  Holonomy algebras arise naturally in the classical description of Yang-Mills
fields and gravity, and it has been suggested, at a heuristic level, that they
may also play an important role in a non-perturbative treatment of the quantum
theory. The aim of this paper is to provide a mathematical basis for this
proposal.
  The quantum holonomy algebra is constructed, and, in the case of real
connections, given the structure of a certain C-star algebra. A proper
representation theory is then provided using the Gel'fand spectral theory. A
corollory of these general results is a precise formulation of the ``loop
transform'' proposed by Rovelli and Smolin. Several explicit representations of
the holonomy algebra are constructed. The general theory developed here implies
that the domain space of quantum states can always be taken to be the space of
maximal ideals of the C-star algebra. The structure of this space is
investigated and it is shown how observables labelled by ``strips'' arise
naturally.
