Stability of multipliers on Banach algebras

Suppose A is a Banach algebra without order. We show that an approximate multiplier T : A → A is an exact multiplier. We also consider an approximate multiplier T on a Banach algebra which need not be without order. If, in addition, T is approximately additive, then we prove the Hyers-Ulam-Rassias stability of T. 1. Introduction and statement of results. It seems that the stability problem of functional equations had been first raised by Ulam (cf. [5, Chapter VI] and [6]): for what metric groups G is it true that a ε-automorphism of G is necessarily near to a strict automorphism? An answer to the above problem has been given as follows. Suppose E 1 and E 2 are two real Banach spaces and f : E 1 → E 2 is a mapping. If there exist δ ≥ 0 and p ≥ 0, p = 1, such that