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Connes–Dixmier Versus Dixmier Traces

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Abstract

This paper is a continuation of Pietsch (Math. Nachr. 285, 1999–2028, 2012) and (Stud. Math. 214, 37–66, 2013). Now we are able to bring in the harvest. Different sets of traces on the Marcinkiewicz operator ideal

$$ \mathfrak M (H):= \Bigg\{T \in \mathfrak L (H) \colon \sup_{1 \le m < \infty} \tfrac 1{\log m +1}\sum_{n=1}^m a_n(T) < \infty \Bigg\} $$

are compared with each other. Their size is measured by means of the density character. In particular, it is shown that the set of Dixmier traces is properly larger than the set of Connes–Dixmier traces, which answers a question posed in Carey–Sukochev (Russ. Math. Surv. 61, 1039–1099, 2006, p. 1062). The proofs are based on considerations about shift-invariant functionals on a suitably chosen sequence space.

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  1. 07749, Jena, Biberweg 7, Germany

    Albrecht Pietsch

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  1. Albrecht Pietsch
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Pietsch, A. Connes–Dixmier Versus Dixmier Traces. Integr. Equ. Oper. Theory 77, 243–259 (2013). https://doi.org/10.1007/s00020-013-2056-2

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