# 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 $f(0)=0$.

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