Noncommutative integral of a spectral triple

Let $(\mathcal A,H,D)$ be a spectral triple of positive metric dimension $p$ , assume $|D|^{-p}\in\mathcal L^{1,\infty}(H)$ after setting it to zero on $\ker D$ , and choose a Dixmier trace $\operatorname{Tr}_{\omega}$ . The associated noncommutative integral is

\int_D a=\operatorname{Tr}_{\omega}\!\left(a|D|^{-p}\right), \qquad a\in\mathcal A,

with $a$ represented on $H$ . This is a positive linear functional on the represented algebra. It is tracial only under additional hypotheses ensuring that commutators with $|D|^{-p}$ have zero Dixmier trace. If every $a|D|^{-p}$ is measurable, its value is independent of $\omega$ ; without measurability, the chosen generalized limit is part of the definition.

Why the critical power is used

Powers $|D|^{-q}$ with $q>p$ are ordinarily trace class, and every Dixmier trace vanishes on them. Powers with $q<p$ need not lie in the Dixmier ideal. The critical power $|D|^{-p}$ has the borderline $1/n$ -type singular-value decay detected by a logarithmic trace. The construction therefore extracts a volume coefficient invisible to the ordinary trace.

When the zeta function has a simple pole and appropriate Tauberian or measurability hypotheses hold, the same functional can be recovered from a residue:

\operatorname{Tr}_{\omega}(a|D|^{-p}) =\frac{1}{p}\operatorname*{Res}_{s=p} \operatorname{Tr}(a|D|^{-s}),

subject to the normalization conventions of the trace and zeta variable.

Classical geometric meaning

For the canonical Dirac spectral triple of a closed $p$ -dimensional Riemannian spin manifold, Connes's trace theorem identifies the Dixmier trace of a classical pseudodifferential operator of order $-p$ with a normalized Wodzicki residue. Consequently

\operatorname{Tr}_{\omega}(f|\not D|^{-p}) =c_p\int_M f\,d\operatorname{vol}_g

for an explicit constant $c_p$ determined by dimension, spinor rank, and normalization. This is the model for interpreting the functional as integration.

Conventions and scope

The symbol $\int_D$ is not standardized. Authors may write $f\,a|D|^{-p}$ , absorb the geometric constant $c_p^{-1}$ , or define the integral by a zeta residue. These normalizations agree only after the constants are stated.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted text. Relevant: Chapter IV, §2 on infinitesimals, Dixmier trace, and the noncommutative integral.
A. Connes, “The Action Functional in Non-Commutative Geometry,” Communications in Mathematical Physics 117 (1988), 673–683. DOI record. Relevant: Theorem 1, identifying the Dixmier trace with the noncommutative residue on critical-order pseudodifferential operators.