SPATIAL DISCRETIZATION OF CUNTZ ALGEBRAS

摘要

The (abstract) Cuntz algebra O_N 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 O_N by the finite sections method. For we represent the Cuntz algebra as a (concrete) algebra of operators on l~2(Z~+) and associate with each operator A in this algebra the sequence AP_n) of its finite sections. The goal of this paper is to examine the structure of the C~*-algebra S(O_N) which is generated by all sequences of this form. Our main results are the fractality of a suitable restriction of the algebra S(O_N) 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 O_N.