Local minimal curves in homogeneous reductive spaces of the unitary group of a finite von Neumann algebra
type: Item
instance of: scholarly article
title: Local minimal curves in homogeneous reductive spaces of the unitary group of a finite von Neumann algebra
zbMATH Open document ID: 1180.58003
author: Eduardo Chiumiento
published in: Integral Equations and Operator Theory
publication date: 2009-07-01T00:00:00Z
full work available at URL: http://sedici.unlp.edu.ar/handle/10915/130677
review text: Let \({\mathcal M}\) be a finite von Neumann algebra, \(U_{\mathcal M}\) its unitary group and \({\mathcal P}\) an (eventually infinite-dimensional) reductive space of \(U_{\mathcal M}\) endowed with a non complete Riemannian metric. This means that the theory of Riemann-Hilbert manifolds does not apply to \({\mathcal P}\). The author introduces a condition on \({\mathcal P}\) called orthogonality that is sufficient for to prove the following result (\({\mathcal V}_R(\rho)= \exp_{\rho}(B_R(0))\) for a ball \(B_R(0)\) and \(\rho \in{\mathcal P}\)): Theorem 3.8. Let \({\mathcal P}\) be an orthogonal homogeneous reductive space and \(\rho \in {\mathcal P}\). Given any \(\rho_1 \in{\mathcal V}_R(\rho)\), there exist a unique geodesic inside \({\mathcal V}_R(\rho)\) which has minimal length among all the piecewise smooth curves inside \({\mathcal V}_R(\rho)\), joining the points \(\rho\) and \(\rho_1\). The result is discussed for several examples.
reviewed by: Mihai Anastasiei
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 5573974
zbMATH Keywords: finite von Neumann algebra
zbMATH Keywords: Finsler metric
zbMATH Keywords: metric geometry
MaRDI profile type: Publication
OpenAlex ID: W2069741340
DOI: 10.1007/S00020-008-1629-Y
description: scientific article; zbMATH DE number 5573974