Continuous bilinear maps on Banach $\star$-algebras

Abstract

Let $A$ be a unital Banach $\star$-algebra with unity $1$, $X$ be a Banach space and $\phi : A \times A \to X$ be a continuous bilinear map. We characterize the structure of $\phi$ where it satisfies any of the following properties: $$a,b \in A, \,\,\, a b^\star = z \, \,(a^\star b=z)\Rightarrow \phi ( a , b^\star ) = \phi ( z, 1 ) \, \, (\phi ( a^\star , b) = \phi ( z, 1 ));$$ $$a,b \in A, \,\,\, a b^\star = z \, \, (a^\star b=z)\Rightarrow \phi ( a , b^\star ) = \phi ( 1, z ) \, \, (\phi ( a^\star , b) = \phi ( 1, z )),$$ where $z\in A$ is fixed.