On bilinear maps on Banach ⋆-algebra

Let A be a unital Banach ⋆-algebra, X be a Banach space and let ϕ : A A ! X be a continuous bilinear map. We consider ϕ with the property that ϕ(a; b⋆) = ϕ(z; 1)(ϕ(a; b⋆) = ϕ(1; z)) at following orthogonality conditions on elements of A: ab⋆ = z (a⋆b = z), and we characterize such bilinear maps.

Keywords:

Bilinear maps , orthogonality , Banach ⋆-algebra