QUADRATIC FUNCTIONAL EQUATIONS ASSOCIATED WITH BOREL FUNCTIONS AND MODULE ACTIONS

For a Borel function psi : R x R -> R satisfying the functional equation psi(s + t, u + v) + psi(s - t, u - v) = 2 psi(s, u) + 2 psi(t, v), we show that it satisfies the functional equation psi(s, t) = s(s - t)psi(1, 0) + st psi(1, 1) + t(t - s)psi(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-star-algebra.