On the domain of singular traces

Details On the domain of singular traces On the domain of singular traces

2002-03-04

arxiv.org/abs/math/0203029v2

International Journal of Mathematics, Vol. 13, No. 6 (2002) 667-674

Mathematics

Operator Algebras

7 pages, LaTeX. Minor corrections, to appear on the International Journal of Mathematics

Scientific paper

10.1142/S0129167X02001447

The question whether an operator belongs to the domain of some singular trace is addressed, together with the dual question whether an operator does not belong to the domain of some singular trace. We show that the answers are positive in general, namely for any (compact, infinite rank) positive operator A we exhibit two singular traces, the first being zero and the second being infinite on A. However, if we assume that the singular traces are generated by a "regular" operator, the answers change, namely such traces always vanish on trace-class, non singularly traceable operators and are always infinite on non trace-class, non singularly traceable operators. These results are achieved on a general semifinite factor, and make use of a new characterization of singular traceability (cf. math.OA/0202108).

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