K-theory and perturbations of absolutely continuous spectra

TL;DR

This paper investigates the K_0 group of commutants of commuting Hermitian operators, revealing the significance of the absolutely continuous spectrum and establishing new structural results under various ideal and continuity assumptions.

Contribution

It provides new insights into the K-theoretic structure of operator commutants, especially highlighting the role of absolute continuity and extending results to higher-dimensional operator tuples.

Findings

In the case n=1, results are comprehensive under certain conditions.

For n>2 and Lorentz (n,1) ideal, the commutant determines a canonical direct summand in K_0.

Weaker properties involving the compact ideal are established under finiteness conditions.

Abstract

We study the K_0 group of the commutant modulo a normed ideal of an n-tuple of commuting Hermitian operators in some of the simplest cases. In case n=1, the results, under some technical conditions are rather complete and show the key role of the absolutely continuous part when the ideal is the trace-class. For a commuting n-tuple, n>2 and the Lorentz (n, 1) ideal, we show under an absolute continuity assumption that the commutant determines a canonical direct summand in K_0. Also, certain properties involving the compact ideal, established assuming quasicentral approximate units mod the normed ideal, have weaker versions which hold assuming only finiteness of the obstruction to quasicentral approximate units.