Description of 2-local derivations and automorphisms on finite dimensional Jordan algebras

TL;DR

This paper studies 2-local derivations and automorphisms on finite-dimensional Jordan algebras, proving that under certain conditions they are equivalent to derivations and automorphisms, respectively.

Contribution

It introduces the concept of 2-local maps on vector spaces and establishes conditions under which these maps are linear and coincide with derivations or automorphisms in Jordan algebras.

Findings

Every 2-local derivation on finite-dimensional formally real Jordan algebras is a derivation.

Every 2-local 1-automorphism of these Jordan algebras is an automorphism.

A sufficient condition for the linearity of 2-local maps on finite-dimensional vector spaces.

Abstract

In the present paper we introduce and investigate the notion of 2-local linear map on vector spaces. A sufficient condition is obtained for linearity of a 2-local linear map on finite dimensional vector spaces. Based on this result we prove that every 2-local derivation on a finite dimensional formally real Jordan algebra is a derivation. Also we show that every 2-local 1-automorphism (i.e. implemented by single symmetries) of mentioned Jordan algebras is an automorphism.