2-Local derivations on matrix algebras over commutative regular algebras

TL;DR

This paper investigates 2-local derivations on matrix algebras over commutative regular algebras, establishing conditions for their existence and showing that all such derivations are actual derivations, with applications to operator algebras.

Contribution

It provides necessary and sufficient conditions for 2-local derivations to differ from derivations and proves that all 2-local derivations on these matrix algebras are derivations.

Findings

Conditions for 2-local derivations not to be derivations.

All 2-local derivations on matrix algebras over commutative regular algebras are derivations.

Applications to algebras of measurable operators in von Neumann algebras.

Abstract

The paper is devoted to 2-local derivations on matrix algebras over commutative regular algebras. We give necessary and sufficient conditions on a commutative regular algebra to admit 2-local derivations which are not derivations. We prove that every 2-local derivation on a matrix algebra over a commutative regular algebra is a derivation. We apply these results to 2-local derivations on algebras of measurable and locally measurable operators affiliated with type I von Neumann algebras.