  We show that noncommutative U(r) gauge theories with a chiral fermion in the
adjoint representation can be constructed on the lattice with manifest
star-gauge invariance in arbitrary even dimensions. Chiral fermions are
implemented using a Dirac operator which satisfies the Ginsparg-Wilson
relation. A gauge-invariant integration measure for the fermion fields can be
given explicitly, which simplifies the construction as compared with lattice
chiral gauge theories in ordinary (commutative) space-time. Our construction
includes the cases where continuum calculations yield a gauge anomaly. This
reveals a certain regularization dependence, which is reminiscent of parity
anomaly in commutative space-time with odd dimensions. We speculate that the
gauge anomaly obtained in the continuum calculations in the present cases can
be cancelled by an appropriate counterterm.
