  We study disordered topological insulators with time reversal symmetry.
Relying on the noncommutative index theorem which relates the Chern number to
the projection onto the Fermi sea and the magnetic flux operator, we give a
precise definition of the $Z_2$ index which is a noncommutative analogue of the
Atiyah-Singer $Z_2$ index. We prove that the noncommutative $Z_2$ index is
robust against any time-reversal symmetric perturbation including disorder
potentials as long as the spectral gap at the Fermi level does not close.
