On the optimal paving over MASAs in von Neumann algebras

TL;DR

This paper demonstrates optimal paving results over singular MASAs in II$_1$ factors, establishing bounds for partitions that minimize the operator norm of certain elements, and discusses related open problems.

Contribution

It proves the existence of optimal pavings over singular MASAs in II$_1$ factors with explicit bounds, and provides examples showing these bounds are sharp.

Findings

Existence of partitions with norm bounds for singular MASAs

Explicit sharp bounds for paving in II$_1$ factors

Discussion of open problems on optimal paving

Abstract

We prove that if $A$ is a singular MASA in a II$_1$ factor $M$ and $\omega$ is a free ultrafilter, then for any $x\in M\ominus A$, with $\|x\|\leq 1$, and any $n\geq 2$, there exists a partition of $1$ with projections $p_1, p_2, ..., p_n\in A^\omega$ (i.e. a {\it paving}) such that $\|\Sigma_{i=1}^n p_i x p_i\|\leq 2\sqrt{n-1}/n$, and give examples where this is sharp. Some open problems on optimal pavings are discussed.