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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Detalles Bibliográficos
Autores principales:	Andruchow, Esteban, Corach, Gustavo, Stojanoff, Demetrio
Publicado:	1998
Materias:	Norm inequalities Positive operators
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
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_00015504_v49_n2_p70_Andruchow
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spelling	paper:paper_00015504_v49_n2_p70_Andruchow2023-06-08T14:20:53Z Löwner's theorem and the differential geometry of the space of positive operators Andruchow, Esteban Corach, Gustavo Stojanoff, Demetrio Norm inequalities 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 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. Fil:Andruchow, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Corach, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Stojanoff, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 1998 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
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Norm inequalities Positive operators
spellingShingle	Norm inequalities Positive operators Andruchow, Esteban Corach, Gustavo Stojanoff, Demetrio Löwner's theorem and the differential geometry of the space of positive operators
topic_facet	Norm inequalities Positive operators
description	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.
author	Andruchow, Esteban Corach, Gustavo Stojanoff, Demetrio
author_facet	Andruchow, Esteban Corach, Gustavo Stojanoff, Demetrio
author_sort	Andruchow, Esteban
title	Löwner's theorem and the differential geometry of the space of positive operators
title_short	Löwner's theorem and the differential geometry of the space of positive operators
title_full	Löwner's theorem and the differential geometry of the space of positive operators
title_fullStr	Löwner's theorem and the differential geometry of the space of positive operators
title_full_unstemmed	Löwner's theorem and the differential geometry of the space of positive operators
title_sort	löwner's theorem and the differential geometry of the space of positive operators
publishDate	1998
url	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
work_keys_str_mv	AT andruchowesteban lownerstheoremandthedifferentialgeometryofthespaceofpositiveoperators AT corachgustavo lownerstheoremandthedifferentialgeometryofthespaceofpositiveoperators AT stojanoffdemetrio lownerstheoremandthedifferentialgeometryofthespaceofpositiveoperators
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Löwner's theorem and the differential geometry of the space of positive operators

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