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On spectral approximation, Følner 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 $Γ$ with a concrete C*-algebra A with a Foelner sequence. We also state a compatibility condition for the action of $Γ$ on A. 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.