On the Araki-Lieb-Thirring inequality in the semifinite von Neumann algebra
type: Item
instance of: scholarly article
title: On the Araki-Lieb-Thirring inequality in the semifinite von Neumann algebra
zbMATH Open document ID: 1431.47004
DOI: 10.1215/20088752-3660864
published in: Annals of Functional Analysis
publication date: 2016-12-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.afa/1474652186
review text: Let \(A\) and \(B\) be positive self-adjoint operators on a Hilbert space, and let \(f\) be any increasing continuous real-valued function on \([0,\infty[\) such that \(f(0) = 0\) and \(t\mapsto f(e^t)\) is a convex function. By the Araki-Lieb-Thirring inequality, for all \(q\geq 1\) the following inequality holds: \[ \text{Tr}f((B^{1/2} AB^{1/2})^q)\leq \text{Tr}f(B^{q/2} A^qB^{q/2}). \] In this paper, the author shows that this inequality and its extension due to \textit{J.-C. Bourin} and \textit{E.-Y. Lee} [Int. J. Math. 27, No. 9, Article ID 1650071, 19 p. (2016; Zbl 1402.15015)] hold also in the setting of von~Neumann algebras, namely, for \(\tau\)-measurable operators associated with a~semifinite von~Neumann algebra. The paper is concluded with a series of submajorization inequalities which are related to Heinz means and some of them are new even in the matrix case.
reviewed by: Lajos Molnár
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 46L52
zbMATH DE Number: 6667758
zbMATH Keywords: Araki-Lieb-Thirring inequality
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: \(\tau\)-measurable operator
author: Yazhou Han
MaRDI profile type: Publication
OpenAlex ID: W2521381557
description: scientific article; zbMATH DE number 6667758