  For a real abelian field and for an odd prime p splitting in the field, we
study a map between the p-parts of the class group and of the quotient of units
modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of
the field. We determine the kernel and the cokernel of this map assuming
Greenberg's conjecture on the vanishing of the lambda-invariant of the
extension.
