  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).
