On spectral approximation, Følner sequences and crossed products

In this article we review first some of the possibilities in which the notions of Følner sequences and quasidiagonality have been applied to spectral approximation problems. We construct then a canonical Følner sequence for the crossed product of a concrete C∗-algebra and a discrete amenable group. We apply our results to the rotation algebra (which contains interesting operators like almost Mathieu operators or periodic magnetic Schrödinger operators on graphs) and the C∗-algebra generated by bounded Jacobi operators.