Löwner's theorem and the differential geometry of the space of positive operators
Let A be a untel C*-algebra and G+ the space of all positive invertible elements of A. In this largely expository paper we collect several geometrical features of G+ which relate its structure with that of Riemannian manifolds with non positive curvature. The main result of the paper is the equivale...
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| Autores principales: | , , |
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1998
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00015504_v49_n2_p70_Andruchow http://hdl.handle.net/20.500.12110/paper_00015504_v49_n2_p70_Andruchow |
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| Sumario: | Let A be a untel C*-algebra and G+ the space of all positive invertible elements of A. In this largely expository paper we collect several geometrical features of G+ which relate its structure with that of Riemannian manifolds with non positive curvature. The main result of the paper is the equivalence of the so-called Löwner-Heinz-Cordes inequality ∥StTt∥ ≤ ∥ST∥t (valid for positive operators S, T on a Hilbert space and t ∈ [0, 1]) with the geometrical fact that for every pair γ, δ of geodesics of G+ the real function t → d(γ(t), δ(t)) is convex, where d denotes the geodesic distance. |
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