On pseudo-amenability of Beurling algebras

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