QUADRATIC FUNCTIONAL EQUATIONS ASSOCIATED WITH BOREL FUNCTIONS AND MODULE ACTIONS

Abstract. For a Borel function ψ: R×R → R satisfying the functional equation ψ(s + t, u + v) + ψ(s − t, u − v) = 2ψ(s, u) + 2ψ(t, v), we show that it satisfies the functional equation ψ(s, t) = s(s − t)ψ(1, 0) + stψ(1, 1) + t(t − s)ψ(0, 1). Using this, we prove the stability of the functional equation f(ax+ ay, bz + bw) + f(ax − ay, bz − bw) = 2abf(x, z) + 2abf(y, w) in Banach modules over a unital C?-algebra. 1