Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras

We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. Using a signum effectively, we can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras. As a direct corollary, we show that to each approximate Jordan homomorphism[inline-graphic not available: see fulltext] from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to[inline-graphic not available: see fulltext].