Topologically irreducible representations and radicals in Banach algebras

by P.G.Dixon

Proc. London Math. Soc. (3), 74 (1997), 174-200.

1991 Mathematics Subject Classification: 46H15, 46H25


It is shown that the topologically irreducible representations of a normed algebra define a certain topological radical in the same way that the strictly irreducible representations define the Jacobson radical and that this radical can be strictly smaller than the Jacobson radical. An abstract theory of `topological radicals' in topological algebras is developed and used to relate this radical to the Baer radical (prime radical). The relations with topologically transitive representations and standard representations in the sense of Meyer are also explored.


Since this paper was written, Douglas Somerset (University of Aberdeen) has shown that every 2-TT representation of a complex PI-algebra A on a normed space is finite-dimensional. This is a substantial improvement on Theorem 5.9, and the proof is shorter.


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