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

TL;DR

This paper explores spectral approximation via Følner sequences in operator algebras, constructing canonical sequences for crossed products of amenable groups with C*-algebras, with applications to rotation and Jacobi operator algebras.

Contribution

It introduces a canonical Følner sequence for crossed products of amenable groups with C*-algebras and discusses compatibility conditions for group actions.

Findings

Constructed a canonical Følner sequence for crossed products.

Illustrated results with examples like rotation algebra and Jacobi operators.

Connected crossed products to generalized band-dominated operators.

Abstract

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 $\Gamma$ with a concrete C*-algebra A with a Foelner sequence. We also state a compatibility condition for the action of $\Gamma$ 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.