Young's Inequality in Semifinite von Neumann Algebras

Details Young's Inequality in Semifinite von Neumann Algebras Young's Inequality in Semifinite von Neumann Algebras

2003-03-25

arxiv.org/abs/math/0303318v1

Mathematics

Operator Algebras

Scientific paper

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 $\t(\cdot)$ 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 $a,b\in M$, $\t(|ab|)\le p^{-1}\t(a^p)+ q^{-1}\t(b^q)$, with equality holding (in the cases where $p^{-1}\t(a^p)+ q^{-1}\t(b^q)<\infty$) if and only if $b^q=a^p$.

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