A result concerning additive functions in Hermitian Banach $^{\ast }$-algebras and an application

Abstract:

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