# 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

## Abstract

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.

## Addendum

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.

## Availability

Reprints of the paper available on request: e-mail P.Dixon with extension
@sheffield.ac.uk. A copy of the preprint in PDF form is available here.

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