REVIEW 3 major objections 4 minor 2 references
Trace Formulas in Noncommutative Geometry
T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The thesis builds a multiple-operator-integral calculus for the abstract pseudodifferential operators of noncommutative geometry, derives a noncommutative Taylor formula and asymptotic heat-trace expansions, and obtains Dixmier trace…
desk verdict The MOI framework in Part I is a real contribution; the quantum ergodicity theorem in Part II rests on an unverified Zelditch lemma. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the multiple operator integral for the abstract pseudodifferential calculus: for symmetric $\Theta$-elliptic operators $H_i\in \mathrm{op}^{h_i}(\Theta)$ with spectral measures $E_i$, and a symbol $\phi$ in the integral projective tensor product $\mathcal{L}^{\beta_0}_\infty(E_0)\ \hat{\otimes}_i\cdots \hat{\otimes}_i\ \mathcal{L}^{\beta_n}_\infty(E_n)$, the integral $T^{H_0,\ldots,H_n}_\phi(X_1,\ldots,X_n)$ integrates $a_0(H_0,\omega)X_1a_1(H_1,\omega)\cdots X_na_n(H_n,\omega)$ as a Bochner integral in every $H^s(\Theta)$. Divided differences $f^{[n]}$—the finite-difference quotients that reduce to $f^{(n)}/n!$ on the diagonal—provide the symbols: for $f\in T^\alpha(\mathbb{R})$ they lie in the corresponding tensor-product classes, and the MOI identity that moving an operator $a$ past an argument $X_j$ costs the commutator $[H_j,a]$ is what drives the Taylor expansion and its remainder estimates. For the second part the carrying device is the (local) Weyl law, $\operatorname{Tr}(e^{-tD^2})\sim C t^{-d/2}$ and $\operatorname{Tr}(a e^{-tD^2})\sim C(a)t^{-d/2}$, which upgrades logarithmic averages of diagonal matrix entries to the Dixmier trace $\operatorname{Tr}_\omega(a\langle D\rangle^{-d})$, together with Widom's lemma on the vanishing of $\operatorname{Tr}(P_\lambda A(1-P_\lambda)BP_\lambda)/\operatorname{Tr}(P_\lambda)$ that yields the Szegő limit theorem.
What would settle it
Take any regular $s$-summable spectral triple satisfying the local Weyl law for all $P\in\mathcal{B}$, with a bounded self-adjoint $V$; a computation that violated the stated remainder $O(t^{N+1-s})$ in Corollary 3.6.6 for some $N$ would refute the heat-trace expansion claim. Similarly, a spectral triple satisfying the Weyl law for which the logarithmic mean of $\operatorname{Tr}(P_{\lambda_n}aP_{\lambda_n})/\operatorname{Tr}(P_{\lambda_n})$ differs from $\operatorname{Tr}_\omega(a\langle D\rangle^{-d})/\operatorname{Tr}_\omega(\langle D\rangle^{-d})$ would refute Theorem 4.1.7.
Extended reading notes
Core claim
On the author's own terms, the discovery is that the theory of multiple operator integrals, which Peller built for bounded operators using integral projective tensor products of $L^\infty$ spaces, survives transplant into the Connes–Moscovici calculus of abstract pseudodifferential operators. The thesis constructs a functional calculus for symmetric $\Theta$-elliptic operators of positive order and for zero-order operators, then defines $T^{H_0,\ldots,H_n}_\phi(X_1,\ldots,X_n)$ by a Bochner integral in every Sobolev space $H^s(\Theta)$, with symbol $\phi$ in the integral projective tensor product $\mathcal{L}^{\beta_0}_\infty(E_0)\ \hat{\otimes}_i\cdots \hat{\otimes}_i\ \mathcal{L}^{\beta_n}_\infty(E_n)$. The payoff is a noncommutative Taylor expansion (Theorem 3.5.5) and asymptotic trace expansions on regular $s$-summable spectral triples (Corollary 3.6.6), together with the counterexample showing the local Weyl law is genuinely necessary. In the second part, spectrally truncated quotients $\operatorname{Tr}(P_\lambda a P_\lambda)/\operatorname{Tr}(P_\lambda)$ are shown to compute the noncommutative integral under a Weyl law, a Szegő limit theorem follows, and the density of states on discrete metric spaces and bounded-geometry manifolds is expressed through Dixmier traces, yielding a Dixmier trace formula for Roe's index on open manifolds.
