  We study Bakry-\'Emery curvature-dimension inequalities for non-local
operators on the one-dimensional lattice and prove that operators with finite
second moment have finite dimension. Moreover, we show that a class of
operators related to the fractional Laplacian fails to have finite dimension
and establish both positive and negative results for operators with sparsely
supported kernels. Moreover, a large class of operators is shown to have no
positive curvature. The results correspond to CD inequalities on locally
infinite graphs.
