Thin II_1 factors with no Cartan subalgebras

TL;DR

This paper constructs examples of s-thin II_1 factors that do not contain any Cartan subalgebras, answering a long-standing open question in operator algebra theory.

Contribution

It provides the first known examples of s-thin II_1 factors lacking Cartan subalgebras, disproving the conjecture that all s-thin factors admit such subalgebras.

Findings

Constructed s-thin II_1 factors without Cartan subalgebras

Disproved the conjecture linking s-thinness to existence of Cartan subalgebras

Answered a major open problem in the theory of von Neumann algebras

Abstract

It is a wide open problem to give an intrinsic criterion for a II_1 factor $M$ to admit a Cartan subalgebra $A$. When $A \subset M$ is a Cartan subalgebra, the $A$-bimodule $L^2(M)$ is "simple" in the sense that the left and right action of $A$ generate a maximal abelian subalgebra of $B(L^2(M))$. A II_1 factor $M$ that admits such a subalgebra $A$ is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II_1 factor $M$ to be s-thin and left open the question whether all s-thin II_1 factors admit a Cartan subalgebra. We answer this question negatively by constructing s-thin II_1 factors without Cartan subalgebras.