Family 300Operator algebrasNo LeanNot yet digested
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 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.
Manuscripts in this family · 2
Approximation Paving over Arbitrary Maximal Abelian Subalgebras
Sep 25, 2026·Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf
We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each , 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 for a universal constant C. No separability or conditional-expectation hypothesis is required.
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras
Sep 25, 2026·Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf
We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every , every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.