On Maximal Regular Ideals and Identities in the Tensor Product of Commutative Banach Algebras

Let A1 and A2 be commutative Banach algebras and A1 ⊙ A2 their algebraic tensor product over the complex numbers C.There is always a t least one norm, namely the greatest cross-norm γ (2), on A1 ⊙ A2 that renders it a normed algebra. We shall write A 1 ⊗α A2 for the α-completion of A1 ⊙ A2 when αis an algebra norm on A1 ⊙ A2.Gelbaum (2; 3), Tomiyama (9), and Gil de Lamadrid (4) have shown that for certain algebra norms α on A1 ⊙ A2 every complex homomorphism on A1 ⊙ A2 is α-continuous. In § 3 of this paper, we present a condition on an algebra norm α which is equivalent to the α-continuity of every complex homomorphism on A1 ⊙ A2.