Let ${\mathcal{A}}$ 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.

中文翻译：

---

具有最小左理想的半简单 BANACH 代数的 2-局部推导

让${\mathcal{A}}$是具有最小左理想的半简单 Banach 代数和$\text{soc}({\mathcal{A}})$成为${\mathcal{A}}$. 我们证明如果$\text{soc}({\mathcal{A}})$是一个基本的理想${\mathcal{A}}$，然后每个 2 局部推导${\mathcal{A}}$是一个推导。作为这个结果的应用，我们可以很容易地证明在一些代数上的每一个 2 局部推导，例如半单模湮灭 Banach 代数、强双三角形子空间格代数和${\数学{J}}$-子空间格代数，是一个推导。