Let $ \mathcal{A} $ be a unital Banach $ \star $-algebra, $X$ be a Banach space and let $ \phi : \mathcal{A} \times \mathcal{A} \to X $ be a continuous bilinear map. We consider $ \phi $ with the property that $ \phi ( a, b^\star ) = \phi ( z, 1 ) (\phi ( a, b^\star ) = \phi ( 1, z ))$ at following orthogonality conditions on elements of $ \mathcal{A} $: $ a b^\star = z $ ($ a^\star b= z $), and we characterize such bilinear maps.
