2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

Abstract

AbstractLet${\mathcal{A}}$be a semisimple Banach algebra with minimal left ideals and$\text{soc}({\mathcal{A}})$be the socle of${\mathcal{A}}$. We prove that if$\text{soc}({\mathcal{A}})$is an essential ideal of${\mathcal{A}}$, then every 2-local derivation on${\mathcal{A}}$is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and${\mathcal{J}}$-subspace lattice algebras, is a derivation.