A short proof of Tomita's theorem

TL;DR

This paper presents a concise proof of Tomita's theorem in von Neumann algebra theory, utilizing the analytic structure of unitary flows and avoiding complex transforms used in previous proofs.

Contribution

The author provides a new, shorter proof of Tomita's theorem that simplifies the existing arguments by focusing on the analytic properties of unitary flows.

Findings

The proof is shorter and more direct than previous proofs.

It avoids the use of operator-valued Fourier and Mellin transforms.

The approach is similar to a special case previously handled by Bratteli and Robinson.

Abstract

Tomita-Takesaki theory associates a positive operator called the "modular operator" with a von Neumann algebra and a cyclic-separating vector. Tomita's theorem says that the unitary flow generated by the modular operator leaves the algebra invariant. I give a new, short proof of this theorem which only uses the analytic structure of unitary flows, and which avoids operator-valued Fourier transforms (as in van Daele's proof) and operator-valued Mellin transforms (as in Zsid\'{o}'s and Woronowicz's proofs). The proof is similar to one given by Bratteli and Robinson in the special case that the modular operator is bounded.