Mixing and weakly mixing abelian subalgebras of type II$_1$ factors

TL;DR

This paper investigates the structure and properties of mixing and weakly mixing maximal abelian subalgebras (masas) in type II₁ factors, providing new characterizations and examples, including non-conjugate mixing masas in free group factors.

Contribution

It offers new necessary and sufficient conditions for the normalizing algebra of a masa and constructs uncountably many non-conjugate mixing masas with specific invariants.

Findings

Characterization of the normalizing algebra of a masa.

Existence of uncountably many non-conjugate mixing masas in free group factors.

Structural results ruling out certain Koopman-realizable measures.

Abstract

This paper studies weakly mixing (singular) and mixing masas in type $\rm{II}_{1}$ factors from a bimodule point of view. Several necessary and sufficient conditions to characterize the normalizing algebra of a masa are presented. We also study the structure of mixing inclusions, with special attention paid to masas of product class. A recent result of Jolissaint and Stalder concerning mixing masas arising out of inclusions of groups is revisited. One consequence of our structural results rules out the existence of certain Koopman-realizable measures, arising from semidirect products, which are absolutely continuous but not Lebesgue. We also show that there exist uncountably many pairwise non--conjugate mixing masas in the free group factors each with Puk\'{a}nszky invariant $\{1,\infty\}$.