Nonpositive curvature in \(p\)-Schatten class
type: Item
instance of: scholarly article
title: Nonpositive curvature in \(p\)-Schatten class
zbMATH Open document ID: 1170.58001
author: Cristian Conde
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-18T00:00:00Z
review text: The cone \({\mathcal P}=M_n(\mathbb C)^+\) of positive invertible matrices, when given the metric induced by the trace of the algebra, and the action of the full linear group induced by the identification \(M_n(\mathbb C)^+\simeq GL(n,\mathbb C)/U(n,\mathbb C)\) (here \(U(n,\mathbb C)\) stands for the group of unitary matrices) is a Riemannian manifold of nonpositive curvature. This positive cone -- a homogeneous space of the full linear group -- is a universal model for (finite dimensional, symmetric) Riemannian manifolds of noncompact type [see \textit{P. B. Eberlein}, Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. Chicago, IL: The University of Chicago Press (1996; Zbl 0883.53003)]. Finsler norms can be introduced on \({\mathcal P}\), usually by ways of an unitarily invariant norm on the algebra \(M_n(\mathbb C)\). What is remarkable is that, if one replaces the Riemannian condition of nonpositive curvature (usually defined via the curvature tensor) in \({\mathcal P}\) by other equivalent notions such as inequalities for comparison triangles (à la Alexandrov or Busemann), or inequalities for the differential of the exponential map (see McAlpin's PhD thesis), then this cone turns out to be nonpositively curved for any unitarily invariant norm. Generalizing the classical setting, one can also consider positive cones of operator algebras, without dimension restrictions (\(C^*\)-algebras, von Neumann algebras). In this paper, the author studies the positive cone \(\Delta_p\) of the operator algebra that is obtained by adjoining the unit to the ideal of compact \(p\)-Schatten operators. The main results of the paper deal with the consequences of the \(p\)-uniform convexity [see \textit{K. Ball, E. A. Carlen} and \textit{E. H. Lieb}, Invent. Math. 115, No.~3, 463--482 (1994; Zbl 0803.47037)] of the tangent norms in this setting, which lead to strong convexity conditions on the geodesic distance in \(\Delta_p\), and in particular results on…
reviewed by: Gabriel Larotonda
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47L07
Mathematics Subject Classification ID: 53C10
zbMATH DE Number: 5566265
zbMATH Keywords: Finsler metric
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: positive cone
zbMATH Keywords: \(p\)-Schatten class
zbMATH Keywords: uniform convexity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.03.036
OpenAlex ID: W1992288050
cites work: Sharp uniform convexity and smoothness inequalities for trace norms
cites work: Q4720067
cites work: Geometric interpolation in \(p\)-Schatten class
cites work: A Geometric Interpretation of Segal's Inequality || e X + Y || ≤ || e X/2 e Y e X/2 ||
cites work: The geometry of the space of selfadjoint invertible elements in a \(C^*\)-algebra
cites work: Convexity of the geodesic distance on spaces of positive operators
cites work: Ball versus distance convexity of metric spaces
cites work: Interpolation zwischen den Klassen 𝔖p von Operatoren in Hilberträumen
cites work: Q4380377
cites work: Norm inequalities in operator ideals
cites work: \(c_ p\)
cites work: Q5848223
DOI: 10.1016/J.JMAA.2009.03.036
description: scientific article; zbMATH DE number 5566265