Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

The kernels of radical homomorphisms and intersections of prime ideals

Author(s): Hung Le Pham
Journal: Trans. Amer. Math. Soc. 360 (2008), 1057-1088.
MSC (2000): Primary 46H40, 46J10; Secondary 46J05, 13C05, 43A20
Posted: July 23, 2007
MathSciNet review: 2346483
Retrieve article in: PDF

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Abstract: We establish a necessary condition for a commutative Banach algebra so that there exists a homomorphism $\theta$ from into another Banach algebra such that the prime radical of the continuity ideal of $\theta$ is not a finite intersection of prime ideals in . We prove that the prime radical of the continuity ideal of an epimorphism from onto another Banach algebra (or of a derivation from into a Banach -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 $\Omega$ for which there exists a homomorphism from $\mathcal C_0(\Omega)$ into a radical Banach algebra whose kernel is not a finite intersection of prime ideals.

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