Thin II_1 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 $A\subset M$ 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.

v2: we proved the absence of Cartan theorem for arbitrary A-valued semicircular systems (v1 only treated the weakly mixing ones)

References in corpus (2)

• Amenable absorption in amalgamated free product von Neumann algebras
• Constructing MASAs with prescribed properties

Cited by in corpus (1)

• Spectral gap characterizations of property (T) for II${}_1$ factors