# A note on the classification of Gamma factors

## TL;DR

This paper demonstrates that classifying Gamma property II_1 factors up to isomorphism cannot be achieved through Borel measurable invariants, highlighting the complexity of their classification.

## Contribution

It proves the non-classifiability of Gamma property II_1 factors by Borel invariants, extending to all full II_1 factors.

## Key findings

- No Borel measurable classification exists for Gamma II_1 factors.
- The non-classifiability extends to all full II_1 factors.
- Highlights the complexity of classifying von Neumann algebra invariants.

## Abstract

One of the earliest invariants introduced in the study of finite von Neumann algebras is the property Gamma of Murray and von Neumann. In this note we prove that it is not possible to classify separable $\rm{II}_1$ factors satisfying the property Gamma up to isomorphism by a Borel measurable assignment of countable structures as invariants. We also show that the same holds true for the full $\rm{II}_1$ factors.