In this paper some results concerning certain functional equations in complex Banach algebras are presented. One of these results is used to prove an abstract generalization of the classical Jordan-Neumann characterization of pre-Hilbert space. This paper is a continuation of our earlier work [6, 7] . Throughout this paper all Banach algebras and vector spaces are over the complex field C. Our terminology and notation will be the same as in [7] . For Banach algebras and Banach ^-algebras we

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... r to [1 and 4]. Our first few results characterize some additive functions. THEOREM 1. Let A be a Banach *-algebra with identity e. Let X and p be automorphisms or antiautomorphisms (i.e. X(ab) = A(6)A(a)) of A (any combination is allowed). If f : A -* A is an additive function such that f(a) = X(a)f(a~1)p(a) for all normal invertible elements a of A, then 2f(b) = X(b)f(e) + f(e)p(b) for all bGA. PROOF. Let us first assume that for the function / (1) /to = 0 holds and let us prove that in this case (2) /(o) = 0 is fulfilled for all o G A. Since / is by the assumption additive, (2) will be proved if we prove that (2) holds for all normal elements. Therefore let a G A be an arbitrary normal element. One can choose rational numbers p and q such that ó-1 and (e -6)_1 exist, where b = pe + qa. Hence f(a) = 0 will be proved if we prove that f(b) = 0. Now according to the requirements of the theorem and (1) we have f{b) = A(6)/(6-1)/i(ft) = X{b)f(b-\e -b))p(b) = X(b)X(b-x(e -b))f(e -b)-1b)p(b~1(e -b))p(b) = X(e-b)f((e-by1-e)p(e-b) = X(e -b)X((e -b)'1)f(e -b)p((e -b)-1)p(e -b) _ = -/(*>)■