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.