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 ${\lim }_{N\to \infty }\frac{1}{LogN}{\sum }_{n=1}^N{s}_n(x)$ 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 $L^{(1,\infty )}/L_0^{(1,\infty )}$, where the space $L_0^{(1,\infty )}$ is the closure of all finite rank operators in the norm $||.|{|}_{(1,\infty )}$.

31 pages, LaTex source, to appear in J. Funct. Anal

Cited by in corpus (17)

• Reconstruction of manifolds in noncommutative geometry
• Dixmier traces and some applications to noncommutative geometry
• Traces of compact operators and the noncommutative residue
• Measures from Dixmier Traces and Zeta Functions
• Measure Theory in Noncommutative Spaces
• Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras
• Semifinite spectral triples associated with graph C*-algebras
• Traces on symmetrically normed operator ideals
• Families of Type {\rm III KMS} States on a Class of $C^{\ast }$-Algebras containing $O_n$ and ${\mathcal{Q}}_{\mathbb{N}}$
• Banach limits and traces on ${\mathcal{L}}_{1,\infty }$
• Dixmier traces are weak${}^{\ast }$ dense in the set of all fully symmetric traces
• The Dixmier trace and asymptotics of zeta functions
• Spectral triple and Sinai - Ruelle - Bowen measures
• Dixmier traces generated by exponentiation invariant generalised limits
• On the distinction between the classes of Dixmier and Connes-Dixmier traces
• Measurable operators and the asymptotics of heat kernels and zeta functions
• Integration on locally compact noncommutative spaces