Spatial discretization of Cuntz algebras

TL;DR

This paper investigates a finite-dimensional approximation of the Cuntz algebra using the finite sections method, analyzing the structure, stability, and spectral properties of the resulting sequence algebra.

Contribution

It introduces a spatial discretization approach for the Cuntz algebra, establishing fractality and stability criteria for the associated sequence algebra.

Findings

Proves fractality of a restricted sequence algebra.

Provides a necessary and sufficient condition for sequence stability.

Analyzes spectral and pseudospectral approximations of Cuntz algebra elements.

Abstract

The (abstract) Cuntz algebra is generated by non-unitary isometries and has therefore no intrinsic finiteness properties. To approximate the elements of the Cuntz algebra by finite-dimensional objects, we thus consider a spatial discretization of this algebra by the finite sections method. For we represent the Cuntz algebra as a (concrete) algebra of operators on a Hilbert space and associate with each operator in this algebra the sequence of its finite sections. The goal of this paper is to examine the structure of the $C^*$-algebra which is generated by all sequences of this form. Our main results are the fractality of a suitable restriction of this sequence algebra and a necessary and sufficient criterion for the stability of sequences in the restricted algebra. These results are employed to study spectral and pseudospectral approximations of elements of the Cuntz algebra.