I am interested in the relation between lattices and noncommutative algebras in the context of noncommutative geometry. In the commutative case, a lattice is a discrete subgroup of a locally compact abelian group, and it can be associated to a commutative C-algebra via the Pontryagin duality. Is there a similar correspondence in the noncommutative setting? More precisely, can one associate a lattice (or some generalization of it) to a noncommutative C-algebra or a von Neumann algebra? If so, what are the properties of such a lattice and how does it relate to the spectral or topological aspects of the noncommutative space?
I would appreciate any references or examples on this topic. Thank you.