Improper filtrations for C*-algebras: spectra of unilateral tridiagonal   operators
  We extend the results of our previous paper "C*-algebras and numerical linear algebra" to cover the case of "unilateral" sections. This situation bears a close resemblance to the case of Toeplitz operators on Hardy spaces, in spite of the fact that the operators here are far from Toeplitz operators. In particular, there is a short exact sequence 0 --> K --> A --> B --> 0 whose properties are essential to the problem of computing the spectra of self adjoint operators. 
William Arveson