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

The $Z_2$ Index of Disordered Topological Insulators with Time Reversal Symmetry