  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.
