The kernels of radical homomorphisms and intersections of prime ideals

Pham, Hung

doi:10.1090/S0002-9947-07-04325-5

The kernels of radical homomorphisms and intersections of prime ideals
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Trans. Amer. Math. Soc. 360 (2008), 1057-1088 Request permission

Abstract:

We establish a necessary condition for a commutative Banach algebra $A$ so that there exists a homomorphism $\theta$ from $A$ into another Banach algebra such that the prime radical of the continuity ideal of $\theta$ 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 $\Omega$ 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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