Dixmier trace - Imported Knowlpedia Content

Definition

Let $H$ be a separable Hilbert space and let $\omega$ be a dilation-invariant generalized limit on bounded sequences. For a positive operator $T$ in the Marcinkiewicz–Macaev ideal $\mathcal M_{1,\infty}(H)$ , its Dixmier trace is

\operatorname{Tr}_{\omega}(T)= \omega\left(\left\{ \frac{1}{\log(1+N)}\sum_{n=1}^{N}\mu_n(T) \right\}_{N\geq1}\right),

where $\mu_n(T)$ are the singular values in decreasing order. The functional extends linearly from positive operators to $\mathcal M_{1,\infty}(H)$ . It is positive, unitarily invariant, tracial, and vanishes on trace-class operators, but its value can depend on $\omega$ .

Why the logarithmic mean appears

For $T\in\mathcal M_{1,\infty}$ , the partial sums of singular values grow at most logarithmically, so the sequence supplied to $\omega$ is bounded. The generalized limit extracts an asymptotic coefficient even when the ordinary limit does not exist. Dilation invariance is the ingredient that makes the result a trace rather than merely a unitarily invariant functional. The construction and the precise admissible generalized limits are developed in Lord–Sukochev–Zanin, Chapters 5–6.

Measurable operators

A positive $T$ is Dixmier measurable when $\operatorname{Tr}_{\omega}(T)$ has the same value for every admissible $\omega$ . A sufficient and standard criterion is convergence of the logarithmic means in the displayed formula; then the Dixmier trace equals that ordinary limit. For a diagonal operator with eigenvalues $1/n$ , the logarithmic means converge to $1$ , so every Dixmier trace gives value $1$ .

Relation to ordinary integration

The canonical operator trace sums singular values and is finite on the trace-class ideal. The Dixmier trace instead detects the critical $1/n$ decay scale and vanishes on that smaller ideal. This singular behavior permits the noncommutative integral of a critical-order infinitesimal in a spectral triple; Connes develops this role in Noncommutative Geometry, Chapter IV.

Conventions and scope

References

S. Lord, F. Sukochev, and D. Zanin, Singular Traces: Theory and Applications, De Gruyter, 2013. DOI record. Relevant: Chapters 5–6 on Dixmier traces and measurable operators.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted record. Relevant: Chapter IV, §2 on the Dixmier trace and the noncommutative integral.