Traces of compact operators and the noncommutative residue
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2013
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Kalton (late), N.
Lord, S.
Potapov, D.
Sukochev, F.
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Advances in Mathematics, 2013; 235:1-55
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Nigel Kalton, Steven Lord, Denis Potapov, Fedor Sukochev
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We extend the noncommutative residue of M.Wodzicki on compactly supported classical pseudo-differential operators of order - d and generalise A. Connes' trace theorem, which states that the residue can be calculated using a singular trace on compact operators. Contrary to the role of the noncommutative residue for the classical pseudo-differential operators, a corollary is that the pseudo-differential operators of order -d do not have a 'unique' trace; pseudo-differential operators can be non-measurable in Connes' sense. Other corollaries are given clarifying the role of Dixmier traces in noncommutative geometry, including the definitive statement of Connes' original theorem. © 2012 Elsevier Ltd.
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Copyright © 2012 Elsevier Ltd. All rights reserved.