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 $\varphi :A\times A\to X$ be a continuous bilinear map. We characterize the structure of $\varphi$ where it satisfies any of the following properties: $$a,b\in A,a{b}^{\star }=z(a^{\star }b=z)\Rightarrow \varphi (a,b^{\star })=\varphi (z,1)(\varphi (a^{\star },b)=\varphi (z,1));$$ $$a,b\in A,a{b}^{\star }=z(a^{\star }b=z)\Rightarrow \varphi (a,b^{\star })=\varphi (1,z)(\varphi (a^{\star },b)=\varphi (1,z)),$$ where $z\in A$ is fixed.

5 pages