  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 $\mathbb C$. 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.
