Strongly solid ${\rm II_1}$ factors with an exotic MASA

TL;DR

This paper constructs a new example of a non-amenable strongly solid II_1 factor with an exotic MASA, demonstrating the existence of such factors without Cartan subalgebras and exploring their unique bimodule properties.

Contribution

It introduces a novel strongly solid II_1 factor with an exotic MASA, expanding understanding of the structure and properties of such factors.

Findings

The constructed II_1 factor is non-amenable and strongly solid.

It contains an exotic MASA with non-coarse, non-discrete bimodule structure.

The factor has the Haagerup property and the complete metric approximation property.

Abstract

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid $\rm{II}_1$ factor $M$ containing an "exotic" maximal abelian subalgebra $A$: as an $A$,$A$-bimodule, $L^2(M)$ is neither coarse nor discrete. Thus we show that there exist $\rm{II}_1$ factors with such property but without Cartan subalgebras. It also follows from Voiculescu's free entropy results that $M$ is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.