Paving arbitrary MASAs: strong topology and quadratic bounds

Source and verification

This article explains family 300 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Paving divides a Hilbert space into diagonal blocks and asks whether the surviving part of an operator is small. Family 300 studies two ways to relax norm paving: perturb the operator in the strong topology, or compress its pinching to a projection approaching the identity strongly. The manuscripts claim both properties for every maximal abelian von Neumann subalgebra. The second obtains the optimal quadratic order in the inverse error. Their different quantifiers and different constructions are central to the comparison. (OpenAI, 2026) (OpenAI, 2026b)

🏷️ Preliminaries: diagonal blocks and strong topology

A von Neumann algebra $M\subseteq \mathcal{B}(H)$ is a unital algebra of bounded operators, closed under adjoints and weak operator limits. A maximal abelian subalgebra, abbreviated MASA, is an abelian unital subalgebra $A$ with $A^{\prime }\cap M=A$. For matrices, the usual diagonal algebra is the basic example.

A projection $p=p^{\ast }=p^2$ selects a closed subspace. An $r$-partition in $A$ consists of mutually orthogonal projections summing to $1$; zero pieces are allowed. Its pinching is

$$S_P(x)=\sum _{i=1}^r{p}_{i}x{p}_i.$$

In a matrix basis diagonalizing $A$, this deletes entries between different parts of a partition of the coordinates. Subtracting $a\in A$ removes a diagonal contribution.

Norm paving requires $\Vert S_P(x)-a\Vert$ to be small. Strong operator convergence only tests individual vectors: $z_j\to z$ strongly means $\Vert (z_j-z)\xi \Vert \to 0$ for every $\xi \in H$. A neighborhood is specified by finitely many vectors $F$ and a positive tolerance $\delta$. Smallness on these tests does not imply small operator norm.

For instance, tail projections on ${\ell }^2(\mathbb{N})$ converge strongly to zero while each has norm one. Accordingly, a projection $q$ can be close to $1$ on the prescribed vectors while discarding an entire subspace.

A normal conditional expectation $E_A:M\to A$ is a positive unital projection preserving increasing bounded suprema and satisfying $A$-bimodularity. It plays the role of conditional averaging. An arbitrary MASA need not admit one.

A normal state $\phi$ is a positive ultraweakly continuous functional with $\phi (1)=1$. It is faithful if $\phi (z^{\ast }z)=0$ forces $z=0$. Its centralizer consists of $b$ satisfying $\phi (bz)=\phi (zb)$ for all $z$. When $\phi$ is faithful and $A$ lies in its centralizer, a $\phi$-preserving normal expectation $E_A$ exists. The state norm

$$\Vert z{\Vert }_{\phi }=\phi (z^{\ast }z{)}^{1/2}$$

controls strong convergence of bounded sequences in the faithful state representation. The state need not be tracial: cyclically rearranging arbitrary products is unavailable.

🏷️ Two paving statements

Approximation paving, as stated in the first manuscript

For each $0<\epsilon <1$, choose an integer $R\le C_{\mathrm{ap}}{\epsilon }^{-6}$, with universal $C_{\mathrm{ap}}$. For every represented inclusion $A\subseteq M$, every self-adjoint $x\in M$, and every finite-vector neighborhood $(F,\delta )$, there are a self-adjoint $y$, an $R$-partition $P$ in $A$, and self-adjoint $a\in A$ such that $\Vert y\Vert \le 3\Vert x\Vert$, $\Vert (y-x)\xi \Vert <\delta$ on $F$, $\Vert a\Vert \le \Vert y\Vert$, and $\Vert S_P(y)-a\Vert \le \epsilon \Vert y\Vert$. (OpenAI, 2026a)

The error is relative to the actual approximant norm. The partition count is chosen before the inclusion, operator, or neighborhood.

Quadratic strong-operator paving, as stated in the companion

Fix $0<\epsilon <1$. Every self-adjoint $x$ in every represented MASA inclusion has the following property. Given $(F,\delta )$, there are an $r_{\epsilon }$-partition $P$ in $A$, self-adjoint $a\in A$ with $\Vert a\Vert \le \Vert x\Vert$, and a projection $q\in M$ such that $\Vert (1-q)\xi \Vert <\delta$ on $F$ and $\Vert q(S_P(x)-a)q\Vert \le \epsilon \Vert x\Vert$. The universal count is

$$\begin{array}{rl}{r}_{\epsilon }& =\lceil 400000000{\epsilon }^{-2}\rceil \\ & +\lceil 4/\epsilon \rceil +1\\ & \le 500000000{\epsilon }^{-2}.\end{array}$$

No separability or conditional-expectation assumption is imposed. (OpenAI, 2026b)

Here $q$ need not belong to $A$. The operator being pinched remains $x$, but the estimate holds on a large compression. Neither theorem asserts norm paving of every original operator in every MASA.

