Singular Traces on Semifinite von Neumann Algebras

Abstract

Singular traces are constructed on a general semifinite von Neumann algebra, thus generalizing the result of Dixmier (C.R. Acad. Sci. Paris262, 1966.) Moreover our technique produces singular traces on type II1 factors. Such traces, though vanishing on all bounded operators, are non trivial 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, ∞), and we prove that the existence of a singular trace which is non trivial on a given operator is equivalent to an eccentricity condition on the singular values function, a result which generalizes the theorem given in (S. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, J. Funct. Anal., to appear.) for B(H).