Skip to main content Accessibility help

Stability of products of equivalence relations

  • Amine Marrakchi (a1)


An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.



Hide All

The author is supported by ERC Starting Grant GAN 637601.



Hide All
[Con76] Connes, A., Classification of injective factors. Cases II1 , II , III𝜆 , 𝜆≠1 , Ann. of Math. (2) 74 (1976), 73115.
[Con85] Connes, A., Factors of type III1 , property L 𝜆 and closure of inner automorphisms , J. Operator Theory 14 (1985), 189211.
[CS76] Connes, A. and Stormer, E., Homogeneity of the state space of factors of type III1 , J. Funct. Anal. 28 (1976), 187196.
[FM77] Feldman, J. and Moore, C. C., Ergodic equivalence relations, cohomology, and von Neumann algebras. I, II , Trans. Amer. Math. Soc. 234 (1977), 289324; 325–359.
[Haa85] Haagerup, U., A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space , J. Funct. Anal. 62 (1985), 160201.
[Haa86] Haagerup, U., Connes’ bicentralizer problem and uniqueness of the injective factor of type III1 , Acta Math. 69 (1986), 95148.
[HMV16] Houdayer, C., Marrakchi, A. and Verraedt, P., Fullness and Connes’ 𝜏 invariant of type III tensor product factors , J. Math. Pures Appl. , to appear. Preprint (2016), arXiv:1611.07914.
[HMV17] Houdayer, C., Marrakchi, A. and Verraedt, P., Strongly ergodic equivalence relations: spectral gap and type III invariants , Ergodic Theory Dynam. Systems , to appear.
[IV15] Ioana, A. and Vaes, S., Spectral gap for inclusions of von Neumann algebras. Appendix to the article Cartan subalgebras of amalgamated free product II1 factors by A. Ioana , Ann. Sci. Éc. Norm. Supér. 48 (2015), 71130.
[JS87] Jones, V. F. R. and Schmidt, K., Asymptotically invariant sequences and approximate finiteness , Amer. J. Math. 109 (1987), 91114.
[McD70] McDuff, D., Central sequences and the hyperfinite factor , Proc. Lond. Math. Soc. (3) 21 (1970), 443461.
[Mar16] Marrakchi, A., Spectral gap characterization of full type III factors , J. Reine Angew. Math. , to appear. Preprint (2016), arXiv:1605.09613.
[Mar18] Marrakchi, A., Strongly ergodic actions have local spectral gap , Proc. Amer. Math. Soc. 146 (2018), 38873893.
[MvN43] Murray, F. and von Neumann, J., Rings of operators. IV , Ann. of Math. (2) 44 (1943), 716808.
[Pop86] Popa, S., A short proof that injectivity implies hyperfiniteness for finite von Newmann algebras , J. Operator Theory 16 (1986), 261272.
[Pop87] Popa, S., The commutant modulo the set of compact operators of a von Neumann algebra , J. Funct. Anal. 712 (1987), 393408.
[Pop95] Popa, S., Free-independent sequences in type $\text{II}_{1}$ factors and related problems, Astérisque 232 (1995), 187–202; Recent advances in operator algebras (Orléans, 1992).
[Pop10] Popa, S., On spectral gap rigidity and Connes’ invariant 𝜒(M) , Proc. Amer. Math. Soc. 138 (2010), 35313539.
[Pop14] Popa, S., Independence properties in subalgebras of ultraproduct II1 factors , J. Funct. Anal. 266 (2014), 58185846.
[WY14] Wu, W. and Yuan, W., A remark on central sequence algebras of the tensor product of II1 factors , Proc. Amer. Math. Soc. 142 (2014), 28292835.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Related content

Powered by UNSILO

Stability of products of equivalence relations

  • Amine Marrakchi (a1)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.