University of Georgia
Mathematics Department Colloquium
Gary Weiss
University of Cincinnatti
November 12, 1998
Elementary sequence and series phenomena with applications to
commutators and ideals in B(H)
Abstract: I will give a brief survey of the history of B(H)-ideals and
B(H)-commutators including the initial elementary ideas that led to
the birth of the study of commutator ideals in the 1970's and some
important elementary open problems. B(H) denotes the bounded linear
operators on a separable, infinite-dimensional, complex Hilbert space
and a B(H)-commutator is an operator of the form AB-BA with A and B
from B(H).
Much of the important work on commutators in recent years
centers around the classification of the (I,J)-commutators of B(H)
(when A,B come from, repectively, two-sided ideals I,J), and the
classification for their linear span [I,J]. The premier result in the
subject was N.J. Kalton's 1988 characterization of [I,J] in the case I
= the trace class and J = B(H) and the case where I = J are the
Hilbert-Schmidt class.
From work joint with K. Dykema, T. Figiel and M. Wodzicki, I
will present a complete generalization of Kalton's characterization to
arbitrary ideals I,J of B(H) and illustrate the role these results
play in the existence of unitarily invariant traces on ideals. For
instance, one surprise is that the standard trace on the trace class
is extendable to a unitarily invariant trace on a strictly larger
ideal.
Some useful new tidbits will be given along the way with a
brief presentation of some new ideal properties arising from the study
of commutators such as a notion of infinite convexity.