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Fractal algebras of discretization sequences

These are the lecture notes for a course at the Summer School on "Applied Analysis" at the Technical University Chemnitz in September 2011. We start with the definition of a fractal algebra and show that the fractal property is enormously useful for several spectral approximation problems, e.g. for the convergence of spectra. These results will be illustrated by sequences in the algebra of the finite sections method for Toeplitz operators. Then we discuss some structural consequences of fractality, which are related with the notion of a compact sequence. Discretized Cuntz algebras will show that idea of fractality is also a very helpful guide in order to analyze concrete algebras of approximation sequences, which illustrates the importance of the idea of {\em fractal restriction}. Our final example is the algebra of the finite sections method for band operators. This algebra is not fractal, but has a related property which we call {\em essential fractality} and which is related with the approximation of points in the essential spectrum.