Let X be a compact subset of
the complex plane and let A be an algebra of functions analytic near X which
contains the polynomials and is complete in its natural topology. This paper is
concerned with determining the spectrum of A and describing A in terms of its
spectrum. It is shown that the spectrum of A is formed from the disjoint union of
certain compact subsets of C (suitably topologized) by making certain identifications.
A is closed under differentiation exactly when no identifications need be performed,
and then A admits a simple, complete description. In particular, if X is connected,
then the completion of A is merely the restriction to X of the algebra of all
functions analytic near the union of X with some of the bounded components of
C − X.
