Young's Inequality in Semifinite von Neumann Algebras

2003/03/25 by Douglas Farenick, Douglas R. Farenick, Farenick, Douglas R. +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Spectral Theory in Mathematical Physics #math.OA #msc:46L05 #msc:47A60

paper · pdf · doi:10.48550/arxiv.math/0303318

arxiv created 2003/03/25 · arxiv updated 2009/11/30

Abstract

This paper formulates Young-type inequalities for singular values (or s-numbers) and traces in the context of von Neumann algebras. In particular, it shown that if \t(⋅) is a faithful semifinite normal trace on a semifinite von Neumann algebra M and if p and q are positive real numbers for which p^-1+q^-1=1, then, for all positive operators a,b∈ M, \t(|ab|)≤ p^-1\t(a^p)+ q^-1\t(b^q), with equality holding (in the cases where p^-1\t(a^p)+ q^-1\t(b^q)<∞) if and only if b^q=a^p.

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• On further refinements for Young inequalities