Abstract
Motivated by construction of states in Lattice-models in Statistical Physics, Folner-nets(sequences) were introduced in the study of amenability of (infinite) discrete groups. Similar structures were introduced by Arveson to study spectral approximation for a class of bounded self-adjoint operators, leading to Szego-type limit theorems. We take this idea further by introducing Folner sequences relative to a suitably chosen unbounded self-adjoint operator, to expand the class of operators that can give rise to Szego-type spectral limit theorems. This allows the study of the spectral approximation for the Weyl-Segal algebra.