Load-bearing premise
The load-bearing premise is that $D^2$ satisfies a (local) Weyl law—eigenvalue growth $\lambda_k(D^2)\sim C k^{2/d}$ with the matching behaviour of $\operatorname{Tr}(a e^{-tD^2})$—so that spectral averaging converges to the noncommutative integral, and, for the Taylor expansions of Part I, that the commutators $\delta^k_H(V)$ decay in operator order by a fixed $\varepsilon>0$; the thesis itself constructs a spectral triple where the global Weyl law holds but the local ones fail, so neither premise is automatic.
Editorial extensions
If this is right
- Heat traces on perturbed spectral triples admit computable asymptotic expansions: $\operatorname{Tr}(P e^{-t(D+V)^2})$ and $\operatorname{Tr}(P e^{-t|D+V|})$ are expanded to arbitrary order in $t$ with coefficients built from commutators of $D$ and $V$, on every regular $s$-summable spectral triple (Corollary 3.6.6).
- The Eckstein–Iochum question is settled: expanding $\operatorname{Tr}(e^{-tD^2})$ alone is insufficient—the diagonal counterexample has no local Weyl law—but expanding $\operatorname{Tr}(P e^{-tD^2})$ for all $P\in\mathcal{B}$ does transfer to the perturbed operator.
- Truncated spectral triples are faithful: under a Weyl law, the noncommutative integral of $a$ is the logarithmic mean of the finite-dimensional quotients $\operatorname{Tr}(P_\lambda a P_\lambda)/\operatorname{Tr}(P_\lambda)$, and Widom's lemma upgrades this to a noncommutative Szegő limit theorem for $f(A)$.
- Classical ergodicity implies quantum ergodicity in NCG: a classically ergodic spectral triple has a density-one subsequence of eigenbasis vectors along which $\langle e_j, a e_j\rangle$ converges to the noncommutative integral of $a$ (Theorem 4.4.11).
- The density of states becomes a Dixmier trace: for Hamiltonians on discrete metric spaces and on manifolds of bounded geometry, the DOS is computed by a Dixmier trace, and this yields a Dixmier trace formula for Roe's index on open manifolds.
Reading between the lines
- Extension: because the MOI construction only needs the abstract Hilbert-scale calculus, the same noncommutative Taylor expansion should hold in any concrete pseudodifferential calculus that fits the $\Theta$-scale format, including filtered manifolds and Lie groupoids, without repeating the symbol-level estimates.
- Extension: the truncated-triple formula links the noncommutative integral to numerically computable finite-dimensional data; one could use it to test ergodicity of noncommutative spaces by computing logarithmic means of eigenbasis matrix entries, which the paper does not do.
- Extension: the Dixmier-trace formula for the density of states suggests defining the DOS as a Dixmier trace even when the spatial average defining it is not known to converge; the paper shows Dixmier measurability is strictly weaker than existence on discrete spaces but leaves the question open for Schrödinger-type operators on manifolds.
- Extension: the commutator-decay hypothesis $\delta^k_H(V)\in\mathrm{op}^{r+k(h-\varepsilon)}(\Theta)$ is stronger than a mere order bound on $V$; if it could be relaxed to allow $\varepsilon=0$ with log-type losses, the Taylor expansion would extend to perturbations of the same order as $H$, which the current statement excludes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This PhD thesis develops trace formulas in noncommutative geometry in two parts. Part I adapts the theory of multiple operator integrals to the abstract pseudodifferential calculus of Connes–Moscovici: it constructs functional calculi for elliptic and zero-order operators, builds multiple operator integrals whose entries may be unbounded pseudodifferential operators (Theorem 3.2.5), derives a noncommutative Taylor expansion (Theorem 3.5.5), and obtains asymptotic trace expansions for perturbations of regular s-summable spectral triples (Corollary 3.6.6), thereby addressing a question of Eckstein and Iochum. Part II studies Connes' integral formula through spectral truncations: Theorem 4.1.7 identifies the noncommutative integral with suitable averaged limits of normalized traces on spectral projections under a Weyl law, Theorem 4.2.2 gives a Szegő limit theorem, and Theorem 4.4.11 claims an NCG analogue of 'ergodic flow implies quantum ergodicity'. The final chapters give Dixmier trace formulas for the density of states on discrete metric spaces and on manifolds of bounded geometry, with an application to Roe's index on open manifolds.
