Some results concerning certain functional equations in complex Banach algebras are presented. One of these results is used to prove the following result:
Let A be a Banach *-algebra with identity e and let X be a vector space which is also a unitary left A-module. Suppose there exists a mapping \(Q: X\to A\) with the properties
(i) \(Q(x+y)+Q(x-y)=2Q(x)+2Q(y)\) for all pairs x,y\(\in X,(ii)\) \(Q(ax)=aQ(x)y^*\) for all \(x\in X\) and all normal invertible \(a\in A.\)
Under these conditions for the mapping B(.,.): \(X\times X\to A\) defined by the relation \[ B(x,y) = 1/4(Q(x+y)-Q(x-y))\quad +\quad i/4(Q(x+iy)- Q(x-iy)) \] the following statements are fulfilled:
1) B(.,.) is additive in both arguments;
2) \(B(ax,y)=aB(x,y)\), \(B(x,ay)=B(x,y)a^*\) for all pairs x,y\(\in X\) and all \(a\in A;\)
3) \(Q(x)=B(x,x)\) for all \(x\in X.\)
The result above is an abstract generalization of the classical Jordan- Neumann characterization of pre-Hilbert space. If A is the complex number field, then the result above reduces to a result first proved by S. Kurepa.