Journal of Operator Theory

Volume 63, Issue 1, Winter 2010 pp. 129-150.

Trace Jensen inequality and related weak majorization in semi-finite von Neumann algebras

Authors: Tetsuo Harada

$1$ and Hideki Kosaki

$2$
Author institution:

$1$ 9-16-201 Hakozaki 1-chome, Higashi-ku, Fukuoka 812-0053, Japan

$2$ Faculty of Mathematics, Kyushu University, Higashi-ku, Fukuoka 812-8581, Japan

Summary: Let

${\mathcal M}$ be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace

$\tau$ , and we assume that

$f(t)$ is a convex function with

$f(0)=0$ . The trace Jensen inequality

$\tau(f(a^*xa)) \leqslant \tau(a^*f(x)a)$ is proved for a contraction

$a \in {\mathcal M}$ and a self adjoint operator

$x \in {\mathcal M}$

$or more generally for a semi-bounded $\tau$-measurable operator$ together with an abundance of related weak majorization-type inequalities. Notions of generalized singular numbers and spectral scales are used to express our results. Monotonicity properties for the map:

$x \in {\mathcal M}_{\mathrm{sa}} \to \tau(f(x))$ are also investigated for an increasing function

$f(t)$ with

$f(0)=0$ .

Keywords: Generalized singular number, Jensen inequality, semi-finite von Neumann algebra,

$\tau$ -measurable operator, trace, weak majorization

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