Two New Complete Invariants of von Neumann Algebras

TL;DR

This paper introduces two new invariants, the oriented context category and the oriented spectral presheaf, which fully characterize von Neumann algebras under certain conditions, advancing the classification theory.

Contribution

It establishes that these two invariants are complete for classifying von Neumann algebras excluding trivial and type I_2 summands.

Findings

The oriented context category is a complete invariant.

The oriented spectral presheaf is a complete invariant.

These invariants distinguish von Neumann algebras uniquely under specified conditions.

Abstract

We show that the oriented context category and the oriented spectral presheaf are complete invariants of a von Neumann algebra not isomorphic to $\mathbb{C}\oplus\mathbb{C}$ and with no direct summand of type $I_2$.