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

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.