The Dixmier Trace and the Noncommutative Integral

Intuition Beginner

Adding up infinitely many positive numbers often gives infinity, and infinity carries no information. But there is a way to read a finite number off a divergent sum: watch how fast the partial sums grow, and report the leading rate rather than the total. The Dixmier trace is exactly such a device, tuned to one specific growth rate — the logarithm.

Picture an operator whose eigenvalues shrink like one-over-position: the size of the $n$-th one is about $1/n$. The running total of the first $N$ of them grows like the logarithm of $N$, which crawls to infinity painfully slowly. Divide the running total by that logarithm and the ratio settles down to a finite number. The Dixmier trace is that settled value. It is blind to any operator whose eigenvalues shrink faster, because those have a finite ordinary total and contribute nothing to the logarithmic rate.

The payoff is that this strange averaged trace turns out to be integration. When you feed it the right operator built from a function on a curved space, it returns the ordinary integral of that function — the area, the volume, the average. So integration on a space can be recovered purely from eigenvalues, with no measure and no coordinates in sight.

Visual Beginner

The Dixmier trace watches the running total of a list of shrinking eigenvalues and divides by how far the logarithm has climbed.

The dictionary reads: eigenvalues that shrink like one-over-position make the running total grow like a logarithm; the ratio of total to logarithm is the Dixmier trace; eigenvalues that shrink faster add only a finite amount and leave the rate untouched, so the Dixmier trace ignores them.

Worked example Beginner

Take the operator on an endless list of slots whose action is to multiply the $n$-th slot by $1/n$, starting at $n=1$. Its eigenvalues, in decreasing order, are $$ 1,\ \tfrac{1}{2},\ \tfrac{1}{3},\ \tfrac{1}{4},\ \ldots $$ Add the first $N$ of them. That sum is the harmonic number, and for large $N$ it is very close to the natural logarithm of $N$: $$ 1 + \tfrac{1}{2} + \cdots + \tfrac{1}{N} \approx \log N. $$ Now form the Dixmier ratio: the running total divided by $\log N$. Since the total is about $\log N$, the ratio is about $1$, and it stays pinned at $1$ as $N$ grows. So the Dixmier trace of this operator is $1$.

Compare an operator whose eigenvalues are $1/n^2$: $1,1/4,1/9,\dots$. Their total is finite (it sums to about $1.645$), so dividing by $\log N$ gives a ratio sliding to zero. Its Dixmier trace is $0$.

What this tells us: the Dixmier trace measures the coefficient of logarithmic growth. The one-over-$n$ operator sits exactly at the borderline where that coefficient is nonzero, and everything that decays faster is invisible to it.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathcal{H}$ be a separable Hilbert space and $T$ a compact operator 02.11.05. Write ${\mu }_0(T)\ge {\mu }_1(T)\ge \cdots \ge 0$ for the singular values of $T$, the eigenvalues of $|T|=(T^{\ast }T{)}^{1/2}$ listed with multiplicity in decreasing order. Define the partial sums $$ \sigma_N(T) = \sum_{n=0}^{N-1} \mu_n(T). $$

The Dixmier ideal (Macaev ideal, weak trace class) is $$ \mathcal{L}^{1,\infty}(\mathcal{H}) = \Big{ T \in \mathcal{K}(\mathcal{H}) : |T|{1,\infty} := \sup{N \ge 2} \frac{\sigma_N(T)}{\log N} < \infty \Big}. $$ Equivalently $T\in {\mathcal{L}}^{1,\infty }$ iff ${\mu }_n(T)=O(1/(n+1))$. It is a two-sided ideal of $\mathcal{B}(\mathcal{H})$, complete in the quasi-norm $\Vert \cdot {\Vert }_{1,\infty }$, and strictly larger than the trace-class ideal ${\mathcal{L}}^1$ (where ${\sigma }_N$ stays bounded). For $T\in {\mathcal{L}}^{1,\infty }$ the function $$ \tau_N(T) = \frac{1}{\log N}, \sigma_N(T) $$ is bounded but generally does not converge as $N\to \infty$.

To extract a number one chooses a dilation- and translation-invariant state $\omega$ on ${\ell }^{\infty }(\mathbb{N})$ — a positive linear functional with $\omega (1)=1$ that agrees with the ordinary limit on convergent sequences and is invariant under the dilation $(a_n)\mapsto (a_{2n})$ and the Cesàro/translation averaging. Such states exist by amenability of the relevant transformation group (Hahn–Banach plus an invariant mean). For a positive $T\in {\mathcal{L}}^{1,\infty }$ define the Dixmier trace $$ \operatorname{Tr}\omega(T) = \omega!\left( \frac{1}{\log N} \sum{n=0}^{N-1} \mu_n(T) \right)_{N}, $$ the value of $\omega$ on the bounded sequence $({\tau }_N(T){)}_N$. The construction extends to all of ${\mathcal{L}}^{1,\infty }$ by linearity over the four-cone decomposition $T=(T_1-T_2)+i(T_3-T_4)$ into positive parts; the additivity needed to make this well defined is a theorem, not an axiom, and rests on the concavity of $T\mapsto {\sigma }_N(T)$ on positive operators.

