  The 1/N expansion in quantum field theory is formulated within an algebraic
framework. For a scalar field taking values in the $N$ by $N$ hermitian
matrices, we rigorously construct the gauge invariant interacting quantum field
operators in the sense of power series in 1/N and the `t Hooft coupling
parameter as members of an abstract *-algebra. The key advantages of our
algebraic formulation over the usual formulation of the 1/N expansion in terms
of Green's functions are (i) that it is completely local so that infra-red
divergencies in massless theories are avoided on the algebraic level and (ii)
that it admits a generalization to quantum field theories on general Lorentzian
curved spacetimes. We expect that our constructions are also applicable in
models possessing local gauge invariance such as Yang-Mills theories.
  The 1/N expansion of the renormalization group flow is constructed on the
algebraic level via a family of *-isomorphisms between the algebras of
interacting field observables corresponding to different scales. We also
consider $k$-parameter deformations of the interacting field algebras that
arise from reducing the symmetry group of the model to a diagonal subgroup with
$k$ factors. These parameters smoothly interpolate between situations of
different symmetry.
