Improper Filtrations for C-algebras: Spectra of Unilateral Tridiagonal Operators

Let A ⊆ B(H) be a C∗-algebra of operators and let P1 ≤ P2 ≤ . . . be an increasing sequence of finite dimensional projections in B(H). In a previous paper [3] we developed methods for computing the spectrum of self adjoint operators T ∈ A in terms of the spectra of the associated sequence of finite dimensional compressions PnTPn. In a suitable context, we showed that this is possible when Pn increases to 1. In this paper we drop that hypothesis and obtain an appropriate generalization of the main results of [3]. Let P+ = limn Pn, H+ = P+H. The set A+ ⊆ B(H+) of all compact perturbations of operators P+T ↾H+ , T ∈ A, is a C ∗-algebra which is somewhat analogous to the Toeplitz C∗-algebra acting on H. Indeed, in the most important examples A is a simple unital C∗-algebra having a unique tracial state, the operators in A are “bilateral”, those in A+ are “unilateral”, and there is a short exact sequence of C ∗-algebras 0 → K → A+ → A → 0 whose features are central to this problem of approximating spectra of operators in A in terms of the eigenvalues of their finite dimensional compressions along the given filtration. This work was undertaken in order to develop an efficient method for computing the spectra of discretized Hamiltonians of one dimensional quantum systems in terms of “unilateral” tridiagonal n×n matrices. The solution of that problem is presented in Theorem 3.4. 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05.