Spatial discretization of Cuntz algebras

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.