Dixmier Traces as Singular Symmetric Functionals and Applications to Measurable Operators

Abstract

This paper introduces a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. By unifying various constructions, and translating the situation of Dixmier traces into the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces, we obtain the results (i) and (ii) below. The results are stated here, for the reader, in terms of the ideal $L^{(1,\infty)}$ of compact operators whose partial sums of singular values are of logarithmic divergence. (i) a positive compact operator $x$ in $L^{(1,\infty)}$ yields the same value for an arbitrary Connes-Dixmier trace (ie. $x$ is measurable in the sense of Connes) if and only if $\lim_{N\to\infty} \frac{1}{Log N}\sum_{n=1}^N s_n(x)$ exists, where $s_n(x)$ are the singular values of the compact operator $x$; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space $L^{(1,\infty)}/L^{(1,\infty)}_0$, where the space $L^{(1,\infty)}_0$ is the closure of all finite rank operators in the norm $||.||_{(1,\infty)}$.