Singular Masas and Measure-Multiplicity Invariant

TL;DR

This paper explores the relationship between the left-right measure and properties of singular masas, providing examples and demonstrating the diversity of such masas in hyperfinite and free group factors.

Contribution

It introduces new examples of singular masas with specific measure properties and shows the existence of many non-conjugate masas with prescribed invariants.

Findings

Existence of Tauer masas with Lebesgue measure class

Uncountably many non-conjugate singular masas in free group factors

Masas with prescribed Pukánszky invariant

Abstract

In this paper we study relations between the \emph{left-right-measure} and properties of singular masas. Part of the analysis is mainly concerned with masas for which the \emph{left-right-measure} is the class of product measure. We provide examples of Tauer masas in the hyperfinite $\rm{II}_{1}$ factor whose \emph{left-right-measure} is the class of Lebesgue measure. We show that for each subset $S\subseteq \mathbb{N}$, there exist uncountably many pairwise non conjugate singular masas in the free group factors with \emph{Puk\'{a}nszky invariant} $S\cup\{\infty\}$.