Almost Commuting Elements in Non-Commutative Symmetric Operator Spaces

Almost Commuting Operators: John von Neumann, in his formulation of the uncertainty principle was the first to consider the classic problem which asks whether two "almost commuting" operators are small perturbations of commuting operators. Von Neumann, however, did not mention any norm. The first appearance of the problem in the literature is given by Rosenthal and Halmos. Although Rosenthal considers the problem with respect to the Hilbert-Schmidt norm, historically most of the results are with respect to the operator norm. In fact, the first positive result was proved by H. Lin with respect to the operator norm. In 2009 Lev Glebsky proved an analog of Lin's theorem with respect to the normalized Hilbert-Schmidt norm. Glebsky's result was refined by Filonov and Kachkovskiy. Motivated by these results, we consider a natural generalization of the problem. In particular, we look at the problem with respect to the normalized Schatten p-norms and have established several theorems that are analogs of Lin and Glebsky's work. Moreover, for p ≠ 2, the corresponding Schatten space is not a Hilbert space and our results use general Banach space techniques along with some tools from classical and harmonic analysis. For p = 2 we recover Filonov and Kachkovskiy's result with the same delta(epsilon) relationship. We have also extended our results to symmetric operator spaces which are spaces of measurable operators associated to a von Neumann algebra equipped with an (n.s.f) trace.