Paving every maximal abelian subalgebra with a uniform number of pieces

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Family 300 · Operator algebras
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Imagine partitioning the coordinates of a matrix into groups, then deleting entries joining different groups. Can the remaining blocks be made small after removing an appropriate diagonal part?

This is the idea of paving, central to the Kadison–Singer problem. For general operator algebras there may be no coordinate basis, but projections in a maximal abelian subalgebra play the role of diagonal coordinate blocks. Ordinary operator-norm paving is too strong in this generality.

Family 300 claims two universal replacements. One allows a nearby bounded operator before norm paving. The other paves the original operator after compressing to a projection arbitrarily close to the identity on any prescribed finite collection of vectors. The latter uses a number of pieces proportional to the inverse square of the error, the optimal general exponent.

2. Background and essential definitions

Let $M\subseteq B(H)$ be a complex von Neumann algebra, and let $A\subseteq M$ be a maximal abelian self-adjoint subalgebra, abbreviated MASA. Its elements commute with one another, and it cannot be enlarged within $M$ while remaining abelian. The ordinary diagonal matrices are the basic finite-dimensional example.

Choose mutually orthogonal projections $p_1,\ldots,p_r\in A$ summing to the identity. Their pinching map is

$$C_P(x)=\sum_{i=1}^r p_i x p_i.$$

It retains the block-diagonal part associated with the partition. Zero projections are allowed, so a bound on the number of pieces can be expressed using one fixed partition length.

Ordinary norm paving asks for $C_P(x)$ to be close in operator norm to some $a\in A$. Operator norm controls all unit vectors at once. The strong operator topology is weaker: a neighborhood of $x$ asks that $(y-x)\xi$ be small only for a specified finite set of vectors $\xi$.

In approximation paving, one first chooses such a nearby self-adjoint $y$, with a uniform norm bound, and then norm-paves $y$. In strong-operator paving, one keeps $x$ but finds a projection $q\in M$ close to the identity on the prescribed vectors and controls

$$q(C_P(x)-a)q.$$

The projection $q$ need not belong to $A$. Being close to the identity strongly does not mean $\|1-q\|$ is small: a nonzero orthogonal projection always has norm one.

3. Development history and previous partial results

Kadison–Singer’s pure-state extension problem concerned the atomic diagonal MASA of $B(\ell^2)$. Anderson formulated its paving counterpart, and Marcus–Spielman–Srivastava solved it through mixed characteristic polynomials.

Moving to arbitrary MASAs changes the problem. Popa–Vaes showed that, in the separable-predual setting, norm paving characterizes type I algebras with a normal expectation onto the MASA. Thus a universal norm-paving theorem for arbitrary MASAs is not the right target.

Popa introduced finite-factor methods, obtaining norm paving in ultraproducts of singular MASAs and hence strong-operator paving in the original inclusions. Popa–Vaes formulated the strong-operator and approximation-paving conjectures and proved major special cases, including type I algebras and certain Cartan inclusions.

The two structural extremes behaved differently. Cartan MASAs have many normalizing symmetries and can be described using measured equivalence relations. Singular MASAs have few such symmetries and admit free-probabilistic methods. Earlier combinations of the methods could multiply partition counts, worsening a quadratic or quartic bound to a sixth-power bound.

Ravichandran–Srivastava obtained asymptotically optimal finite-dimensional multipaving estimates. M2 uses their finite polynomial bound while coordinating the Cartan and complementary parts with one partition. This avoids multiplying the number of colors and is central to the claimed quadratic rate.

4. What the new manuscripts claim

M1: norm paving after a bounded strong perturbation

For every $0<\varepsilon<1$, M1 claims a universal integer $R(\varepsilon)\leq C_{\mathrm{ap}}\varepsilon^{-6}$ with the following property. For any MASA $A\subseteq M\subseteq B(H)$, self-adjoint $x\in M$, finite $F\subseteq H$, and $\delta>0$, there are a self-adjoint $y\in M$, an $R(\varepsilon)$-piece partition in $A$, and self-adjoint $a\in A$ such that

$$\|y\|\leq3\|x\|,\qquad \|(y-x)\xi\|<\delta\quad(\xi\in F),$$

and

$$\|a\|\leq\|y\|,\qquad \|C_P(y)-a\|\leq\varepsilon\|y\|.$$

The error is relative to the approximant’s own norm. The count is fixed before choosing the inclusion, operator or strong neighborhood. No separability or conditional-expectation assumption is imposed.

M1 also obtains a quadratic approximation-paving bound when the MASA is the range of a normal conditional expectation, using the companion strong-operator theorem. It does not claim that quadratic approximation rate for arbitrary expectation-free MASAs.

