K-theory and perturbations of absolutely continuous spectra
Abstract
We study the K0 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 K0. 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.
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