Singular Traces and Compact Operators. I

Abstract

We give a necessary and sufficient condition on a positive compact operator $T$ for the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value on $T$. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states on ${\ell }^{\infty }(\mathbb{N})$ using tools of non standard analysis in an essential way.

20 pages, AMS TeX, SFB 237-Preprint Nr.180-May 93

Cited by in corpus (20)

• Dimensions and singular traces for spectral triples, with applications to fractals
• Dixmier traces and some applications to noncommutative geometry
• Spectral triples for the Sierpinski Gasket
• A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem
• Traces on operator ideals and arithmetic means
• Measure Theory in Noncommutative Spaces
• Commuting and noncommuting infinitesimals
• Fractals in Noncommutative Geometry
• Dimensions and spectral triples for fractals in R^N
• An infinite dimensional Schur-Horn theorem and majorization theory with applications to operator ideals
• Singular traces, dimensions, and Novikov-Shubin invariants
• On the domain of singular traces
• Novikov-Shubin invariants and asymptotic dimensions for open manifolds
• Asimptotic dimension and Novikov-Shubin invariants for open manifolds
• A noncommutative Sierpinski Gasket
• The canonical trace and the noncommutative residue on the noncommutative torus
• Majorization and arithmetic mean ideals
• Spectral triple and Sinai - Ruelle - Bowen measures
• Set theory and C*-algebras
• Soft ideals and arithmetic mean ideals