Approximation and quadratic strong-operator paving. Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with $O(\varepsilon^{-2})$ blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.
released 2026-09-25 | 3 theorems · 19 lemmas · 27 proofs · 20,639 words |
PLAY LEVEL 1 »(pdf)
We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each $0\lt \varepsilon\lt 1$, every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most ε times the approximant's own norm. The number of projections is at most $C\varepsilon^{-6}$ for a universal constant C. No separability or conditional-expectation hypothesis is required.
released 2026-09-25 | 2 theorems · 22 lemmas · 29 proofs · 25,624 words |
PLAY LEVEL 2 »(pdf)
We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every $0\lt \varepsilon \lt 1$, every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most $5\times10^8\varepsilon ^{-2}$ projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.