\title{Commutative Banach Algebras \\ and Modular Representation Theory}
\author{David J. Benson}
\address{Institute of Mathematics, University of Aberdeen, Aberdeen
  AB24 3UE, Scotland, United Kingdom}

\begin{abstract}
In a recent paper of Benson and Symonds,
a new invariant was introduced for modular representations of a finite
group. An interpretation was given as a spectral radius with respect
to a Banach algebra completion of the representation ring.
Our purpose here is to take these notions further, and investigate
the structure of the resulting Banach algebras.
Some of the material in that paper is repeated
here in greater generality, and for clarity of exposition.

We give an axiomatic definition of an abstract representation ring, 
and representation ideal. The completion is then a commutative
Banach algebra, and the techniques of Gelfand from the 1940s are
applied in order to study the space of algebra homomorphisms to $\bC$.
One surprising consequence of this investigation 
is that the Jacobson radical and the nil
radical of a (complexified) representation ring always coincide.

These notes are intended for representation theorists. So background
material on commutative Banach algebras is given in detail, whereas
representation theoretic background is more condensed.
\end{abstract}