Under a normal-expectation hypothesis, the manuscripts use the established Popa–Vaes conversion between these two formulations. Together with the companion theorem, it yields quadratic approximation paving for expected MASAs. The arbitrary-MASA approximation theorem above retains its sixth-power bound.

🏷️ The normalizer decomposition

First assume a faithful normal state central over $A$, and reduce to separable predual. A partial normalizer is a partial isometry carrying one corner of $A$ onto another by conjugation. Let $N$ be the von Neumann algebra they generate. There is a state-preserving expectation $E_N$, and

$$\begin{array}{rl}x-E_{A}x& =y+w,\\ y& =E_{N}x-E_{A}x,\\ w& =x-E_{N}x.\end{array}$$

For $\Vert x\Vert \le 1$, both terms have norm at most two.

The inclusion $A\subseteq N$ is Cartan: normalizers generate the ambient algebra. It can be represented by a countable measured equivalence relation on a probability space. Operators become matrices along equivalence classes; changing the root may conjugate them by a diagonal unitary.

The other term lies in $\ker E_N$. Its left-right interaction with $A$ is diffuse. More precisely, for $w\in \ker E_N$ the positive map

$$K_w(f)=E_A(w^{\ast }fw),f\in A,$$

has an atomless kernel: conditional on almost every output point, the associated measure on input points gives zero mass to singletons.

The manuscript proves this structural lemma by contradiction. A measurable family of atoms would produce a nonzero intertwiner $fb=b\theta (f)$ in $\ker E_N$. Polar decomposition and a type I corner argument express that intertwiner through partial normalizers, placing it in $N$. Since $N\cap \ker E_N=\{0\}$, the atoms cannot exist. Finite compositions of these kernels remain atomless, with fixed null exceptional sets handled before composition. (OpenAI, 2026b)

🏷️ A single coloring for both terms

The quadratic gain comes from using the same partition on $y$ and $w$. Independently paving them and refining the two partitions would multiply the numbers of pieces.

The Cartan construction attaches independent labels to the sites of each equivalence class: a relative Bernoulli extension. For a finite-degree graph supporting a zero-diagonal Hermitian contraction $T$, the manuscript’s coloring theorem produces $r$ colors with uniform conditional marginals. For any prescribed ball radius $m$ and positive $\alpha ,\eta$, it gives, outside a set of measure less than $\eta$,

$$\begin{array}{c}\Vert T_{B(v,m),c}\Vert \le 2\beta t+\alpha ,\\ \beta =80,t>3/\sqrt{r}.\end{array}$$

Here $T_{B(v,m),c}$ is the matrix on the radius-$m$ ball after deleting entries of different colors. The graph degree and edge measure-distortion bounds affect the construction, not the final color count.

Its finite input uses real-rooted mixed determinantal polynomials for both signs of the matrix. The sum of positive roots measures the cost of a partial coloring. Rooted spectral measures distribute this cost over vertices; assignment inequalities allow additional vertices to receive colors while controlling its increase.

The underlying relation may be nonsingular rather than measure preserving. The proof therefore passes to a Maharam extension with an extra real height coordinate, where invariant mass transport is available. Weight-deletion estimates and cutoff limits are needed to return a uniform cost to the original probability space. These lengthy polynomial and measured-coloring arguments are inputs to this exposition; they have not been independently verified here. (OpenAI, 2026b)

Place the extension algebra $\tilde{N}$ and $M$ in the amalgamated free product $\mathcal{K}=M{\ast }_N\tilde{N}$. Its faithful normal model state is $\Phi =\phi {|}_N\circ E_N^{\mathcal{K}}$, extending $\phi$, where $E_N^{\mathcal{K}}$ is the expectation onto $N$. “Free over $N$” means that the expectation of an alternating product of centered letters from the two algebras is zero.