Significance. If the results are valid, the thesis makes a substantial contribution. The construction of multiple operator integrals for abstract pseudodifferential operators is a genuine generalization of Peller's theory and provides a common language for operator integral arguments in NCG; the noncommutative Taylor expansion and the trace expansions are parameter-free theorems with explicit constants, not fitted quantities. The counterexample in Example 3.6.5 is concrete and checkable, and it precisely shows the limitations of Weyl-law assumptions. Part II offers a fresh bridge between NCG and quantum ergodicity and gives general Dixmier trace formulas for the density of states, including a Roe-index application. The proofs of the core Part I results are detailed, and the Borel lemma and Hadamard three-line argument are written out. The main weakness is the proof of the Part II flagship ergodicity theorem, which rests on an unverified transfer of Zelditch's Lemma 2.1 to noncommutative C*-dynamical systems.
major comments (3)
- [Section 4.4, Theorem 4.4.11] The proof of Theorem 4.4.11 is a one-line reduction: 'Classical ergodicity ... means precisely that (S*A,R,σ_t) has a unique vacuum state ... hence ... a consequence of [Zel96, Lemma 2.1].' The lemma is not stated, and its hypotheses are not checked for the noncommutative C*-dynamical system (S*A,R,σ_t). The thesis itself notes that Zelditch's results are mostly formulated for 'quantised abelian' systems, and Example 4.4.12(2) explicitly corrects a Zelditch corollary in the noncommutative setting. A unique invariant vector in L^2 gives mean ergodic convergence (Proposition 4.4.15), which is not obviously equivalent to density-one convergence of matrix coefficients for every eigenbasis. To make Theorem 4.4.11 load-bearing, the author must either state and prove the needed version of Zelditch's lemma for noncommutative S*A, or prove the density-one convergence directly; as written, the claimed NCG analogue of 'ergodic flow implies quantum ergodicity' is unsupported.
- [Section 4.1, Definition 4.1.3 and Theorem 4.1.7] The Weyl law assumption is the load-bearing premise of Part II: Theorem 4.1.7, Theorem 4.2.2, and Theorem 4.4.11 all depend on the global and local growth of the eigenvalue counting function. The thesis correctly demonstrates in Example 3.6.5 that the local Weyl law is not automatic even when the global Weyl law holds. This is not an internal inconsistency, but the presentation should more prominently state that the local Weyl law is a genuine additional hypothesis whose verification for a given spectral triple is a separate problem, and the reader should be pointed to the precise places where failure of the local law invalidates the conclusions. In particular, the proof of Theorem 4.1.7 uses the local trace only through the constants C(a), so the statement should make explicit that the result is conditional on the local Weyl law for the specific a in question.
- [Section 3.2, Theorem 3.2.5] The independence of the multiple operator integral from the chosen integral representation of the symbol is asserted by reference to [ACDS09, Lemma 4.3]. Since the present setting allows unbounded self-adjoint operators H_i and the entries are abstract pseudodifferential operators rather than bounded operators, the transfer of that argument is not automatic. The text indicates the rank-one operators θ_{η,ξ} lie in op^{-∞}(Θ), but the proof of independence in [ACDS09] is formulated for bounded Hilbert-space operators. Please expand the argument here, or at least spell out why the unbounded case follows from the same computations on H^s for every s. This point is central because all later MOI identities and trace expansions inherit the well-definedness of the construction.
minor comments (4)
- [Example 3.6.8] The text says 'we can choose ε = 0' in applying Corollary 3.6.6, but Corollary 3.6.6 states the error term for arbitrary ε>0. Please justify why the endpoint ε=0 is admissible in this example or rephrase to avoid the appearance of an inconsistent choice.
- [Corollary 3.6.6] The symbol B is overloaded: it denotes both the algebra generated by A and D and, later in the same corollary, the operator |D+V|-|D|. Please use different notation for one of these objects to avoid confusion.
- [Section 4.4, Proposition 4.4.6] The statement that τ defines a finite positive trace on S*A is labeled 'well-known' and the proof is only sketched. Since the quotient by K(H) is involved, a short argument showing well-definedness on the quotient (i.e., vanishing on compact perturbations) would improve readability and make the subsequent L^2 construction self-contained.
- [Chapter 3, Lemma 3.3.1] The proof of part 1 of Lemma 3.3.1 first establishes the integral representation for compactly supported smooth functions and then invokes density of C_c^∞ in T_α(R). Please add a sentence explaining why the divided differences f^[n] depend continuously on f in the relevant norm, since this is not entirely immediate from the definition of the ⊠_i seminorms.
