  It is known that there exist an infinite number of inequivalent quantizations
on a topologically nontrivial manifold even if it is a finite-dimensional
manifold. In this paper we consider the abelian sigma model in (1+1) dimensions
to explore a system having infinite degrees of freedom. The model has a field
variable $ \phi : S^1 \to S^1 $. An algebra of the quantum field is defined
respecting the topological aspect of this model. A central extension of the
algebra is also introduced. It is shown that there exist an infinite number of
unitary inequivalent representations, which are characterized by a central
extension and a continuous parameter $ \alpha $ $ ( 0 \le \alpha < 1 ) $. When
the central extension exists, the winding operator and the zero-mode momentum
obey a nontrivial commutator.
