Joint Continuity of Bi-multiplicative Functionals

For Banach algebras A and B, we show that if A=A×B is unital, then each bi-multiplicative mapping from A into a semisimple commutative Banach algebra D is jointly continuous. This conclusion generalizes  a famous result due to Silov, concerning the automatic continuity of homomorphisms between Banach algebras. We also prove that every n-bi-multiplicative functionals on A is continuous if and only if it is continuous for the case n=2.