The Role of C∗-algebras in Infinite Dimensional Numerical Linear Algebra

This paper deals with mathematical issues relating to the computation of spectra of self adjoint operators on Hilbert spaces. We describe a general method for approximating the spectrum of an operator A using the eigenvalues of large finite dimensional truncations of A. The results of several papers are summarized which imply that the method is effective in most cases of interest. Special attention is paid to the Schrödinger operators of one-dimensional quantum systems. We believe that these results serve to make a broader point, namely that numerical problems involving infinite dimensional operators require a reformulation in terms of C∗-algebras. Indeed, it is only when the given operator A is viewed as an element of an appropriate C∗-algebra A that one can see the precise nature of the limit of the finite dimensional eigenvalue distributions: the limit is associated with a tracial state on A. For example, in the case where A is the discretized Schrödinger operator associated with a one-dimensional quantum system, A is a simple C∗-algebra having a unique tracial state. In these cases there is a precise asymptotic result. 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05.