Joint Continuity of Bi-multiplicative Functionals

For Banach algebras A and B, we show that if U = A x B is unital, then each bi-multiplicative mapping from U 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 U is continuous if and only if it is continuous for the case n = 2.