Conditional expectations and operator decompositions
type: Item
instance of: scholarly article
title: Conditional expectations and operator decompositions
zbMATH Open document ID: 0829.58004
DOI: 10.1007/BF02108305
author: Horacio Porta
author: Lázaro Recht
published in: Annals of Global Analysis and Geometry
publication date: 1996-01-17T00:00:00Z
review text: For a unital \(C^*\)-algebra \({\mathcal A}\) and a conditional expectation \(\Phi\) from \({\mathcal A}\) onto a subalgebra \({\mathcal B}\) consider the decomposition \({\mathcal A}= {\mathcal B}\oplus H\), the corresponding selfadjoint parts \({\mathcal A}^s= {\mathcal B}^s \oplus H^s\), and the corresponding images under the exponential function: \(G^+= \exp ({\mathcal A}^s)\) the set of positive invertibles of \({\mathcal A}\), \({\mathcal B}^+\) the set of positive invertibles of \({\mathcal B}\), and \(C= \{\exp h\mid h\in H\}\). Here is the main result: The maps \[ {\mathcal B}^+ \times C\ni (b,c) \mapsto (bcc^* b^*)^{1/2}\in G^+, \qquad {\mathcal B}^+ \times C \ni (bc)\mapsto bcb\in G^+ \] are analytic diffeomorphisms, and so is \[ {\mathcal B}^+\times C\times U\ni (b, c, u)\mapsto bcu\in G, \] where \(G\) and \(U\) denote the set of invertibles and of unitaries of \({\mathcal A}\), respectively.
Mathematics Subject Classification ID: 58B20
zbMATH DE Number: 762878
zbMATH Keywords: operator decompositions
zbMATH Keywords: conditional expectation
MaRDI profile type: Publication
cites work: A Geometric Interpretation of Segal's Inequality || e X + Y || ≤ || e X/2 e Y e X/2 ||
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full work available at URL: https://doi.org/10.1007/bf02108305
OpenAlex ID: W2023081483
description: scientific article; zbMATH DE number 762878