The functional ${\mathrm{Tr}}_{\omega }$ is a positive trace: it is linear, positive, unitarily invariant, satisfies ${\mathrm{Tr}}_{\omega }(ST)={\mathrm{Tr}}_{\omega }(TS)$, and is singular — it vanishes on every trace-class operator, since for $T\in {\mathcal{L}}^1$ the partial sums ${\sigma }_N(T)$ stay bounded and ${\sigma }_N/\log N\to 0$. A positive operator $T\in {\mathcal{L}}^{1,\infty }$ is called measurable when ${\mathrm{Tr}}_{\omega }(T)$ is independent of the choice of $\omega$; one then writes the common value $$ \fint T := \operatorname{Tr}\omega(T), $$ the noncommutative integral of $T$. Measurability holds exactly when the sequence ${\tau }_N(T)$ converges in the Cesàro–logarithmic sense, i.e. when $$ \frac{1}{\log N}\sum{n=0}^{N-1}\mu_n(T) \longrightarrow L $$ for an ordinary limit $L$; then $\text{fint}T=L$ for every $\omega$.

Counterexamples to common slips

• The Dixmier trace is not the ordinary trace restricted to ${\mathcal{L}}^{1,\infty }$, and it is not normal: it cannot be computed as a supremum of its values on finite-rank truncations. A finite-rank operator has ${\sigma }_N$ bounded, so ${\mathrm{Tr}}_{\omega }$ vanishes on it; yet there are $T\in {\mathcal{L}}^{1,\infty }$ with ${\mathrm{Tr}}_{\omega }(T)=1$. The trace lives entirely "at infinity".
• Membership in ${\mathcal{L}}^{1,\infty }$ is a condition on the singular values, the eigenvalues of $|T|$, not on the eigenvalues of $T$. A non-normal $T$ can have all eigenvalues zero (quasinilpotent) while ${\mu }_n(T)\sim 1/n$, so ${\mathrm{Tr}}_{\omega }(T)$ need not equal any sum of eigenvalues of $T$ itself.
• Not every $T\in {\mathcal{L}}^{1,\infty }$ is measurable. The Cesàro–logarithmic average ${\tau }_N(T)$ can oscillate between two values along $N=2^k$ and $N=2^{2^k}$; different invariant states $\omega$ then read off different limits, and ${\mathrm{Tr}}_{\omega }(T)$ genuinely depends on $\omega$. Measurability is the special, geometrically relevant case.

The ideal ${\mathcal{L}}^{1,\infty }$ is the borderline member of the Lorentz scale ${\mathcal{L}}^{p,\infty }=\{T:{\mu }_n(T)=O(n^{-1/p})\}$ between the Schatten classes. For $p>1$ the ideal ${\mathcal{L}}^{p,\infty }$ contains ${\mathcal{L}}^p$ properly and supports no analogue of the Dixmier trace, because ${\sum }_{n<N}{\mu }_n$ converges; for $p<1$ the partial sums grow polynomially and no logarithmic renormalisation isolates a finite limit. Only at $p=1$ does the divergence sit exactly at the rate $\log N$ that an invariant mean can tame, which is why the noncommutative integral is intrinsically a phenomenon of the weak-${\mathcal{L}}^1$ ideal. Inside ${\mathcal{L}}^{1,\infty }$ the operators with ${\mu }_n(T)=o(1/n)$ form a closed subideal ${\mathcal{L}}_0^{1,\infty }$ on which every ${\mathrm{Tr}}_{\omega }$ vanishes; the Dixmier trace therefore factors through the quotient ${\mathcal{L}}^{1,\infty }/({\mathcal{L}}_0^{1,\infty }+{\mathcal{L}}^1)$, recording only the coefficient of the borderline $1/n$ tail.

