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On spectral approximation, Følner sequences and crossed products. (English) Zbl 1280.43001

Summary: 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 \(\Gamma\) with a concrete \(C^\ast\)-algebra \(\mathcal A\) with a Følner sequence. We also state a compatibility condition for the action of \(\Gamma\) on \(\mathcal A\). 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 \(C^\ast\)-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.

MSC:

43A45 Spectral synthesis on groups, semigroups, etc.
46L55 Noncommutative dynamical systems
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