The kernels of radical homomorphisms and intersections of prime ideals

Abstract

We establish a necessary condition for a commutative Banach algebraAAso that there exists a homomorphismθ\thetafromAAinto another Banach algebra such that the prime radical of the continuity ideal ofθ\thetais not a finite intersection of prime ideals inAA. We prove that the prime radical of the continuity ideal of an epimorphism fromAAonto another Banach algebra (or of a derivation fromAAinto a BanachAA-bimodule) is always a finite intersection of prime ideals. Under an additional cardinality condition (and assuming the Continuum Hypothesis), this necessary condition is proved to be sufficient. En route, we give a general result on norming commutative semiprime algebras; extending the class of algebras known to be normable. We characterize those locally compact metrizable spacesΩ\Omegafor which there exists a homomorphism fromC0(Ω)\mathcal C_0(\Omega )into a radical Banach algebra whose kernel is not a finite intersection of prime ideals.