Approximation Paving over Arbitrary Maximal Abelian Subalgebras

We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each $0<\epsilon <1$, 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 $C{\epsilon }^{-6}$ for a universal constant C. No separability or conditional-expectation hypothesis is required.