The Popa-Vaes paving conjecture over arbitrary maximal abelian subalgebras: strong-operator, approximation and quadratic paving

Quadratic paper, Theorem 1.1: for $0<\varepsilon<1$ and $r_\varepsilon\le5\times10^8\varepsilon^{-2}$, every self-adjoint $x$ in any von Neumann algebra $M$ is $(\varepsilon,r_\varepsilon)$ so-pavable over every MASA $A\subseteq M$, with no separability or conditional expectation assumed; corollary: quadratic norm paving in Ocneanu ultrapowers of expected countably decomposable MASAs. Approximation paper, Theorem 1.1: every self-adjoint $x$ is a strong limit of self-adjoint $y$ with $\|y\|\le3\|x\|$, each norm-pavable to error $\varepsilon\|y\|$ with at most $C\varepsilon^{-6}$ projections. Not shown: norm paving of the original operator (false in general), optimal numerical constants, or a quadratic rate for approximation paving.

Quadratic paper, Theorem 1.1: for $0<\varepsilon<1$ and $r_\varepsilon\le5\times10^8\varepsilon^{-2}$, every self-adjoint $x$ in any von Neumann algebra $M$ is $(\varepsilon,r_\varepsilon)$ so-pavable over every MASA $A\subseteq M$, with no separability or conditional expectation assumed; corollary: quadratic norm paving in Ocneanu ultrapowers of expected countably decomposable MASAs. Approximation paper, Theorem 1.1: every self-adjoint $x$ is a strong limit of self-adjoint $y$ with $\|y\|\le3\|x\|$, each norm-pavable to error $\varepsilon\|y\|$ with at most $C\varepsilon^{-6}$ projections. Not shown: norm paving of the original operator (false in general), optimal numerical constants, or a quadratic rate for approximation paving.