Quasi-multipliers on Banach algebras related to locally compact semigroups

Abstract

Let ${\mathcal {S}}$ be a compactly cancellative foundation semigroup with identity and $M_a({\mathcal {S}})$ be its semigroup algebra. In this paper, we give some characterizations for ${{\mathfrak {Q}}}{{\mathfrak {M}}}(M_a({\mathcal {S}}))$, the quasi-multipliers of $M_a({\mathcal {S}})$. It is shown that ${{\mathfrak {Q}}}{{\mathfrak {M}}}(M_a({\mathcal {S}}))$ may be identified by $M({\mathcal {S}})$. We deal with the quasi-multipliers on the dual Banach algebra $L_{0}^{\infty }({\mathcal {S}};M_a({\mathcal {S}}))$ and prove that its quasi-multipliers is again $M({\mathcal {S}})$. We also discuss the bilinear mappings ${\mathfrak {m}} :M_a({\mathcal {S}})^{*} \times M_a({\mathcal {S}})^{*} \longrightarrow M_a({\mathcal {S}})^{*}$ which commutes with translations and convolutions.