Bi-derivations and quasi-multipliers on module extensions Banach algebras

This paper characterize two bi-linear maps bi-derivations and quasi-multipliers on the module extension Banach algebra $A\oplus_1 X$, where $A$ is a Banach algebra and $X$ is a Banach $A$-module. Under some conditions, it is shown that if every bi-derivation on $A\oplus_1 A$ is inner, then the quotient group of bounded bi-derivations and inner bi-derivations, is equal to space of quasi-multipliers of $A$. Moreover, it is proved that $\mathrm{QM}(A \oplus_1 A)=\mathrm{QM}(A)\oplus (\mathrm{QM}(A)+\mathrm{QM}(A)')$, where $\mathrm{QM}(A)'=\{m\in \mathrm{QM}(A):m(0,a)=m(a,0)=0\}$.