Kyoto Journal of Mathematics

Thin II1 factors with no Cartan subalgebras

Anna Sofie Krogager and Stefaan Vaes

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Abstract

It is a wide open problem to give an intrinsic criterion for a II1 factor M to admit a Cartan subalgebra A. When AM is a Cartan subalgebra, the A-bimodule L2(M) is simple in the sense that the left and right actions of A generate a maximal abelian subalgebra of B(L2(M)). A II1 factor M that admits such a subalgebra A is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II1 factor M to be s-thin and left open the question whether all s-thin II1 factors admit a Cartan subalgebra. We answer this question negatively by constructing s-thin II1 factors without Cartan subalgebras.

Article information

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

Dates
Received: 14 December 2016
Revised: 13 June 2017
Accepted: 14 June 2017
First available in Project Euclid: 26 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1569463628

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

Subjects
Primary: 46L54: Free probability and free operator algebras
Secondary: 46L36: Classification of factors 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

Keywords
Cartan subalgebras $\mathrm{II}_{1}$ factors A-valued semicircular systems measures on small sets

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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