Thin II1 factors with no Cartan subalgebras

Abstract

It is a wide open problem to give an intrinsic criterion for a ${\mathrm{II}}_1$ 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 actions of $A$ generate a maximal abelian subalgebra of $B(L^2(M))$. A ${\mathrm{II}}_1$ factor $M$ that admits such a subalgebra $A$ is said to be $s$-thin. Very recently, Popa discovered an intrinsic local criterion for a ${\mathrm{II}}_1$ factor $M$ to be $s$-thin and left open the question whether all $s$-thin ${\mathrm{II}}_1$ factors admit a Cartan subalgebra. We answer this question negatively by constructing $s$-thin ${\mathrm{II}}_1$ factors without Cartan subalgebras.