Klyachko’s Theorem in Semi-finite von Neumann Algebras

Abstract

Let $\mathfrak{A}$ be a von Neumann algebra and $\bar{\mathfrak{A}}$ be the set of all $\tau$-measurable operators. For positive elements $A$ and $B$ in $\bar{\mathfrak{A}}$ we prove that

where $\mu (\cdot )$ denotes the generalized $s$-number and $I,J$, and $K$ are on an analogue of the Klyachko list.