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2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 110 / Issue 3 / June 2021
- Published online by Cambridge University Press:
- 24 July 2020, pp. 321-332
- Print publication:
- June 2021
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- Article
- Export citation
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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.
