Derivations, local and 2-local derivations of standard operator algebras

TL;DR

This paper investigates derivations in standard operator algebras on Banach spaces, proving that local and 2-local derivations are actually derivations, thus extending classical results in operator algebra theory.

Contribution

It extends classical results by showing that local and 2-local derivations on standard operator algebras are necessarily derivations.

Findings

Every derivation from A into B(X) is inner.

Every local derivation on B(X) is a derivation.

Every 2-local derivation from A into B(X) is a derivation.

Abstract

Let X be a Banach space over field F (R or C). Denote by B(X) the set of all bounded linear operators on X and by F(X) the set of all finite rank operators on X. A subalgebra A of B(X) is called a standard operator algebra if A contain F(X). We give a brief proof of a well-known result that every derivation from A into B(X) is inner. There is another classical result that every local derivation on B(X) is a derivation. We extend the result by proving that every local derivation from A into B(X) is a derivation. Based on these two results, we prove that every 2-local derivation from A into B(X) is a derivation.