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Summary: THE ROLE OF C
-ALGEBRAS IN INFINITE
DIMENSIONAL NUMERICAL LINEAR ALGEBRA
William Arveson
Department of Mathematics
University of California
Berkeley, CA 94720 USA
6 June, 1993
Abstract. 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 Schršodinger 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 Schršodinger operator
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