Singular traces on semifinite von Neumann algebras
Some time ago, \textit{J. Dixmier} showed the existence of non normal traces on \({\mathcal B}({\mathcal H})\) [C. R. Acad. Sci., Paris, Sér. A 262, 1107-1108 (1966; Zbl 0141.12902)]. These traces now play a role in A. Connes' non commutative geometry. Here the authors extend Dixmier's construction to obtain singular traces on general semifinite von Neumann algebras. From the abstract: ``Moreover, our technique produces singular traces on type \(\text{II}_1\) factors. Such traces, though vanishing on all bounded operators, are nontrivial on the *-algebra of affiliated unbounded operators. On a semifinite factor, we show that all traces are given by a dilation invariant functional on the cone of positive decreasing functions on \([0,\infty)\), and we prove that the existence of a singular trace which is nontrivial on a given operator is equivalent to an eccentricity condition on the singular values function, \dots''.
scientific article; zbMATH DE number 850629