For a color projection $p$ with conditional expectation $\lambda 1$, the companion proves

$$\Vert pwp\Vert \le (\lambda +2\sqrt{\lambda })\Vert w\Vert .$$

There is a short mechanism behind this estimate. In the free-product Hilbert space, let $Q$ project onto words beginning with a centered $M$-letter. Then

$$\begin{array}{c}(1-Q)w(1-Q)=0,\\ QpQ=\lambda Q.\end{array}$$

Expand $pwp$ using $Q+(1-Q)$ on both sides of $w$. Three terms remain; their bounds are $\lambda \Vert w\Vert$ and twice $\sqrt{\lambda }\Vert w\Vert$. Uniform color marginals give $\lambda =1/r$. Orthogonality of the color corners turns this into a bound for their sum without an additional factor $r$.

🏷️ From model colors to actual diagonal projections

The model colors live in an extension, so another step is essential. Approximate them by functions of finitely many label symbols. Randomly assign these symbols to pieces of increasingly fine partitions of the original $A$. This evaluates every cylinder function in $A$ and preserves projection partitions.

The transfer statement concerns each fixed scalar moment: its expected value under this evaluation converges to the corresponding model moment. All factors in a word use the same label assignment.

To explain why diffuse kernels matter, expand centered cylinder functions in finite Fourier series. Label averaging leaves patterns where matching partition indices cancel the Fourier exponents. A shortest linked pair in such a cyclic word permits state Cauchy–Schwarz:

$$\begin{array}{c}\left|\sum _J\phi (X_J{Y}_J)\right|\\ \le {\left(\sum _J\phi (X_J{X}_J^{\ast })\right)}^{1/2}\\ \cdot {\left(\sum _J\phi (Y_J^{\ast }{Y}_J)\right)}^{1/2}.\end{array}$$

One square sum stays bounded. The other tests whether two endpoints fall into the same partition piece under an atomless composed kernel. First refine the interior partition while keeping endpoint data coarse; only then refine the coarse endpoint partition. The limiting collision set is a graph with zero conditional mass. Reversing this order would discard the uniformity needed for the argument.

Nontracial states require further control: a bounded sequence small in state norm cannot be multiplied on the right by arbitrary varying bounded operators without justification. The proof uses modular analytic approximation and a class of right-multiplier sequences that preserve bounded state-norm-null sequences. Constants and bounded centralizer sequences belong to this class. Its closure properties also justify the continuous functional calculus used next. (OpenAI, 2026b)

🏷️ Clipping, moments, and the large projection

Fix $r=\lceil 20000^2{\epsilon }^{-2}\rceil$ first. Approximate the Cartan term $y$ by finite-support $y_n$, with $\Vert y_n\Vert \le 2$. Apply the coloring theorem at radius $n$, with

$$\begin{array}{c}t=4/\sqrt{r},\alpha =\epsilon /32,\\ \eta =(\epsilon /16{)}^{2n}.\end{array}$$

For the pinched approximant ${\hat{Y}}_n$, integration of rooted matrix coefficients gives

$$\Phi ({\hat{Y}}_n^{2n})\le (\epsilon /4{)}^{2n}+2^{2n}(\epsilon /16{)}^{2n}.$$

This uses the fact that paths of length $n$ remain in the radius-$n$ ball. It does not require $\Phi$ to be a trace.

Clip the Cartan pinch to $[-\epsilon /2,\epsilon /2]$. The tail estimate makes this correction state-norm small. The right-multiplier calculus transfers that conclusion from ${\hat{Y}}_n$ to the pinch of the original $y$. The pinch of $w$ has norm less than $\epsilon /4$ by free compression, using the same colors.

Thus the model pinch $Z_n$ of $x-E_{A}x$ is state-norm close to a bounded operator of norm at most $3\epsilon /4$. For every fixed positive integer $k$,

$$\underset{n}{\mathrm{limsup}}\Phi (Z_n^{2k})\le (3\epsilon /4{)}^{2k}.$$

Given a desired lost state mass $\rho >0$, choose $k$ so large that $(3/4{)}^{2k}<\rho /4$. Then choose $n$, cylinder approximations, and finally a fine diagonal evaluation. Moment transfer supplies a genuine partition in $A$ with residual $z$ satisfying $\phi (z^{2k})<\rho {\epsilon }^{2k}$.

The spectral projection $q=1_{[-\epsilon ,\epsilon ]}(z)$ obeys

$$\phi (1-q)<\rho ,\Vert qzq\Vert \le \epsilon .$$

The order matters: the number of colors was fixed before $\rho$ and before all moment and approximation choices. No exponentially fast strong approximation of the original operator is assumed.

