  We present a possible definition of a mobility gap for a many-body quantum
system, in analogy to definitions of dynamical localization for single particle
systems. Using this definition, we construct "corrected" quasi-adiabatic
continuation operators. Under an appropriate definition of a unique ground
state, we show how to introduce virtual fluxes. Armed with these results, we
can directly carry over previous results in the case of a spectral gap. We
present a proof of decay of correlation functions and we present a proof of
Hall conductance quantization under very mild density-of-states assumptions
defined later. We also generalize these definitions to the case of a "bulk
mobility gap", in the case of a system with boundaries, and present a proof of
Hall conductance quantization on an annulus under appropriate assumptions.
  Further, we present a new "optimized" quasi-adiabatic continuation operator
which simplifies previous estimates and tightens bounds in certain cases. This
is presented in an appendix which can be read independently of the rest of the
paper as it also improves estimates in the case of systems with a spectral gap.
This filter function used decays in time at least as fast as ${\cal
O}(\exp(-t^\alpha))$ for all $\alpha<1$, a class of decay called subexponential
(a tighter description of what is possible is below). Using this function it is
possible to tighten recent estimates of the Hall conductance quantization for
gapped systems\cite{hall} to a decay which is subexponential in system size.