Key theorem with proof Intermediate+

Theorem (Connes' trace theorem). Let $M$ be a closed Riemannian manifold of dimension $n$ and let $P$ be a classical (one-step polyhomogeneous) pseudodifferential operator of order $-n$ acting on sections of a vector bundle $E\to M$, with principal symbol ${\sigma }_{-n}(P)(x,\xi )$. Then $P$, as an operator on $L^2(M,E)$, lies in ${\mathcal{L}}^{1,\infty }$, is measurable, and its Dixmier trace equals a universal constant times the Wodzicki residue: $$ \operatorname{Tr}\omega(P) ;=; \frac{1}{n},(2\pi)^{-n} \int{S^*M} \operatorname{tr}, \sigma_{-n}(P)(x,\xi), d\xi, dx ;=; \frac{1}{n,(2\pi)^n},\operatorname{Res}_W(P), $$ *independently of $\omega$, where $S^{\ast }M$ is the cosphere bundle and ${\mathrm{Res}}_W$ is the Wodzicki residue.* ^{[Connes 1988; Connes Ch. IV]}

Proof. Since $P$ has order $-n$ on an $n$-manifold, its singular values obey the Weyl-type law for elliptic-order operators: the counting function $N(\lambda )=\#\{j:{\mu }_j(P)>\lambda \}$ grows like $C{\lambda }^{-1}$ as $\lambda \to 0$, with $$ C = (2\pi)^{-n}, \frac{1}{n}\int_{S^*M} \operatorname{tr},\sigma_{-n}(P), d\xi, dx, $$ because the symbol of order $-n$ controls the leading small-eigenvalue asymptotics through the symbolic volume in phase space. Inverting the counting function gives ${\mu }_j(P)\sim C/j$ as $j\to \infty$, so ${\sigma }_N(P)={\sum }_{j<N}{\mu }_j(P)\sim C\log N$. Hence $P\in {\mathcal{L}}^{1,\infty }$ with ${\tau }_N(P)={\sigma }_N(P)/\log N\to C$ as an honest limit. A convergent sequence has the same value under every invariant state, so $P$ is measurable and ${\mathrm{Tr}}_{\omega }(P)=C$ for all $\omega$.

The Weyl constant deserves a word of geometric content. For an elliptic operator the number of eigenvalues below a threshold is, to leading order, the symplectic volume of the region of phase space where the symbol lies below that threshold, divided by $(2\pi {)}^n$ — the semiclassical correspondence "one quantum state per Planck cell". For $P$ of order $-n$ the relevant region in each cotangent fibre is governed by ${\sigma }_{-n}(P)(x,\xi )$, a function homogeneous of degree $-n$ in $\xi$, and integrating it over the cosphere bundle and dividing by the radial Jacobian produces exactly the constant $C$ above. The factor $1/n$ is the radial integral ${\int }_1^{\infty }{r}^{-1}dr/\log$ normalisation that converts the homogeneous-degree-$(-n)$ symbol into a logarithmically divergent phase-space volume, matching the $\log N$ growth of ${\sigma }_N(P)$ term for term.

It remains to identify $C$ with the Wodzicki residue. The Wodzicki residue of a classical $\Psi$DO $Q$ of order $-n$ is by definition $$ \operatorname{Res}W(Q) = \int{S^*M} \operatorname{tr}, \sigma_{-n}(Q)(x,\xi), d\xi, dx, $$ the integral over the cosphere bundle of the trace of the symbol component of degree exactly $-n$. This is the unique (up to scale) trace on the algebra of classical $\Psi$DOs, and it is local: it depends only on one symbol coefficient. Comparing with the displayed constant $C$ gives ${\mathrm{Tr}}_{\omega }(P)=C=(2\pi {)}^{-n}{n}^{-1}{\mathrm{Res}}_W(P)$. The two normalising conventions — the $1/(n(2\pi {)}^n)$ — are bookkeeping for the volume of the unit cosphere and the Weyl constant; with them the equality is exact. $\square$

Bridge. This theorem builds toward the entire integration calculus of noncommutative geometry, and it appears again in 39.06.06 where the local index formula expands the index pairing as a sum of Dixmier-trace (Wodzicki-residue) terms over the dimension spectrum. The foundational reason it works is exactly the Weyl law: order $-n$ forces the singular values onto the borderline ${\mu }_j\sim C/j$, which is the unique decay rate the Dixmier trace can see, so the analytic averaging and the geometric symbol integral are forced to agree. This is exactly the dual of the heat-kernel picture of 39.06.01, where the same symbol data produced the residue of the zeta function ${\zeta }_{|D|}(s)$ at $s=n$; the Dixmier trace and that zeta residue compute one number two ways. Putting these together, the operator-theoretic side (${\mathcal{L}}^{1,\infty }$ and ${\mathrm{Tr}}_{\omega }$) and the symbolic side (the Wodzicki residue) are two faces of the integral, and the bridge is the Weyl law that pins both to the coefficient of logarithmic eigenvalue growth.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has compact operators, the trace of a trace-class operator, and the spectral theorem, but it lacks the singular-value rearrangement ${\mu }_n(T)$, the Macaev ideal ${\mathcal{L}}^{1,\infty }$ with its quasi-norm, the dilation-invariant state $\omega$ on ${\ell }^{\infty }(\mathbb{N})$, and therefore the Dixmier trace, measurability, and Connes' trace theorem.

