K-Theory and Asymptotically Commuting Matrices
Terry A. Loring · Canadian Journal of Mathematics · 1988
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.