Approximation and quadratic strong-operator paving

We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each 0< ε< 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 Cε^{-6} for a universal constant C. No separability or conditional-expectation hypothesis is required.

We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every 0< ε < 1, every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most 5\times10^8ε ^{-2} projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.