Lifting fixed points of completely positive semigroups

2011/06/13 by Bebe Prunaru, Prunaru, Bebe
Mathematics · #46L55 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L55

paper · pdf · doi:10.48550/arxiv.1106.2521

4 pages, a new theorem is added, showing that part 3 of Theorem 1 holds for any commutative semigroup

openalex publication_date 2011/06/13 · arxiv created 2011/07/13 · arxiv updated 2011/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \ϕ_s\_{s∈ S} be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra N. Assume there exists a semigroup \α_s\_{s∈ S} of weak*-continuous *-endomorphisms of some larger von Neumann algebra M⊃ N and a projection p∈ M with N=pMp such that α_s(1-p)≤ 1-p for every s∈ S and ϕ_s(y)=pα_s(y)p for all y∈ N. If inf_{s∈ S}α_s(1-p)=0 then we show that the map E:M→ N defined by E(x)=pxp for x∈ M induces a complete isometry between the fixed point spaces of \α_s\_{s∈ S} and \ϕ_s\_{s∈ S}.

Related

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• Dilations of semigroups on von Neumann algebras and noncommutative L^p-spaces
• Representations of product systems over semigroups and dilations of commuting CP maps
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• Minimal E₀-semigroups