On pseudo-amenability of Beurling algebras

Amenability and pseudo-amenability of $ \ell^{1}(S,\omega) $ is characterized, where $S$ is a left (right) zero semigroup or it is a rectangular band semigroup. The equivalence conditions to amenability of $\ell^{1}(S,\omega)$ are provided, where $ S $ is a band semigroup. For a locally compact group $G$, pseudo-amenability of $ \ell^{1}(G,\omega) $ is also discussed.