Published online by Cambridge University Press: 16 September 2019
Let $S$ be a discrete inverse semigroup,
$l^{1}(S)$ the Banach semigroup algebra on
$S$ and
$\mathbb{X}$ a Banach
$l^{1}(S)$-bimodule which is an
$L$-embedded Banach space. We show that under some mild conditions
${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result.