HNN extensions and unique group measure space decomposition of II factors

Authors:
Pierre Fima and Stefaan Vaes

Journal:
Trans. Amer. Math. Soc. **364** (2012), 2601-2617

MSC (2010):
Primary 46L36; Secondary 20E06, 28D15, 46L54

DOI:
https://doi.org/10.1090/S0002-9947-2012-05415-8

Published electronically:
January 6, 2012

MathSciNet review:
2888221

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for a fairly large family of HNN extensions , the group measure space II factor given by an arbitrary free ergodic probability measure preserving action of has a unique group measure space Cartan subalgebra up to unitary conjugacy. From this we deduce new examples of W-superrigid group actions, i.e. where the II factor entirely remembers the group action from which it was constructed.

**[CH08]**I. Chifan and C. Houdayer, Bass-Serre rigidity results in von Neumann algebras.*Duke Math J.***153**(2010), 23-54. MR**2641939****[Co76]**A. Connes, Classification of injective factors.*Ann. of Math. (2)***104**(1976), 73-115. MR**0454659 (56:12908)****[Ga99]**D. Gaboriau, Coût des relations d'équivalence et des groupes.*Invent. Math.***139**(2000), 41-98. MR**1728876 (2001f:28030)****[HPV10]**C. Houdayer, S. Popa and S. Vaes, A class of groups for which every action is W-superrigid. To appear in*Groups Geom. Dyn.*`arXiv:1010.5077`**[Io10]**A. Ioana, W-superrigidity for Bernoulli actions of property (T) groups.*J. Amer. Math. Soc.***24**(2011), no. 4, 1175-1226. MR**2813341****[IPP05]**A. Ioana, J. Peterson and S. Popa, Amalgamated free products of weakly rigid factors and calculation of their symmetry groups.*Acta Math*.**200**(2008), 85-153. MR**2386109 (2009a:46119)****[Pa99]**F. Paulin, Propriétés asymptotiques des relations d'équivalences mesurées discrètes.*Markov Process. Related Fields***5**(1999), 163-200. MR**1762172 (2001m:37010)****[Pe09]**J. Peterson, Examples of group actions which are virtually W-superrigid.*Preprint.*`arXiv:1002.1745`**[Po91]**S. Popa, Markov traces on universal Jones algebras and subfactors of finite index.*Invent. Math*.**111**(1993), 375-405. MR**1198815 (94c:46128)****[Po01]**S. Popa, On a class of type factors with Betti numbers invariants.*Ann. of Math*.**163**(2006), 809-899. MR**2215135 (2006k:46097)****[Po03]**S. Popa, Strong rigidity of II factors arising from malleable actions of -rigid groups, I.*Invent. Math.***165**(2006), 369-408. MR**2231961 (2007f:46058)****[Po04]**S. Popa, Strong rigidity of II factors arising from malleable actions of -rigid groups, II.*Invent. Math.***165**(2006), 409-452. MR**2231962 (2007h:46084)****[Po05]**S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of -rigid groups.*Invent. Math.***170**(2007), 243-295. MR**2342637 (2008f:37010)****[Po06a]**S. Popa, On the superrigidity of malleable actions with spectral gap.*J. Amer. Math. Soc*.**21**(2008), 981-1000. MR**2425177 (2009e:46056)****[Po06b]**S. Popa, Deformation and rigidity for group actions and von Neumann algebras. In*Proceedings of the International Congress of Mathematicians (Madrid, 2006)*, Vol. I, European Mathematical Society Publishing House, 2007, pp. 445-477. MR**2334200 (2008k:46186)****[PV06]**S. Popa and S. Vaes, Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups.*Adv. Math*.**217**(2008), 833-872. MR**2370283 (2009c:37004)****[PV09]**S. Popa and S. Vaes, Group measure space decomposition of factors and W-superrigidity, Invent. Math.**182**(2010), no. 2, 371-417. MR**2729271****[RX05]**E. Ricard and Q. Xu, Khintchine type inequalities for reduced free products and applications.*J. Reine Angew. Math.***599**(2006), 27-59. MR**2279097 (2009h:46110)****[Se83]**J.-P. Serre, Arbres, amalgames, .*Astérisque*, No. 46, Société Mathématique de France, Paris, 1977. MR**0476875 (57:16426)****[Ue04]**Y. Ueda, HNN extensions of von Neumann algebras.*J. Funct. Anal.***225**(2005), 383-426. MR**2152505 (2006k:46100)****[Ue07]**Y. Ueda, Remarks on HNN extensions in operator algebras.*Illinois J. Math.***52**(2008), 705-725. MR**2546003 (2010h:46093)****[Va06]**S. Vaes, Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa).*Séminaire Bourbaki*, exp. no. 961.*Astérisque***311**(2007), 237-294. MR**2359046 (2009k:46112)****[Va07]**S. Vaes, Explicit computations of all finite index bimodules for a family of factors.*Ann. Sci. École Norm. Sup.***41**(2008), 743-788. MR**2504433****[VDN92]**D.V. Voiculescu, K.J. Dykema and A. Nica, Free random variables.*CRM Monograph Series***1**, American Mathematical Society, Providence, RI, 1992. MR**1217253 (94c:46133)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
46L36,
20E06,
28D15,
46L54

Retrieve articles in all journals with MSC (2010): 46L36, 20E06, 28D15, 46L54

Additional Information

**Pierre Fima**

Affiliation:
Department of Mathematics, K. U. Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium

Address at time of publication:
Institute of Mathematics of Jussieu, University Denis Diderot Paris 7, 175, rue du Chevaleret, F-75013 Paris, France

Email:
pierre.fima@wis.kuleuven.be, pfima@math.jussieu.fr

**Stefaan Vaes**

Affiliation:
Department of Mathematics, K. U. Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium

Email:
stefaan.vaes@wis.kuleuven.be

DOI:
https://doi.org/10.1090/S0002-9947-2012-05415-8

Received by editor(s):
June 15, 2010

Received by editor(s) in revised form:
July 6, 2010

Published electronically:
January 6, 2012

Additional Notes:
The first author was supported by ERC Starting Grant VNALG-200749.

The second author was partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0231.07 of the Research Foundation, Flanders (FWO) and K. U. Leuven BOF research grant OT/08/032.

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.