Abstract
We prove that a commutative unital Banach algebra which is a valuation ring must reduce to the field of complex numbers, which implies that every homomorphism from l ∞ onto a Banach algebra is continuous. We show also that if b ϵ [ b Rad B] − for some nonnilpotent element b of the radical of a commutative Banach algebra B, then the set of all primes of B cannot form a chain, and we deduce from this result that every homomorphism from b (K) onto a Banach algebra is continuous.
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