  We study symmetric diffusion operators on metric measure spaces. Our main
question is whether or not the restriction of the operator to a suitable core
continues to be essentially self-adjoint or $L^p$-unique if a small closed set
is removed from the space. The effect depends on how large the removed set is,
and we provide characterizations of the critical size in terms of capacities
and Hausdorff dimension. As a key tool we prove a truncation result for
potentials of nonnegative functions. We apply our results to Laplace operators
on Riemannian and sub-Riemannian manifolds and on metric measure spaces
satisfying curvature dimension conditions. For non-collapsing Ricci limit
spaces with two-sided Ricci curvature bounds we observe that the self-adjoint
Laplacian is already fully determined by the classical Laplacian on the regular
part.
