  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.

Operator Algebras