On spectral approximation, Følner sequences and crossed products

In this article we study Følner sequences for operators and mention their relation to spectral approximation problems. We construct a canonical Følner sequence for the crossed product of a discrete amenable group ΓΓ with a concrete C∗-algebra AA with a Følner sequence. We also state a compatibility condition for the action of ΓΓ on AA. We illustrate our results with two examples: 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. 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.