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Journal of the London Mathematical Society Advance Access originally published online on August 28, 2007
Journal of the London Mathematical Society 2007 76(1):165-180; doi:10.1112/jlms/jdm018
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© 2007 London Mathematical Society

Reduction of discontinuity for derivations on Fréchet algebras

Marc P. Thomas

Mathematics and Computer Science Departments
California State University, Bakersfield
9001 Stockdale Highway
Bakersfield, CA 93311-1022
USA

We consider strictly irreducible representations with which the discontinuity of a derivation on a (locally multiplicatively convex) Fréchet algebra must be associated. Only those strictly irreducible representations which are compatible with the topology of the algebra are considered. The main results show that when consideration is fixed upon each seminorm, the exceptional set of primitive ideals supporting the discontinuity must be a finite set, with each ideal being the kernel of some finite-dimensional irreducible representation. This result is the best possible, as can be seen by considering the radical Fréchet algebra constructed by Charles Read with identity adjoined which has a derivation with separating ideal that is the entire algebra, and one could take (countable) Fréchet products of his counterexample. It is also proved that derivations on commutative Fréchet algebras, the structure spaces of which are compact metric in the weak* topology, have only finitely many such exceptional points overall.

marc{at}cs.csubak.edu

2000 Mathematics Subject Classification Primary 46H05.

Received November 10, 2005; revised August 8, 2006; published online August 28, 2007.


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