The intended statement reads schematically:

import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Analysis.Normed.Operator.Compact

/-- The Dixmier ideal: compact operators whose singular-value partial sums
grow at most logarithmically. (Schematic: `singularValue` as a decreasing
rearrangement and the dilation-invariant state `omega` are not yet in
Mathlib.) -/
structure DixmierTrace (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] where
  ideal      : Set (H →L[ℂ] H)              -- morally L^{1,∞}
  singular   : (H →L[ℂ] H) → ℕ → ℝ          -- μ_n(T), decreasing
  omega      : (ℕ → ℝ) →ₗ[ℝ] ℝ              -- dilation-invariant state
  trWamega   : (H →L[ℂ] H) → ℝ              -- Tr_ω(T) = omega (σ_N / log N)
  vanishesOnTraceClass : ∀ T, True          -- T ∈ L¹ → Tr_ω T = 0
  isTrace    : ∀ S T, True                  -- Tr_ω(ST) = Tr_ω(TS)

Connes' trace theorem — ${\mathrm{Tr}}_{\omega }(P)=\frac{1}{n(2\pi {)}^n}{\mathrm{Res}}_W(P)$ for a classical $\Psi$DO of order $-n$ — is a far-horizon target requiring the Wodzicki residue on the algebra of classical pseudodifferential operators alongside the singular-trace machinery.

Advanced results Master

The Dixmier construction sits inside a richer theory of singular traces and admits several equivalent reformulations, each computing the same noncommutative integral by a different limiting procedure ^{[Connes Ch. IV]}.

The heat-kernel formula. For positive $T\in {\mathcal{L}}^{1,\infty }$ the Dixmier trace can be read off the small-time behaviour of a heat-type trace. If $\mathrm{Tr}(e^{-t/T})\sim a/t$ as $t\downarrow 0$ — more usefully, for $T=|D{|}^{-n}$ one uses $\mathrm{Tr}(e^{-t|D{|}^2})$ — then $T$ is measurable and ${\mathrm{Tr}}_{\omega }(T)$ is the coefficient of the leading singularity. Concretely, for the canonical triple the Seeley–DeWitt expansion $\mathrm{Tr}(e^{-t|D{|}^2})\sim a_0{t}^{-n/2}+\cdots$ gives ${\mathrm{Tr}}_{\omega }(|D{|}^{-n})=\frac{a_0}{\Gamma (n/2+1)}$, tying the noncommutative volume to the first heat coefficient $a_0=(4\pi {)}^{-n/2}\mathrm{vol}(M)\mathrm{rank}(S)$. The mechanism is a Laplace–Mellin interchange: $|D{|}^{-n}=\frac{1}{\Gamma (n/2)}{\int }_0^{\infty }{t}^{n/2-1}{e}^{-t|D{|}^2}dt$ has its small-$t$ divergence controlled by $a_0{t}^{-n/2}$, and the leading divergence of the Cesàro–logarithmic average of singular values is fixed by precisely this coefficient. The subleading heat coefficients $a_2,a_4,\dots$ — curvature integrals of increasing order — produce the lower points of the dimension spectrum and the lower terms of the spectral action, but the noncommutative volume reads only $a_0$.

The zeta-function formula. When the zeta function ${\zeta }_T(s)=\mathrm{Tr}(T^s)$ has a simple pole at $s=1$ with residue $r$, then $T$ is measurable and ${\mathrm{Tr}}_{\omega }(T)=r={\mathrm{Res}}_{s=1}\mathrm{Tr}(T^s)$. For a spectral triple of dimension $n$, ${\mathrm{Tr}}_{\omega }(a|D{|}^{-n})={\mathrm{Res}}_{s=n}\mathrm{Tr}(a|D{|}^{-s})$, identifying the noncommutative integral with the leading zeta residue and hence with the top of the dimension spectrum 39.06.01. The heat-kernel and zeta formulas are Mellin transforms of one another.

Measurability and the role of the symbol. Connes' trace theorem makes precise that geometric operators are automatically measurable: any classical $\Psi$DO of order $-n$ has ${\tau }_N$ convergent, so the $\omega$-dependence is invisible on the operators that arise from geometry. The pathological non-measurable $T\in {\mathcal{L}}^{1,\infty }$ of the exercises require artificially oscillating singular values, which no symbol of fixed order produces. Thus on the algebra generated by a regular spectral triple the integral $\text{fint}={\mathrm{Tr}}_{\omega }$ is canonical, and the choice of $\omega$ is a red herring outside deliberately engineered counterexamples.

