Journal of Operator Theory

Volume 63, Issue 1, Winter 2010 pp. 3-46.

Traces on operator ideals and arithmetic means

Authors: Victor Kaftal (1) and Gary Weiss (2)
Author institution: (1) Department of Mathematics, University of Cincinnati, Cincinnati, OH, 45221-0025, USA
(2) Department of Mathematics, University of Cincinnati,\break Cincinnati, OH, 45221-0025, USA

Summary: We investigate the codimension of commutator spaces $[I, B(H)]$ of operator ideals on a separable Hilbert space, i.e., `How many traces can an ideal support?'' We conjecture that the codimension can be only zero, one, or infinity. The conjecture is proven for all ideals not contained in the largest arithmetic mean at infinity stable ideal and not containing the smallest am-stable ideal, for all soft-edged ideals (i.e., $I=se(I) = IK(H)$) and all soft-complemented ideals (i.e., $I=scI= I/K(H)$), which include most classical operator ideals. We apply some of the methods developed to two problems on elementary operators studied by V. Shulman.

Keywords: Traces, operator ideals, commutators, arithmetic mean

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