Paving over arbitrary MASAs in von Neumann algebras (Q2353506)

Let \(\mathcal B(\ell^2)\) be the space of all bounded operators on the Hilbert space \(\ell^2\) and denote by \(\mathcal D\) the diagonal MASA (maximal abelian \(*\)-subalgebra) of \(\mathcal B(\ell^2)\). \textit{J. Anderson} [Trans. Am. Math. Soc. 249, 303--329 (1979; Zbl 0408.46049)] reformulated the Kadison-Singer problem into what is now known as the paving conjecture: For every \(\varepsilon > 0\), there exists a natural number \(n\) such that for every self-adjoint \(T \in \mathcal B(\ell^2)\) with zero diagonal, there exists a partition of 1 with projections \(P_1, \dots, P_n \in \mathcal D\) such that \(\| \sum P_k T P_k \| \leq \varepsilon \| T \|\). \textit{A. W. Marcus} et al. [Ann. Math. (2) 182, No. 1, 327--350 (2015; Zbl 1332.46056)] settled this question in the affirmative and obtained an estimate for the number \(n\). Inspired by this result, the authors study pavings in a more general setting. Let \(A\) be a MASA in a von Neumann algebra \(M\). We say that an element \(x \in M\) is \((\varepsilon, n\)) so-pavable over \(A\) if for every strong neighborhood \(\mathcal V\) of 0 in \(M\) there exist projections \(p_1, \dots, p_n \in A\), an element \(a \in A\) and a projection \(q \in M\) such that \(\| a \| \leq \| x \| \), \(\sum p_k = 1\), \(\| q (\sum p_k x p_k - a)q \| \leq \varepsilon \| x \|\) and \(1-q \in \mathcal V\). If, for every \(\varepsilon > 0\), there exists a natural number \(n\) such that every self-adjoint \(x \in M\) is \((\varepsilon,n)\) so-pavable over \(A\), then \(A\) has the uniform so-paving property. The authors show that all MASAs in \(\mathcal B(\ell^2)\) and every Cartan subalgebra of an amenable von Neumann algebra have the uniform so-paving property. Finally, they conjecture that any MASA in any von Neumann algebra has the uniform so-paving property.