Joint Continuity of Bi-multiplicative Functionals

For Banach algebras \mathcal{A} and \mathcal{B}, we show that if \mathfrak{A}=\mathcal{A}\times \mathcal{B} is unital, then each bi-multiplicative mapping from \mathfrak{A} into a semisimple commutative Banach algebra \mathcal{D} is jointly continuous. This conclusion generalizes  a famous result due to\check{\text{S}}ilov, concerning the automatic continuity of homomorphisms between Banach algebras. We also prove that every n-bi-multiplicative functionals on \mathfrak{A} is continuous if and only if it is continuous for the case n=۲. 

Keywords:

Jointly continuous , Bi-multiplicative functional , Almost bi-multiplicative