Thin II1 factors with no Cartan subalgebras

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 ⊂ 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.

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