M2: quadratic strong-operator paving of the original operator

M2 gives

$$r_\varepsilon=lceil400000000\varepsilon^{-2}\rceil +\lceil4/\varepsilon\rceil+1 \leq500000000\varepsilon^{-2}.$$

For every inclusion and self-adjoint $x$ as above, and every finite-vector neighborhood, it claims a partition with $r_\varepsilon$ pieces, self-adjoint $a\in A$, and projection $q\in M$ satisfying

$$\|(1-q)\xi\|<\delta\quad(\xi\in F),$$

$$\|a\|\leq\|x\|,\qquad \|q(C_P(x)-a)q\|\leq\varepsilon\|x\|.$$

Again there is no separability or expectation hypothesis. The exponent two is optimal in general by an existing finite-factor obstruction that survives the permitted compressions. The numerical constant is not optimized.

For countably decomposable expected MASAs, M2 also transfers the quadratic bound to norm paving in the associated Ocneanu von Neumann ultrapower. That corollary retains both its countable-decomposability and normal-expectation assumptions.

5. Why the advance matters

Importance: 5/5. If correct, the pair resolves the general approximation-paving and quadratic strong-operator-paving conjectures. It extends the paving framework far beyond the atomic diagonal setting of Kadison–Singer.

The uniform partition count is substantive. The number of pieces is chosen from the desired error before the algebra, representation and finite-vector tests are known. Arbitrarily large or nonseparable representations do not require more colors.

The quadratic exponent places a sharp quantitative boundary on strong-operator paving, even though its displayed constant is large. Approximation paving gives a different useful conclusion: the entire nearby operator is norm-paved, without discarding a corner. Neither statement should be conflated with universal norm paving of the original operator.

6. The main idea

Separate normalizing structure from diffuse behavior

When a suitable faithful normal state centralizes the MASA, let $N$ be the algebra generated by its partial normalizers. Conditional expectations give the decomposition

$$x-E_A(x)=\bigl(E_N(x)-E_A(x)\bigr)+\bigl(x-E_N(x)\bigr).$$

The first term belongs to a Cartan-type measured-relation setting. The second has diffuse, or atomless, left-right kernels. These kernels describe how the operator connects points of the abelian algebra without concentration on discrete matching pairs.

The challenge is to pave both contributions while retaining a uniform count. M2 arranges for the same color projections to handle them, rather than first paving one term and refining every resulting piece for the other.

Transfer finite matrix choices to measurable colorings

Mixed determinantal or characteristic polynomials encode finite paving choices through their roots. The manuscripts use spectral-root estimates to assign colors while controlling an accumulated cost.

For measured equivalence relations, the choices must be measurable and compatible across overlapping finite regions. Nonsingular actions need not preserve the initial measure, so an auxiliary real coordinate converts the problem to an invariant mass-transport setting. Uniform estimates then descend the color choices back to the original space.

M1 uses buffered pinning and sparse updates to construct its approximation paving. M2 works with a relative Bernoulli extension, giving independent site labels and uniform conditional color marginals. The latter support both the measured Cartan estimate and a free-compression estimate for the diffuse term.

Return from the auxiliary model to actual projections

M2 puts the algebra and its Bernoulli model in an amalgamated free product. It then transfers fixed mixed moments back to projections in the original MASA. Atomless kernels make unwanted collisions disappear, while modular analytic approximation handles the lack of a tracial state.

Clipping produces bounded approximants, and a sufficiently high moment estimate yields a spectral cut $q$ with the required strong-neighborhood control. The number of colors is fixed before choosing these later approximation parameters.

Remove expectation and separability assumptions

The arbitrary-MASA arguments split the representation into a corner supporting centralizing normal functionals and a complementary corner. The first admits state-based methods; the second uses almost disjoint conjugate states and additional randomization.

Careful corner assembly controls off-diagonal terms. In M2, disjoint palettes make counts add across corners. Separable reductions preserve the bounds before returning to arbitrary Hilbert spaces. M1 additionally uses a norming vector and final clipping so the paving error is measured against $\|y\|$, as its theorem requires.

7. Scope, limitations, and open questions

The input operators are self-adjoint. The main quadratic result concerns one operator at a time; it does not claim a partition count independent of the size of an arbitrarily prescribed family of operators that must all be paved simultaneously.

M1’s general bound is sixth-power approximation paving, with norm inflation at most three. M2’s quadratic bound controls a compression of the original operator. The expected-MASA conversion gives quadratic approximation paving only under its stated normal-expectation hypothesis.

The original operators need not admit norm paving at all. Strongly large projections may discard a nonzero subspace, and strong approximants need not be close in operator norm. These allowances are exactly what makes the general formulations plausible.

The two main proofs are independent, while particular corollaries use established transfer results or the companion theorem. Improving constants, determining optimal approximation-paving rates without expectations, and understanding simultaneous-family bounds remain separate quantitative questions.

8. Sources and further reading

• Popa and Vaes, Paving over arbitrary MASAs in von Neumann algebras, for the conjectures, definitions and transfer results.
• Popa and Vaes, On the optimal paving over MASAs in von Neumann algebras, for sharp singular-MASA bounds.
• Ravichandran and Srivastava, Asymptotically Optimal Multi-Paving, for the finite-dimensional input used by M2.

Source manuscripts

• M1. OpenAI, Approximation Paving over Arbitrary Maximal Abelian Subalgebras (September 25, 2026).
• M2. OpenAI, Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras (September 25, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.