Homogeneous manifolds from noncommutative measure spaces

• Autores: Andruchow, Esteban; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés
• Año de publicación: 2010
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: FINITE VON NEUMANN ALGEBRA; FINSLER METRIC; GEODESIC; HOMOGENEOUS SPACE; PATH METRIC SPACE; P-NORM; QUOTIENT METRIC; UNITARY GROUP
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

Let M be a finite von Neumann algebra with a faithful trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group U of M, endowed with a Finsler quotient metric induced by the p-norms of τ, ||x||_p=τ(|x|^p)^{1/p}, p ≥ 1. The main results include the following. The unitary group carries on a rectifiable distance d_p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d_p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d_{O,p}$,. For p ≥2, we prove that the distances d_p and d_{O , p} coincide. Based on this fact, we show that the metric space (O,d_p) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of UM with the p-norm.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de La Plata; Argentina
Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; Argentina
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; Argentina