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

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     $\mathfrak {c}_0$-singular functional     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: