2-Local and local derivations on Jordan matrix rings over commutative involutive rings

TL;DR

This paper proves that 2-local inner derivations on Jordan rings of self-adjoint matrices over commutative involutive rings are actual derivations, extending to infinite-dimensional cases and spatial derivations.

Contribution

It establishes that all 2-local and local derivations in these Jordan algebras are genuine derivations, providing new insights into their structure.

Findings

Every 2-local inner derivation on the Jordan ring is a derivation.

Every 2-local spatial derivation on infinite-dimensional Jordan algebras is a spatial derivation.

Every local spatial derivation on these Jordan algebras is a derivation.

Abstract

In the present paper we prove that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is a derivation. We also apply our technique to various Jordan algebras of infinite dimensional self-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. It is also proved that every local spatial derivation on the same Jordan algebras is a derivation.