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K-Theory and Asymptotically Commuting Matrices

Published online by Cambridge University Press:  20 November 2018

Terry A. Loring*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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To shed light on the following unsolved problem, several authors have considered related problems. The problem is that of finding commuting approximants to pairs of asymptotically commuting self-adjoint matrices:

Suppose that Hn and Kn are self-adjoint matrices of dimension m(n), with ║Hn║, ║Kn║ ≦ 1, which commute asymptotically in the sense that

Must there then exist commuting self-adjoint matrices H′n and K′n for which

One may alter the conditions imposed on Hn and Kn, for example, by requiring Hn to be normal and Kn to be self-adjoint, and ask whether commuting approximants H′n and K′n can be found satisfying the same conditions. Some of these related problems have been solved. This paper will examine their solutions from a K-theoretic point of view, illustrating the difficulty inherent in modifying them to work for the original problem.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1988

References

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9. Voiculescu, D., Asymptotically commuting finite rank unitary operators without commuting approximants, Acta Sci. Math.. 45 (1983), 429431.Google Scholar
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