Variational and convexity properties of families of involutions
type: Item
instance of: scholarly article
title: Variational and convexity properties of families of involutions
zbMATH Open document ID: 0831.46062
DOI: 10.1007/BF01203097
author: Horacio Porta
author: Lázaro Recht
published in: Integral Equations and Operator Theory
publication date: 1996-02-18T00:00:00Z
review text: Let \(\mathcal S\) be the set of all \(C^*\)-algebra structures that share a given underlying topological algebra \(\mathcal A\), which is called Star on \(\mathcal A\). The problem is to study natural geometric structures of the set \(\mathcal S\). The following are some results shown in the article. 1) The set \(\mathcal S\) has a natural structure of an analytic manifold. Let \(\text{Int}({\mathcal A})\) be the group of inner automorphisms of \(\mathcal A\). Then \[ \sigma\to L_\phi \sigma,\quad \phi\in \text{Int}({\mathcal A}) \] is an analytic map, where \(L_\phi \sigma:= \phi \sigma \phi^{- 1}\). Let \(M\) be the typical orbit of this action: \[ M= \{L_\phi \sigma;\;\phi\in \text{Int}({\mathcal A})\}, \] which is called a constellation in the sky of \(\mathcal A\). We also have \[ T_\sigma M\simeq \{\text{ad}(x); \text{ad}(x)\sigma= 0,\;x\in {\mathcal A}\}, \] where \(T_\sigma M\) is the tangent space at \(\sigma\in M\). The orbit \(M\) is equipped with a reductive homogeneous space structure and a Finsler metric. At the tangent space \(T_\sigma M\) the metric is defined by means of a fixed conditional expectation \(\Phi\) onto the center \({\mathcal Z}({\mathcal A})\). 2) A presentation of each orbit \(M\) as the base of a fibration of the space \(G^\sigma\) of \(\sigma\) positive invertible elements of \(\mathcal A\) is given, where \(\sigma\) is any element of \(M\). As a consequence, we get that (1) geodesics in \(M\) are curves of minimal length. (2) any pair of stars in \(M\) can be joined by a unique geodesic. 3) The geodesic distance function of \(M\) is a convex function, that is, for any fixed \(\sigma\in M\), the function \(f_\sigma(\theta):= \text{dist}(\theta, \sigma)\), \(\theta\in M\) satisfies an inequality \[ f_\sigma(\gamma(t))\leq (1- t) f_\sigma(\gamma(0))+ tf_\sigma(\gamma(1)), \] for each geodesic \(\gamma(t)\) in \(M\).
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 58E10
zbMATH DE Number: 764057
zbMATH Keywords: conditional expectation
zbMATH Keywords: \(C^*\)-algebra structures
zbMATH Keywords: Star
zbMATH Keywords: analytic manifold
zbMATH Keywords: orbit
zbMATH Keywords: constellation
zbMATH Keywords: Finsler metric
zbMATH Keywords: fibration
zbMATH Keywords: geodesics
zbMATH Keywords: geodesic distance function
reviewed by: Yasuhiro Asoo
MaRDI profile type: Publication
cites work: The geometry of the space of selfadjoint invertible elements in a \(C^*\)-algebra
cites work: The geometry of spaces of projections in \(C^*\)-algebras
cites work: A Geometric Interpretation of Segal's Inequality || e X + Y || ≤ || e X/2 e Y e X/2 ||
cites work: Convexity of the geodesic distance on spaces of positive operators
cites work: Q3736359
cites work: Q5565587
cites work: Q4014029
cites work: Minimality of Geodesics in Grassmann Manifolds
cites work: Conditional expectations and operator decompositions
cites work: Q3997905
full work available at URL: https://doi.org/10.1007/bf01203097
OpenAlex ID: W1992650630
description: scientific article; zbMATH DE number 764057