The Wodzicki residue as the unique trace. The deeper structural fact behind Connes' theorem is Wodzicki's: on the algebra ${\Psi }^{\mathbb{Z}}(M)$ of classical (integer-order) pseudodifferential operators on a connected closed manifold of dimension $n\ge 1$, the residue ${\mathrm{Res}}_W$ is, up to scalar, the only trace. The operator trace $\mathrm{Tr}$ exists only on the ideal of order $<-n$ operators (the trace-class ones) and does not extend to a trace on the whole algebra; the Wodzicki residue extends it as the unique singular trace. Connes' trace theorem is precisely the statement that the Dixmier trace, restricted to order-$(-n)$ operators, realises this unique residue analytically.

The uniqueness has a homological cause: the quotient of ${\Psi }^{\mathbb{Z}}(M)$ by its commutator subspace is one-dimensional, so any trace is a scalar multiple of ${\mathrm{Res}}_W$. The residue is defined through the term of symbol-degree exactly $-n$, the only degree whose integral over the cosphere fibre is reparametrisation-invariant: degrees above $-n$ give divergent integrals discarded by the regularisation, and degrees below contribute terms that are total derivatives in $\xi$ and integrate to zero. This locality — the residue sees a single homogeneous component of the complete symbol — is what makes it computable from local geometry and is mirrored, on the operator side, by the fact that ${\mathrm{Tr}}_{\omega }$ depends only on the $1/n$ tail of the singular values. The two localities are the same statement read through the symbol calculus and through the singular-value calculus respectively, which is the content of the equality ${\mathrm{Tr}}_{\omega }=c_n^{-1}{\mathrm{Res}}_W$.

Non-normality and why it is forced. A trace $\phi$ on $\mathcal{B}(\mathcal{H})$ is normal if $\phi ({\sup }_{\alpha }{T}_{\alpha })={\sup }_{\alpha }\phi (T_{\alpha })$ for bounded increasing nets of positives — equivalently, if it is determined by its values on finite-rank projections. The ordinary trace is normal; the Dixmier trace is not, and this is not a defect but a necessity. Any normal trace on $\mathcal{B}(\mathcal{H})$ is a multiple of the ordinary trace, hence either zero or infinite on the rank-one projections of an infinite orthonormal family, and so it cannot take a finite nonzero value on an operator like $\mathrm{diag}(1/n)$ whose ordinary trace diverges. To assign such an operator a finite integral one must break normality, and the invariant-mean limit ${\lim }_{\omega }$ is exactly the controlled way to do so: it is additive and unitarily invariant but discards all finite-rank information. The price — dependence on $\omega$ for non-measurable operators — is the residue of the choice involved in summing a divergent series, and it disappears precisely on the geometric operators where a canonical summation exists.

The integral on the noncommutative torus. The construction is not merely a re-description of classical integration: it defines integration on genuinely pointless spaces. For the noncommutative two-torus ${\mathcal{A}}_{\theta }$, generated by unitaries $U,V$ with $VU=e^{2\pi i\theta }UV$, the flat triple has $|D{|}^{-2}\in {\mathcal{L}}^{1,\infty }$ with ${\mathrm{Tr}}_{\omega }(a|D{|}^{-2})=\frac{1}{2\pi }\tau (a)$, where $\tau (\sum c_{mn}{U}^m{V}^n)=c_{00}$ is the canonical (unique) tracial state on ${\mathcal{A}}_{\theta }$. The Dixmier trace recovers exactly the trace $\tau$ that plays the role of "integration against Haar measure", even though for irrational $\theta$ the algebra has no characters and the space has no points. This is the prototype showing $\text{fint}$ is the right notion of integral whenever a metric (the Dirac operator) is present.

The spectral action and the integral. The noncommutative integral is the linear functional underlying the spectral action $S=\mathrm{Tr}\chi (D^2/{\Lambda }^2)$ 39.06.01: the heat expansion of this trace is a sum of terms ${\Lambda }^{n-k}\text{fint}{a}_k$ where each $\text{fint}{a}_k={\mathrm{Tr}}_{\omega }(a_k|D{|}^{-(n-k)})$ is a Dixmier integral of a curvature scalar, producing the cosmological, Einstein–Hilbert, and Yang–Mills–Higgs terms in turn. The integral $\text{fint}$ is the bridge from spectral data to action functionals: it is what makes "the action depends only on the spectrum" a calculation rather than a slogan, and it returns in the local index formula of 39.06.06 as the coefficient functional on the dimension spectrum.

