Gauge groups of E_0-semigroups obtained from Powers weights

Jankowski, Christopher A.; Markiewicz, Daniel

doi:10.48550/arxiv.1102.5671

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“…Nevertheless, Theorem 3.7 implies that when the gauge group of an E 0 -semigroup of type I is locally compact then it is a Lie group. That is also the case for many of the E 0 -semigroups of type II 0 analyzed recently by [APP06] and [JM11]. In fact, to our knowledge in all cases when the gauge group has been proven to be locally compact, it has also turned out to be a Lie group.…”

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Continuous families of E_0-semigroups

Hirshberg¹,

Markiewicz²

2011

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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...

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Continuous families of E_0-semigroups

Hirshberg¹,

Markiewicz²

2011

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“…The following records facts that are implicit in the proof of Theorem 4.17 in Powers [15] (for a proof, see Proposition 2.11 of [10]):…”

Section: Boundary Weight Mapsmentioning

confidence: 99%

“…In this section we review the concept and notation associated with generalized Schur maps introduced in [10], and which will be used in the remainder of the paper.…”

Section: Boundary Weight Mapsmentioning

confidence: 99%

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E_0-semigroups and q-purity

Jankowski,

Markiewicz,

Powers

2011

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An E 0 -semigroup is called q-pure if it is a CP-flow and its set of flow subordinates is totally ordered by subordination. The range rank of a positive boundary weight map is the dimension of the range of its dual map. Let K be a separable Hilbert space. We describe all q-pure E 0 -semigroups of type II 0 which arise from boundary weight maps with range rank one over K. We also prove that no q-pure E 0 -semigroups of type II 0 arise from boundary weight maps with range rank two over K. In the case when K is finite-dimensional, we provide a criterion to determine if two boundary weight maps of range rank one over K give rise to cocycle conjugate q-pure E 0 -semigroups.

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Gauge groups of E_0-semigroups obtained from Powers weights

Cited by 2 publications

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Continuous families of E_0-semigroups

Continuous families of E_0-semigroups

E_0-semigroups and q-purity

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