Spectral characterization of sums of commutators II

Abstract

For countably generated ideals, \Jc, of B(\Hil), geometric stability is necessary for the canonical spectral characterization of sums of (\Jc,B(\Hil))--commutators to hold. This answers a question raised by Dykema, Figiel, Weiss and Wodzicki. There are some ideals, \Jc, having quasi--nilpotent elements that are not sums of (\Jc,B(\Hil))--commutators. Also, every trace on every geometrically stable ideal is a spectral trace