Synthesis. The Dixmier trace is the central insight that turns "integration" into a spectral-asymptotic measurement, and putting these reformulations together is exactly the content of the theory: the partial-sum limit, the heat-kernel leading coefficient, and the zeta residue are three computations of one number, and Connes' trace theorem is the foundational reason they coincide with the symbol integral. The Dixmier trace generalises the Wodzicki residue from $\Psi$DOs to arbitrary ${\mathcal{L}}^{1,\infty }$ operators, and is dual to the ordinary trace in the precise sense that $\mathrm{Tr}$ owns the trace-class ideal while ${\mathrm{Tr}}_{\omega }$ owns the larger Macaev ideal and vanishes where $\mathrm{Tr}$ lives. This is exactly the analytic engine behind the noncommutative integral $\text{fint}a={\mathrm{Tr}}_{\omega }(a|D{|}^{-n})$: in the commutative case it recovers ${\int }_{M}ad{\mathrm{vol}}_g$, and in the noncommutative case it defines what integration means, so the bridge from classical Riemannian volume to spectral integration is the identification of the volume with a singular trace.

Full proof set Master

Proposition (${\mathcal{L}}^{1,\infty }$ is a two-sided ideal). Let $T\in {\mathcal{L}}^{1,\infty }$ and $S\in \mathcal{B}(\mathcal{H})$. The Ky Fan inequality ${\sigma }_N(ST)\le \Vert S\Vert {\sigma }_N(T)$ and ${\sigma }_N(TS)\le \Vert S\Vert {\sigma }_N(T)$ shows $\Vert ST{\Vert }_{1,\infty },\Vert TS{\Vert }_{1,\infty }\le \Vert S\Vert \Vert T{\Vert }_{1,\infty }$, so the set is closed under left and right multiplication by bounded operators. For sums, the singular-value subadditivity ${\sigma }_N(T_1+T_2)\le {\sigma }_N(T_1)+{\sigma }_N(T_2)$ gives $\Vert T_1+T_2{\Vert }_{1,\infty }\le \Vert T_1{\Vert }_{1,\infty }+\Vert T_2{\Vert }_{1,\infty }$. Hence ${\mathcal{L}}^{1,\infty }$ is a two-sided ideal, complete in $\Vert \cdot {\Vert }_{1,\infty }$. $\square$

Proposition (${\mathrm{Tr}}_{\omega }$ vanishes on ${\mathcal{L}}^1$). If $T\in {\mathcal{L}}^1$ is positive then ${\sigma }_N(T)={\sum }_{n<N}{\mu }_n(T)\le \Vert T{\Vert }_1<\infty$ uniformly in $N$. Therefore ${\tau }_N(T)={\sigma }_N(T)/\log N\le \Vert T{\Vert }_1/\log N\to 0$, and since $\omega$ agrees with the ordinary limit on convergent sequences, ${\mathrm{Tr}}_{\omega }(T)=0$. By linearity over the four positive cones the same holds for all $T\in {\mathcal{L}}^1$. $\square$

Proposition (additivity of ${\mathrm{Tr}}_{\omega }$ on positives). For positive $S,T\in {\mathcal{L}}^{1,\infty }$ the singular values obey the two-sided estimate $$ \sigma_N(S) + \sigma_N(T) ;-; \delta_N ;\le; \sigma_{N}(S+T) ;\le; \sigma_N(S) + \sigma_N(T), $$ where the upper bound is subadditivity and the deficit ${\delta }_N$ satisfies ${\delta }_N=o(\log N)$ because ${\sigma }_{2N}(S+T)\ge {\sigma }_N(S)+{\sigma }_N(T)$ (the eigenvalues of a sum of positives interlace those of the summands). Dividing by $\log N$ and using $\log (2N)/\log N\to 1$ forces ${\tau }_N(S+T)-{\tau }_N(S)-{\tau }_N(T)\to 0$. Applying $\omega$, the error vanishes and ${\mathrm{Tr}}_{\omega }(S+T)={\mathrm{Tr}}_{\omega }(S)+{\mathrm{Tr}}_{\omega }(T)$. This additivity on positives is what lets the four-cone extension define a genuine linear functional. $\square$

Proposition (${\mathrm{Tr}}_{\omega }$ is a trace). For $T\in {\mathcal{L}}^{1,\infty }$ and unitary $U$, ${\mu }_n(UT{U}^{\ast })={\mu }_n(T)$, so ${\mathrm{Tr}}_{\omega }(UT{U}^{\ast })={\mathrm{Tr}}_{\omega }(T)$; since every $S\in \mathcal{B}(\mathcal{H})$ is a linear combination of four unitaries, ${\mathrm{Tr}}_{\omega }(ST{S}^{-1})={\mathrm{Tr}}_{\omega }(T)$ whenever $S$ is invertible, and the commutator-vanishing ${\mathrm{Tr}}_{\omega }(ST)={\mathrm{Tr}}_{\omega }(TS)$ follows from $|{\sigma }_N(ST)-{\sigma }_N(TS)|=O(1)$ (Exercise 7) divided by $\log N$ tending to $0$. $\square$

