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A New View at Dixmier Traces on \(\mathfrak {L}_{1,\infty } (H)\)

  • Albrecht PietschEmail author
Article
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Abstract

Dixmier traces are usually defined by means of functionals on \(\mathfrak {l}_\infty (\mathbb {N})\) that are invariant under the forward dilation
$$\begin{aligned} D_+: (\sigma _1,\sigma _2,\sigma _3,\sigma _4,\dots \;) \mapsto (\sigma _1,\sigma _1,\sigma _2,\sigma _2,\dots \;) \end{aligned}$$
and \(\mathfrak {c}_0\)-singular, which means that they vanish at all null sequences.
The backward dilation
$$\begin{aligned} D_-: (\sigma _1,\sigma _2,\sigma _3,\sigma _4,\dots \;) \mapsto (\sigma _2,\sigma _4,\sigma _6,\sigma _8,\dots \;) \end{aligned}$$
can be used, as well. Note that \(D_-\)-invariant functionals automatically vanish on \(\mathfrak {c}_0(\mathbb {N})\).

As observed by Sukochev and coauthors, any requirement concerning dilation invariance can be dropped when working on \(\mathfrak {L}_{1,\infty } (H)\). It just suffices to assume that the generating functionals vanish on \(\mathfrak {c}_0(\mathbb {N})\). So it seems to be a good idea to present the theory of Dixmier traces on the bases of an adapted definition. We carry out this project. Although the fundamental results remain unchanged, their interplay becomes a quite different appearance.

Even more is possible. Looking at the conclusions

Hahn–Banach Theorem   Open image in new window   \(\mathfrak {c}_0\)-singular functional   Open image in new window   Dixmier trace, we may wonder whether the intermediate step is necessary. Indeed, there exists a sublinear function on the underlying operator ideal \(\mathfrak {L}_{1,\infty } (H)\) that directly yields all proper Dixmier traces via the Hahn–Banach Theorem. So the theory of Dixmier traces can be turned upside down:

Open image in new window

Keywords

Operator ideal Dixmier trace Extended limit Banach limit Dilation limit Shift-invariance Dilation-invariance 

Mathematics Subject Classification

Primary 47B10 47B37 Secondary 46B45 46A45 

Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.JenaGermany

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