Joint Continuity of Bi-multiplicative Functionals

Abstract

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 $n=2$.