Characterization of $2$-local derivations and local Lie derivations on some algebras

We prove that each $2$-local derivation from the algebra $M_n(\mathscr A)$ ($n>2$) into its bimodule $M_n(\mathscr M)$ is a derivation, where $\mathscr A$ is a unital Banach algebra and $\mathscr M$ is a unital $\mathscr A$-bimodule such that each Jordan derivation from $\mathscr A$ into $\mathscr M$ is an inner derivation, and that each $2$-local derivation on a $C^*$-algebra with a faithful traceable representation is a derivation. We also characterize local and $2$-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, the Jiang–Su algebra, and UHF algebras.