The Kernels of Radical Homomorphisms and Intersections of Prime Ideals
نویسنده
چکیده
We establish a necessary condition for a commutative Banach algebra A so that there exists a homomorphism θ from A into another Banach algebra such that the prime radical of the continuity ideal of θ is not a finite intersection of prime ideals in A. We prove that the prime radical of the continuity ideal of an epimorphism from A onto another Banach algebra (or of a derivation from A into a Banach A-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 spaces Ω for which there exists a homomorphism from C0(Ω) into a radical Banach algebra whose kernel is not a finite intersection of prime ideals.
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