# 2-local $ (\upvarphi, \uppsi) $-Derivations on Finite von Neumann Algebras

## TL;DR

This paper introduces the concept of $(, \u03a8)$-finite von Neumann algebras and proves that under certain conditions, (approximately) 2-local $(, \u03a8)$-derivations on such algebras are actual $(, \u03a8)$-derivations.

## Contribution

The paper defines $(, \u03a8)$-finite von Neumann algebras and establishes that (approximately) 2-local $(, \u03a8)$-derivations are genuine $(, \u03a8)$-derivations under specific conditions.

## Key findings

- Introduction of $(, \u03a8)$-finite von Neumann algebras
- Characterization of 2-local $(, \u03a8)$-derivations
- Conditions ensuring 2-local derivations are actual derivations

## Abstract

In this paper, I introduce the concept of $ (\upvarphi, \uppsi) $-finite von Neumann algebras and I show that if $ \mathscr{M} $ is a finite and $ (\upvarphi, \uppsi) $-finite von Neumann algebra togather with condition $ \{ \big( \Delta(u+v)-\Delta(u)-\Delta(v)\big)^{*}\} \subseteq \uppsi(\mathscr{M}) $, then each (approximately) 2-local $ (\upvarphi, \uppsi) $-derivation $ \delta $ on $ \mathscr{M} $, is a $ (\upvarphi, \uppsi) $-derivation.