Wodzicki Residue in Noncommutative Geometry

• Wodzicki residue is a unique trace functional on classical pseudodifferential operators defined via an integral over the symbol that vanishes on commutators.
• It plays a central role in noncommutative geometry by linking microlocal analysis with global spectral invariants through results like Connes’ Trace Theorem.
• Extensions to non-classical and Laplacian-modulated operators reveal non-measurable cases, challenging the uniqueness of traces in advanced operator theory.

The Wodzicki residue, also known as the noncommutative residue, is a canonical trace functional defined on the algebra of classical pseudodifferential operators of critical order. It is uniquely characterized by vanishing on commutators and forms a foundational component of noncommutative geometry, connecting microlocal analysis with global spectral invariants via singular traces. Connes’ trace theorem establishes an explicit equivalence between the Wodzicki residue and Dixmier traces on weak trace class ideals, providing a noncommutative integral in the framework of spectral triples.

1. Definition and Local Formula

Let $M$ be a compact $d$-dimensional Riemannian manifold without boundary. For a classical pseudodifferential operator $P$ of order $-d$, the full symbol in local coordinates $(x, \xi)$ admits an asymptotic expansion: $$\sigma(P)(x,\xi) \sim \sum_{j=0}^{\infty} \sigma_{-d-j}(P)(x,\xi),$$ with $\sigma_{-d-j}(P)$ homogeneous of degree $-d-j$ in $\xi$ for $|\xi|\ge1$.

The Wodzicki residue is given by

$$\mathrm{Res}_W(P) := \frac{1}{(2\pi)^d} \int_M \int_{|\xi|=1} \sigma_{-d}(P)(x,\xi)\;dS(\xi)\;dx,$$

where $dS(\xi)$ is the surface measure on the unit sphere in $T^*_xM$.

Key properties:

• $\mathrm{Res}_W$ vanishes on commutators: $\mathrm{Res}_W([A,B])=0$.
• It is the unique trace (up to normalization) on the algebra of classical pseudodifferential operators of order $-d$ (Kalton et al., 2012).

2. Extensions Beyond Compact Manifolds

For operators acting on $\mathbb{R}^d$ with total symbol $p(x,\xi)$ compactly supported in $x$, define for $n\in\mathbb{N}$: $$\mathrm{Res}_n(P) := d \int_{|\xi|\le n^{1/d}} \int_{\mathbb{R}^d} p(x,\xi)\;dx\,d\xi,$$ which satisfies the asymptotic

$$\mathrm{Res}_n(P) = \mathrm{Res}_W(P) \log n + O(1).$$

Thus, the sequence $\{\mathrm{Res}_n(P)\}_{n\ge1}$ defines an equivalence class in $\ell^\infty / c_0$, and for classical compactly supported symbols, this residue reduces to the constant class associated to $\mathrm{Res}_W(P)$ (Kalton et al., 2012).

If the symbol is non-classical, the residue can yield non-constant elements in $\ell^\infty/c_0$, signaling new phenomena in trace theory.

3. Connes’ Trace Theorem and Dixmier Traces

The ideal $\mathcal{L}_{1,\infty}$ consists of compact operators with singular values $s_n(T)=O(n^{-1})$. Dixmier traces $\mathrm{Tr}_\omega$ on this ideal are positive, unitarily invariant, and singular, characterized by

$$\mathrm{Tr}_\omega(\mathrm{diag}(1, 1/2, 1/3, \ldots)) = 1.$$

For any classical $P$ of order $-d$ on a closed manifold,

$$\sum_{j=1}^n \lambda_j(P) = \frac{\mathrm{Res}_W(P)}{(2\pi)^d} \log n + O(1),$$

with $\{\lambda_j(P)\}$ the ordered eigenvalues. The foundational result is:

Connes’ Trace Theorem:

For every Dixmier trace $\mathrm{Tr}_\omega$ on $\mathcal{L}_{1,\infty}$,

$$\mathrm{Tr}_\omega(P) = \frac{1}{d(2\pi)^d} \mathrm{Res}_W(P).$$

This formula remains valid for compactly supported operators of order $-d$ on $\mathbb{R}^d$ (Kalton et al., 2012).

The proof uses spectral asymptotics and the commutator subspace characterization of traces on $\mathcal{L}_{1,\infty}$.

4. Laplacian-Modulated Operators and Generalized Residues

Moving beyond the classical calculus, consider Laplacian-modulated operators on $L^2(\mathbb{R}^d)$: those $T$ with $\|T(1-\Delta)^{d/2}\|_{\mathrm{HS}}=O(1)$, where $(1-\Delta)^{d/2}$ is the strong enough weight for modulation. Their symbols $p_T(x,\xi)$ satisfy

$$\int_{|\xi|\ge t}\!\int_{\mathbb{R}^d} |p_T(x,\xi)|^2 dx\,d\xi = O(t^{-d}), \quad t\to\infty.$$

For $T$ Laplacian-modulated and in $\mathcal{L}_{1,\infty}$, the residue is defined as a class in $\ell^\infty/c_0$: $$\mathrm{Res}(T) = \left[ d (2\pi)^d \sum_{j=1}^n (T e_j, e_j) \right]_{n=1}^\infty,$$ where $\{e_j\}$ is the orthonormal basis of eigenfunctions of $(1-\Delta)^{d/2}$.

The corresponding trace formula is: $$\mathrm{Tr}_\omega(T) = \frac{1}{d(2\pi)^d} \omega(\mathrm{Res}(T)),$$ with $\omega$ a dilation-invariant state on $\ell^\infty$ (Kalton et al., 2012).

Measurability in the sense of Connes means that $\mathrm{Res}(T)$ must be scalar in $\ell^\infty/c_0$; for classical $\Psi^{-d}$, this is always satisfied, but for non-classical Laplacian-modulated operators, non-scalar residues yield non-measurability.

5. Non-Measurable Operators and Failure of Uniqueness

Example 6.17 in (Kalton et al., 2012) constructs a compactly supported (non-classical) $\Psi^{-d}$ on $\mathbb{R}^d$, $Q$, whose residue sequence

$$\left[ \sin(\log\log(n^{1/d})) \right]_{n=1}^\infty$$

does not represent a constant modulo $c_0$. Consequently,

• The value $\mathrm{Tr}_\omega(Q)$ depends on the choice of Dixmier trace $\omega$.
• Pseudodifferential operators of order $-d$ do not, in general, have a unique trace; only the classical ones are measurable in Connes' sense.

This illustrates the breakdown of trace uniqueness when passing from the classical to the broader class of Laplacian-modulated operators.

6. Singular Traces, Noncommutative Integration, and Spectral Triples

In noncommutative geometry as developed by Connes, spectral triples $(\mathcal{A}, H, D)$ provide the analytic framework, with $D$ a self-adjoint operator whose spectrum exhibits eigenvalue growth $\lambda_n \sim n^{1/d}$. The Dixmier trace is then interpreted as the noncommutative integral: $a \mapsto \mathrm{Tr}_\omega(a |D|^{-d}),$ serving as a linear functional on $\mathcal{A}$.

Whenever $a|D|^{-d}$ is classical of order $-d$, this noncommutative integral matches, up to the universal constant $1/(d(2\pi)^d)$, the Wodzicki residue of the associated operator (Kalton et al., 2012).

A plausible implication is that singular traces supply a general “noncommutative integration” method for operators whose Schwartz kernels decay at the critical rate $|\xi|^{-d}$ at infinity, broadening the scope of noncommutative geometry beyond the field of classical pseudodifferential analysis. Only in the classical, measurable (scalar residue) case does the trace become canonical and independent of the choice of singular trace.