On the extension of positive maps to Haagerup non-commutative L^p-spaces

2024/04/05 by Christian Le Merdy, Merdy, Christian Le, Safoura Zadeh +1
Mathematics · Medicine · #Advanced Operator Algebra Research #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders

paper · pdf · doi:10.48550/arxiv.2404.04400

Abstract

Let M be a von Neumann algebra, let φ be a normal faithful state on M and let L^p(M,φ) be the associated Haagerup non-commutative L^p-spaces, for 1≤ p≤∞. Let D∈ L¹(M,φ) be the density of φ. Given a positive map T\colon M→ M such that φ∘ T≤ C₁φ for some C₁≥ 0, we study the boundedness of the L^p-extension T_p,θ\colon D^(1-θ)/(p) M D^\fracθp→ L^p(M,φ) which maps D^(1-θ)/(p) x D^\fracθp to D^(1-θ)/(p) T(x) D^\fracθp for all x∈ M. Haagerup-Junge-Xu showed that T_p,\frac12 is always bounded and left open the question whether T_p,θ is bounded for θ\not=\frac12. We show that for any 1≤ p<2 and any θ∈ [0,2^-1(1-√(p-1))]∪[2^-1(1+√(p-1)), 1], there exists a completely positive T such that T_p,θ is unbounded. We also show that if T is 2-positive, then T_p,θ is bounded provided that p≥ 2 or 1≤ p<2 and θ∈[1-p/2,p/2].

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