Author: Roch, Steffen, Technical University Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, Germany (roch@mathematik.tu-darmstadt.de).
Spatial discretization of Cuntz algebras, pp. 1097-1132.
ABSTRACT. The (abstract) Cuntz algebra is generated by non-unitary isometries and has therefore no intrinsic finiteness properties. To approximate its elements by finite-dimensional objects, we thus consider a spatial discretization of the Cuntz algebra by the finite section method. For we represent the Cuntz algebra as a (concrete) algebra of operators on a Hilbert space, choose a suitable basis, and associate with each operator in the Cuntz algebra the sequence of its finite sections with respect to the chosen basis. The goal of this paper is to examine the structure of the algebra which is generated by all sequences of this form. Our main results are the fractality of a suitable restriction of this algebra and a necessary and sufficient criterion for the stability of sequences in the restricted algebra. These results are employed to study spectral and pseudo-spectral approximations of elements of the Cuntz algebra.