On spectral approximation, F{\o}lner sequences and crossed products

In this article we study Foelner sequences for operators and mention their relation to spectral approximation problems. We construct a canonical Foelner sequence for the crossed product of a discrete amenable group $\mathrm{\Gamma }$ with a concrete C*-algebra A with a Foelner sequence. We also state a compatibility condition for the action of $\mathrm{\Gamma }$ on A. We illustrate our results with two examples: the rotation algebra (which contains interesting operators like almost Mathieu operators or periodic magnetic Schr"odinger operators on graphs) and the C*-algebra generated by bounded Jacobi operators. These examples can be interpreted in the context of crossed products. The crossed products considered can be also seen as a more general frame that included the set of generalized band-dominated operators.