Constructing MASAs with prescribed properties

TL;DR

This paper introduces an iterative method for constructing maximal abelian *-subalgebras (MASAs) with specific properties in II$_1$ factors, utilizing local characterizations and the intertwining technique.

Contribution

It develops a new iterative approach for constructing MASAs with prescribed properties in II$_1$ factors, especially those with an s-MASA, and proves the existence of many non-intertwinable singular and semiregular s-MASAs.

Findings

Established a local characterization for II$_1$ factors with s-MASA.

Proved the existence of uncountably many non-intertwinable singular s-MASAs.

Demonstrated the method's compatibility with intertwining techniques.

Abstract

We consider an iterative procedure for constructing maximal abelian $^*$-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 {\it s-MASA}, $A\subset M$ (i.e., for which $A \vee JAJ$ is maximal abelian in $\Cal B(L^2M)$), and use this strategy to prove that any factor in this class has uncountably many non-intertwinable singular (respectively semiregular) s-MASAs.