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.

1991 Mathematics Subject Classification: 46H15, 46H25, 16Nxx.