Proposition (Connes' trace theorem, singular-value form). Let $P$ be a positive classical $\Psi$DO of order $-n$ on a closed $n$-manifold. The Weyl law for the elliptic-order asymptotics gives the eigenvalue-counting estimate $N(\lambda )=\#\{j:{\mu }_j(P)>\lambda \}\sim C{\lambda }^{-1}$ with $C=(2\pi {)}^{-n}{n}^{-1}{\int }_{S^{\ast }M}\mathrm{tr}{\sigma }_{-n}(P)$, hence ${\mu }_j(P)\sim C/j$ and ${\sigma }_N(P)\sim C\log N$. Therefore ${\tau }_N(P)\to C$, $P$ is measurable, and ${\mathrm{Tr}}_{\omega }(P)=C=\frac{1}{n(2\pi {)}^n}{\mathrm{Res}}_W(P)$. The general (not necessarily positive) case follows by writing $P$ in terms of positive parts and using linearity, the symbol integral being additive in ${\sigma }_{-n}$. $\square$

Proposition (the Dixmier trace recovers the trace on ${\mathcal{A}}_{\theta }$). Let ${\mathcal{A}}_{\theta }$ be the smooth noncommutative torus with derivations ${\delta }_1,{\delta }_2$ and flat Dirac operator $D={\delta }_1\otimes {\gamma }^1+{\delta }_2\otimes {\gamma }^2$ on $\mathcal{H}=L^2({\mathcal{A}}_{\theta },\tau )\otimes {\mathbb{C}}^2$, where $\tau$ is the canonical trace. The eigenvalues of $|D|$ on the Fourier mode $U^m{V}^n$ are $2\pi \sqrt{m^2+n^2}$ with multiplicity $2$, so the eigenvalues of $|D{|}^{-2}$ are $(2\pi {)}^{-2}(m^2+n^2{)}^{-1}$, and the lattice-point count $\#\{(m,n):m^2+n^2\le R^2\}\sim \pi R^2$ gives ${\mu }_k(|D{|}^{-2})\sim (2\pi {)}^{-2}\cdot 2\pi /k=1/(2\pi k)$. For $a=\sum c_{mn}{U}^m{V}^n$ the operator $a|D{|}^{-2}$ has the same leading singular-value asymptotics weighted by the diagonal symbol, yielding ${\sigma }_N(a|D{|}^{-2})\sim \frac{1}{2\pi }{c}_{00}\log N=\frac{1}{2\pi }\tau (a)\log N$. Hence ${\mathrm{Tr}}_{\omega }(a|D{|}^{-2})=\frac{1}{2\pi }\tau (a)$, the canonical trace up to the constant, even though ${\mathcal{A}}_{\theta }$ has no characters for irrational $\theta$. $\square$

Proposition (measurability via the zeta residue). Let positive $T\in {\mathcal{L}}^{1,\infty }$ have ${\zeta }_T(s)=\mathrm{Tr}(T^s)$ holomorphic for $\mathrm{Re}s>1$ with a simple pole of residue $r$ at $s=1$. The Dirichlet series ${\sum }_n{\mu }_n(T{)}^s$ then satisfies the Karamata Tauberian hypotheses, giving ${\sigma }_N(T)\sim r\log N$. Hence ${\tau }_N(T)\to r$, $T$ is measurable, and ${\mathrm{Tr}}_{\omega }(T)=r$ for all $\omega$. The converse implication (log-asymptotic $\Rightarrow$ simple pole) follows from the Mellin transform $\Gamma (s)\zeta (\text{related})={\int }_0^{\infty }{t}^{s-1}{\sum }_n{e}^{-t/{\mu }_n}dt$ analysed near its rightmost singularity. $\square$

Connections Master

• Spectral triples and the reconstruction theorem 39.06.01 — the noncommutative integral $\text{fint}a={\mathrm{Tr}}_{\omega }(a|D{|}^{-n})$ is the finiteness/absolute-continuity axiom of a spectral triple made explicit; the Dixmier trace is the singular trace named there as ${\mathrm{Tr}}_{\omega }$, and the dimension spectrum is the pole set whose top residue this unit computes.

• Compact operators 02.11.05 — the singular values ${\mu }_n(T)$ of $|T|$ and the Schatten ideals are the substrate; ${\mathcal{L}}^{1,\infty }$ is the weak-${\mathcal{L}}^1$ ideal sitting just above trace class, and the spectral theory of compact operators is what makes the rearranged eigenvalue list well defined.

• Connes-Moscovici local index formula 39.06.06 — the local index formula expands the index pairing as a finite sum of Wodzicki-residue (Dixmier-trace) terms indexed by the dimension spectrum; this unit supplies the residue functional that the index cochain is built from.

• Atiyah-Singer index theorem 03.09.10 — Connes' trace theorem identifies the Dixmier trace of an order-$(-n)$ operator with the symbol integral that computes the topological index; the classical index theorem is the commutative specialisation of the residue-cochain pairing.

