ON CENTRALIZERS OF SEMISIMPLE $H^{\ast}−$ALGEBRAS

Abstract

In this paper we prove the following result. Let $A$ be a semisimple $H^{\ast}-$algebra and let $T: A \rightarrow A$ be an additive mapping satisfying the relation $2T(x^{m+n+1}) = x^{m} T(x) x^{n} + x^{n} T(x) x^{m}$, for all $x \in A$ and some nonnegative integers $m,n$ such that $ m+n \neq 0$. In this case $T$ is a left and a right centralizer.