  We prove that uniformly small short-range perturbations do not close the bulk
gap above the ground state of frustration-free quantum spin systems that
satisfy a standard local topological quantum order condition. In contrast with
earlier results, we do not require a positive lower bound for finite-system
Hamiltonians uniform in the system size. To obtain this result, we adapt the
Bravyi-Hastings-Michalakis strategy to the GNS representation of the
infinite-system ground state.
