  The index, which is given in terms of the number of zero modes of the Dirac
operator with definite chirality, plays a central role in various topological
aspects of gauge theories. We investigate its properties in non-commutative
geometry. As a simple example, we consider the U(1) gauge theory on a
discretized 2d non-commutative torus, in which general classical solutions are
known. For such backgrounds we calculate the index of the overlap Dirac
operator satisfying the Ginsparg-Wilson relation. When the action is small, the
topological charge defined by a naive discretization takes approximately
integer values, and it agrees with the index as suggested by the index theorem.
Under the same condition, the value of the index turns out to be a multiple of
N, the size of the 2d lattice. By interpolating the classical solutions, we
construct explicit configurations, for which the index is of order 1, but the
action becomes of order N. Our results suggest that the probability of
obtaining a non-zero index vanishes in the continuum limit, unlike the
corresponding results in the commutative space.