Circularity Check
No significant circularity: the trace formulas and Taylor expansions are derived from independently stated functional-calculus and MOI estimates, not from their conclusions.
full rationale
The central derivations are self-contained rather than circular. Chapter 2 builds a functional calculus for abstract pseudodifferential operators from Θ-ellipticity and resolvent estimates, with no use of the trace formulas it later feeds. Chapter 3 constructs MOIs by adapting Peller's integral representations and proves the noncommutative Taylor expansion from MOI identities (Proposition 3.4.1) and divided-difference estimates (Lemma 3.3.1); the commutator decay hypothesis δ^k_H(V)∈op^{r+k(h-ε)} is an input controlling the remainder order, not a restatement of the expansion. The heat-trace and spectral-action expansions (Theorem 3.6.3, Corollary 3.6.6) then follow by estimating trace-class norms of the Taylor remainders. Part II likewise has no fitted constants: the constants in Theorem 4.1.7 are fixed by the assumed Weyl law and by the cited [LSZ21, Cor. 8.1.3] heat-trace/Dixmier-trace relation, and the Szegő and ergodicity statements are imported from Widom and Zelditch as external lemmas. The proof of Theorem 4.4.11 is terse: it reduces to [Zel96, Lemma 2.1] after identifying 'classically ergodic' with Zelditch's unique-vacuum-state hypothesis, and the paper does not spell out the verification of that lemma's hypotheses for noncommutative S*A; but that is a support/completeness concern, not circularity, since the lemma is not one of the paper's own fitted inputs and the conclusion is not assumed in the hypotheses. The self-citations to [HMN24], [HM24b], [Aza+22], [HM24a] are to the papers on which the thesis is based, and the relevant proofs are reproduced in the text.
Assumptions & free parameters
assumptions (9)
- standard math Borel functional calculus for unbounded self-adjoint operators, including fractional powers Θ^s of the scale operator (Definition 2.1.1, §1.6).
- standard math Complex interpolation for the Hilbert scale (Stein–Weiss) and the Hadamard three-line theorem (Proposition 2.1.2, proof of Proposition 2.2.1).
- standard math Hardy–Littlewood Tauberian theorem and Hardy's regular transformation theory for series means (Proposition 4.1.1, Lemma 4.1.5).
- standard math Peller's bounded-operator multiple operator integral theory, including the integral projective tensor product and independence of representation ([Pel06; Pel16], [ACDS09, Lemma 4.3]).
- domain assumption Θ-ellipticity as the abstract ellipticity condition: existence of a parametrix of order -r (Definition 2.1.7).
- domain assumption Weyl law and local Weyl law for D^2: Tr(e^{-tD^2}) ~ C t^{-d/2} and Tr(ae^{-tD^2}) ~ C(a) t^{-d/2} (Definition 4.1.3).
- domain assumption Commutator decay condition δ^k_H(V) ∈ op^{r+k(h-ε)}(Θ) for some ε>0 (Theorem 3.5.5), and regularity of the spectral triple (δ^n(a) bounded, Theorem 4.4.3).
- domain assumption Separability of the Hilbert space H and of the closure of A in B(H) in Theorem 4.4.11 (Definition 1.6 and §4.4).
- domain assumption Existence of the density of states measure, via Riesz–Markov–Kakutani, whenever the local trace limits exist (Section 1.5, following Simon [Sim82]).
invented entities (2)
-
Θ-ellipticity (Definition 2.1.7)
independent evidence
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Noncommutative cotangent sphere S*A with trace τ and automorphism flow σ_t (Section 4.4)
independent evidence
Cite this review
Pith. "Pith review of Trace Formulas in Noncommutative Geometry." pith.science (2026). https://pith.science/paper/Q2SE7G3G
@misc{pith2026250621950,
author = {Pith},
title = {Pith review of: Trace Formulas in Noncommutative Geometry},
year = {2026},
howpublished = {\url{https://pith.science/paper/Q2SE7G3G}},
note = {Machine review of arXiv:2506.21950}
}
read the original abstract
Trace formulas appear in many forms in noncommutative geometry (NCG). In the first part of this thesis, we obtain results for asymptotic expansions of trace formulas like heat trace expansions by adapting the theory of Multiple Operator Integration to NCG. More broadly, this construction provides a natural language for operator integrals in NCG, which systematises and simplifies operator integral arguments throughout the literature. Towards this end, we construct a functional calculus for abstract pseudodifferential operators and generalise Peller's construction of multiple operator integrals to this abstract pseudodifferential calculus. In the process, we obtain a noncommutative Taylor formula. In the second part of this thesis, we shift our attention to Dixmier trace formulas. First, we provide an approximation of the noncommutative integral for spectrally truncated spectral triples in the Connes--Van Suijlekom paradigm of operator system spectral triples. Our approximation has a close link to Quantum Ergodicity, which we will use to state an NCG analogue of the fundamental result that ergodic geodesic flow implies quantum ergodicity. Furthermore, we provide a Szeg\H{o} limit theorem in NCG. Next, we provide a Dixmier trace formula for the density of states, a measure originating in solid state physics that can be associated with an operator on a geometric space. We first provide this formula in the setting of discrete metric spaces, and then in the setting of manifolds of bounded geometry. The latter leads to a Dixmier trace formula for Roe's index on open manifolds.
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Works this paper leans on
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arXiv 1982
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