• Riemannian volume and the integral 02.11.03 — the Riemannian volume ${\int }_{M}d{\mathrm{vol}}_g$ is recovered as ${\mathrm{Tr}}_{\omega }(|D{|}^{-n})$ up to the constant $c_n$, so the spectral-measure machinery for the unbounded $|D|$ is exactly what makes integration a trace on the operator side; the discrete spectrum and resolvent of $|D|$ supplied there are the very singular-value list this unit averages.

Historical & philosophical context Master

Jacques Dixmier constructed the first example of a non-normal (singular) trace on $\mathcal{B}(\mathcal{H})$ in a 1966 note, Existence de traces non normales ^{[Dixmier 1966]}, answering a question about whether all traces on the bounded operators are normal — that is, computable as suprema over finite-rank projections. His construction used exactly the logarithmic divergence of ${\sum }_{n<N}{\mu }_n$ on the Macaev ideal and an invariant mean to extract a limit. For two decades the object was a curiosity in the theory of operator ideals, studied alongside the Lorentz and Marcinkiewicz spaces by Gohberg, Krein, and others. The Macaev ideal ${\mathcal{L}}^{1,\infty }$ in which it lives had appeared in Macaev's 1961 work on the spectral theory of non-self-adjoint operators, and the systematic theory of symmetrically normed ideals — the framework in which ${\mathcal{L}}^{1,\infty }$ is the limiting Lorentz space — was developed by Gohberg and Krein in their 1965 monograph. That the resulting trace is genuinely non-normal, and so escapes the von Neumann classification of normal traces on $\mathcal{B}(\mathcal{H})$, was the structural surprise: it showed that the algebra of bounded operators carries traces invisible to the predual.

Alain Connes brought it into geometry. In The action functional in noncommutative geometry (1988) ^{[Connes 1988]} he proved that for a classical pseudodifferential operator of order $-n$ on an $n$-manifold the Dixmier trace equals (up to an explicit constant) the Wodzicki residue, the unique trace on the algebra of $\Psi$DOs that Mariusz Wodzicki had isolated in his 1984 thesis and 1987 paper. This identification — operator-theoretic singular trace equals symbolic noncommutative residue — turned ${\mathrm{Tr}}_{\omega }$ into the integral of noncommutative geometry, developed systematically in Connes' 1994 book ^{[Connes Ch. IV]} and its physical application to the spectral action with Chamseddine. The textbook account of measurability and the $\omega$-dependence is in Gracia-Bondía, Várilly, and Figueroa (2001) ^{[Gracia-Bondía-Várilly-Figueroa]}; the definitive modern treatment of singular traces, including the precise heat-kernel and zeta-function formulas and the resolution of subtle $\omega$-dependence questions, is the monograph of Lord, Sukochev, and Zanin (2013) ^{[Lord-Sukochev-Zanin]}.

Bibliography Master

• Connes, A., Noncommutative Geometry, Academic Press, 1994. Ch. IV §2.
• Dixmier, J., "Existence de traces non normales", Comptes Rendus Acad. Sci. Paris 262 (1966), A1107–A1108.
• Connes, A., "The action functional in noncommutative geometry", Communications in Mathematical Physics 117 (1988), 673–683.
• Wodzicki, M., "Noncommutative residue. Chapter I: Fundamentals", in K-theory, Arithmetic and Geometry, Lecture Notes in Math. 1289, Springer (1987), 320–399.
• Gohberg, I. C. & Krein, M. G., Introduction to the Theory of Linear Non-selfadjoint Operators, Translations of Mathematical Monographs 18, AMS, 1969 (Russian orig. 1965).
• Gracia-Bondía, J. M., Várilly, J. C. & Figueroa, H., Elements of Noncommutative Geometry, Birkhäuser, 2001. Ch. 7.
• Connes, A. & Marcolli, M., Noncommutative Geometry, Quantum Fields and Motives, AMS Colloquium Publications 55, 2008. §1.
• Lord, S., Sukochev, F. & Zanin, D., Singular Traces: Theory and Applications, De Gruyter Studies in Mathematics 46, 2013. Ch. 6–11.

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Operator-algebras spine, noncommutative-geometry chapter. The Dixmier trace ${\mathrm{Tr}}_{\omega }$ on the Macaev ideal ${\mathcal{L}}^{1,\infty }$, measurability, Connes' trace theorem identifying it with the Wodzicki residue, and the noncommutative integral $\text{fint}a={\mathrm{Tr}}_{\omega }(a|D{|}^{-n})$ recovering ${\int }_{M}a\sqrt{g}$. Builds on spectral triples (39.06.01) and compact-operator singular values (02.11.05); feeds the local index formula (39.06.06).