Krein's trace theorem revisited

TL;DR

This paper presents a novel proof of Krein's Trace Theorem that avoids complex analysis, applicable to type II von Neumann algebras with unbounded perturbations from their predual.

Contribution

It provides the first proof of Krein's Trace Theorem that does not rely on complex analysis, extending its validity to a broader class of von Neumann algebras.

Findings

Proof applicable to σ-finite von Neumann algebras of type II

Valid for unbounded perturbations from the predual

First proof avoiding complex analysis

Abstract

We supply the first proof of Krein's Trace Theorem which does not use complex analysis. Our proof holds for~$\sigma$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of~$\mathcal{M}$.