Constructing MASAs with prescribed properties

Abstract

We consider an iterative procedure for constructing maximal abelian ${}^{\ast }$-subalgebras (MASAs) satisfying prescribed properties in II${}_1$ factors. This method pairs well with the intertwining by bimodules technique and with properties of the MASA and of the ambient factor that can be described locally. We obtain such a local characterization for II${}_1$ factors $M$ that have an s-MASA, $A\subset M$ (i.e., for which $A\vee JAJ$ is maximal abelian in $\mathcal{B}\left(L^{2}M\right)$), and use this strategy to prove that any factor in this class has uncountably many nonintertwinable singular (resp., semiregular) s-MASAs.