A result concerning additive functions in Hermitian Banach *-algebras and an application

Abstract

Let A \mathcal {A} be a complex hermitian Banach ∗ * -algebra with an identity element e e . Suppose there exists an additive function f : A → A f:\mathcal {A} \to \mathcal {A} such that f ( a ) = − a ∗ a f ( a − 1 ) f\left ( a \right ) = - {a^ * }af\left ( {{a^{ - 1}}} \right ) holds for all normal invertible elements a ∈ A a \in \mathcal {A} . We prove that in this case f f is of the form f ( a ) = f ( i e ) k f\left ( a \right ) = f\left ( {ie} \right )k , where a = h + i k a = h + ik . Using this result we generalize S. Kurepa’s extension of Jordan-Neumann characterization of pre-Hilbert space.