasize that in this paper all cocycles will be unitary cocycles. We will denote by Cβ the set of all β-cocycles. An E0-semigroup α is cocycle equivalent to β if there exists a β-cocycle Ut such that αt(X) = Utβt(X)U t for all t ≥ 0 and X ∈ B(H). We will denote by Eβ be the set of all E0-semigroups acting on B(H) which are cocycle equivalent to the E0-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 metrizable topology, and a Polish group is a topological group whose topology is Polish. Let R+ denote the half-open interval [0,∞). Let us endow C(R+,U(H)) 2000 Mathematics Subject Classification. 46L55. I.H. was partially supported by Israel Science Foundation grant 1471/07 and D.M. was partially supported by U.S.-Israel Binational Science Foundation grant 2008295.