## Kyoto Journal of Mathematics

### Thin $\mathrm{II}_{1}$ 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.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 4 (2019), 815-867.

Dates
Revised: 13 June 2017
Accepted: 14 June 2017
First available in Project Euclid: 26 September 2019

https://projecteuclid.org/euclid.kjm/1569463628

Digital Object Identifier
doi:10.1215/21562261-2019-0028

#### Citation

Krogager, Anna Sofie; Vaes, Stefaan. Thin $\mathrm{II}_{1}$ factors with no Cartan subalgebras. Kyoto J. Math. 59 (2019), no. 4, 815--867. doi:10.1215/21562261-2019-0028. https://projecteuclid.org/euclid.kjm/1569463628

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