🏷️ Removing the central-state hypothesis

Let $p\in A$ be the supremum of supports of all $A$-central normal positive functionals. Given the finite test vectors, choose one such support $e\le p$ so that $p-e$ is small on them. The corner $eMe$ has a faithful central state; its lost state mass can be made small enough for all prescribed vector tests.

On the corner $1-p$, no nonzero central normal functional exists. The companion uses the symmetric Haagerup-space map

$$T(z)=h_{\phi }^{1/4}z{h}_{\phi }^{1/4}$$

for a chosen normal state with density $h_{\phi }$. Two randomizations produce small symmetric $L^2$ norm with only $\lceil 4/\epsilon \rceil$ colors. Such smallness does not directly imply small $\Vert z{\Vert }_{\phi }$. A positive-cone overlap estimate, Jordan decomposition of functionals, and a subsequent spectral cut produce the required large compression.

Use disjoint color sets on $e$, $1-p$, and the discarded corner $p-e$. These projections need not be central in $M$, but pinching now kills all cross-corner terms. Consequently the color counts add, giving the displayed $r_{\epsilon }$.

🏷️ The approximation-paving construction

The first manuscript combines separate upper and lower Cartan pavings with a diffuse-part paving. Their common refinement has

$$\begin{array}{c}{R}_{\mathrm{st}}(\gamma )=(r+1{)}^{2}n,\\ r,n=O({\gamma }^{-2}),\end{array}$$

which explains the sixth power. In a suitable state-null sequence quotient, the residual has norm at most $\gamma$. Clipping it gives block-diagonal self-adjoint corrections $w_j$ satisfying

$$\begin{array}{c}\Vert w_j\Vert \le 2,\Vert w_j{\Vert }_{\phi }\to 0,\\ \Vert S_{P_j}(x-w_j)-E_{A}x\Vert \le \gamma .\end{array}$$

The quotient uses the normalizer of the null ideal; the nontracial null sequences are not treated as an ideal in all bounded operator sequences.

On the corner without central normal functionals, almost disjoint conjugate states yield positive contractions $s$ with large prescribed state mass and $p_{i}s{p}_i\le p_i/m$. Then $sxs$ approximates $x$ strongly and has pinching norm at most $1/m$ for $\Vert x\Vert \le 1$.

The final assembly retains the original off-diagonal blocks until pinching. Its correction has norm at most two, giving $\Vert y\Vert \le 3$ for normalized $x$. Add a unit test vector with $\Vert x{\xi }_0\Vert >3/4$ and approximate it within $1/4$; this ensures $\Vert y\Vert >1/2$. With $\gamma =\epsilon /4$, clip the diagonal witness to $[-\Vert y\Vert ,\Vert y\Vert ]$. The final error is at most $2\gamma \le \epsilon \Vert y\Vert$, proving the stated normalization. The case $x=0$ is immediate. (OpenAI, 2026a)

🏷️ Sharp exponent and scope of consequences

The companion invokes the Popa–Vaes finite-factor obstruction to show that quadratic order is necessary. For $n<{\epsilon }^{-2}$, choose $\epsilon <\eta <n^{-1/2}$ and a centered self-adjoint unitary $x$ whose every $n$-piece pinch has $\Vert S_P(x){\Vert }_2>\eta$.

Trace orthogonality yields

$$\Vert S_P(x)-a{\Vert }_2^2=\Vert S_P(x){\Vert }_2^2+\Vert a{\Vert }_2^2.$$

For $z=S_P(x)-a$ with $\Vert a\Vert \le 1$, one has $\Vert z\Vert \le 2$. If $\Vert qzq\Vert \le \epsilon$, splitting off the two discarded corner terms gives

$$\Vert S_P(x){\Vert }_2\le \epsilon +4\sqrt{\tau (1-q)}.$$

Testing the strong neighborhood on the trace vector makes the last term smaller than $\eta -\epsilon$, a contradiction. The exponent is optimal in this uniform sense; the large numerical constant is not optimized.

For a countably decomposable expected MASA, the companion also imports a transfer theorem giving norm paving in the associated Ocneanu ultrapower with the same error and count. This consequence is about the ultrapower, not norm paving in every original inclusion.

