Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(4): 1101-1110

Online first article October 11, 2023      Printed October 31, 2023

Copyright © The Korean Mathematical Society.

Approximately local derivations on $\ell^{1}$-Munn algebras with applications to semigroup algebras

Ahmad Alinejad, Morteza Essmaili, Hatam Vahdati

University of Tehran; Kharazmi University; Kharazmi University


At the present paper, we investigate bounded approximately local derivations of $\ell^{1}$-Munn algebra ${\mathbb M}_{I}(\mathcal{A})$, where $I$ is an arbitrary non-empty set and $\mathcal A$ is an approximately locally unital Banach algebra. Indeed, we show that if ${_\mathcal A}B(\mathcal A ,{\mathcal A}^{\ast})$ and $B_{\mathcal A}(\mathcal A ,{\mathcal A}^{\ast})$ are reflexive, then every bounded approximately local derivation from ${\mathbb M}_{I}(\mathcal A)$ into any Banach ${\mathbb M}_{I}(\mathcal A)$-bimodule $ X$ is a derivation. Finally, we apply this result to study bounded approximately local derivations of the semigroup algebra $\ell^{1}(S)$, where $S$ is a uniformly locally finite inverse semigroup.

Keywords: Approximately local derivation, $\ell^{1}$-Munn algebra, inverse semigroup

MSC numbers: 47B47, 43A20