🏷️ Key steps and proof outlines

Clipping separates two limits that cannot be taken together

The companion’s completion argument uses moments of increasing order to control the approximants, but transfers only a fixed moment back to the original MASA. Here is the bridge between these operations. Its bound on ${\hat{Y}}_n$ implies

$$\Phi \left(1_{\{|{\hat{Y}}_n|>\epsilon /2\}}\right)\le 2^{-2n}+4^{-2n}.$$

This is spectral Markov: on the indicated spectral set, $|{\hat{Y}}_n{|}^{2n}\ge (\epsilon /2{)}^{2n}$. Since $\Vert {\hat{Y}}_n\Vert \le 2$, removing that tail by the clipping function $f$ makes $\Vert {\hat{Y}}_n-f({\hat{Y}}_n){\Vert }_{\Phi }$ tend to zero. The proof now uses continuity of functional calculus in a carefully controlled sequence class, rather than attempting to estimate the difference of the $2n$th powers of $Y_n$ and ${\hat{Y}}_n$. The latter would demand an unavailable rate of approximation. (OpenAI, 2026b)

To see the sequence-class argument, define ${\mathcal{R}}_{\Phi }$ to contain bounded sequences $R_n$ for which $D_n{R}_n$ is state-norm null whenever $D_n$ is bounded and state-norm null. It is closed under sums, products, uniform norm limits, and addition of a bounded state-norm-null sequence. For self-adjoint sequences $U_n,V_n$ in this class, use

$$U_n^j-V_n^j=\sum _{h=0}^{j-1}{U}_n^h(U_n-V_n)V_n^{j-1-h}.$$

The right factor preserves state-norm convergence by definition, and left multiplication costs only the operator norm. This proves the desired continuity for fixed polynomials; uniform polynomial approximation proves it for continuous $f$. The color projections are in the centralizer and hence in this class, as are the pinches made from them and fixed operators.

Therefore $Y_n-f(Y_n)$ is state-norm null. Adding the diffuse pinch $W_n$ gives $Z_n$ close in state norm to $Q_n=f(Y_n)+W_n$, with $\Vert Q_n\Vert \le 3\epsilon /4$. Apply the same telescoping identity at a fixed exponent $2k$. After choosing $k$ to fit the desired lost mass, choose $n$, then cylinder approximations, then the evaluation partition. The average evaluated even moment is small and nonnegative, so at least one deterministic evaluation has the required bound. This last probabilistic selection produces actual projections in $A$.

A strong approximation needs tests on both the vector and its image

The first manuscript’s corner without central functionals supplies a positive contraction $s$ such that a chosen state gives $s$ nearly full mass and $p_{i}s{p}_i\le p_i/m$. To approximate $x$ on test vectors ${\xi }_j$, choose that state from the vector functionals of both ${\xi }_j$ and $x{\xi }_j$. Since $(1-s{)}^2\le 1-s$, large state mass makes $(1-s)$ small on all these vectors. Then

$$sxs-x=sx(s-1)+(s-1)x$$

proves the required strong approximation with $m$ held fixed. For a self-adjoint contraction $x$,

$$-s^2\le sxs\le s^2\le s,$$

so every compressed block is bounded between $-p_i/m$ and $p_i/m$. The block sum has norm at most $1/m$, because its blocks are orthogonal. (OpenAI, 2026a)

The final normalization uses one additional unit vector on which $\Vert x{\xi }_0\Vert >3/4$. Approximating it within $1/4$ forces $\Vert y\Vert >1/2$. If the provisional diagonal $d$ has error $\gamma$ from the pinch of $y$, then $\Vert d\Vert \le \Vert y\Vert +\gamma$. Clipping $d$ to $[-\Vert y\Vert ,\Vert y\Vert ]$ costs at most another $\gamma$. With $\gamma =\epsilon /4$, the resulting error is at most $\epsilon /2\le \epsilon \Vert y\Vert$. This proves the theorem’s relative error, including the dependence on the actual approximant norm.

🏷️ Links and reading scope

• Family 298 uses free-probability models to study matrix constraints; here amalgamated freeness controls the diffuse part of a pinching.
• The collection guide places these manuscripts in the operator-analysis sequence.

The selected-source reading covers both introductions, the first manuscript’s state-topology setup and general assembly, selected expected-MASA arguments, and the companion’s reductions, atomless-kernel mechanism, selected transfer arguments, and completion. The full measured-coloring proofs, all polynomial estimates, and cited external structure theorems were not independently audited.

🏷️ Manuscripts and verification scope

• Approximation Paving over Arbitrary Maximal Abelian Subalgebras — pinned manuscript and companion source.
• Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras — pinned manuscript and companion source.

No family scope page was present at lean/docs/300.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Approximation Paving over Arbitrary Maximal Abelian Subalgebras.

🐻  OpenAI 2026b. Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras.