Duke Mathematical Journal

Bass-Serre rigidity results in von Neumann algebras

Ionut Chifan and Cyril Houdayer

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Abstract

We obtain new Bass-Serre-type rigidity results for ${\rm II}_1$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any nonamenable factor arising as an amalgamated free product of von Neumann algebras $\mathcal{M}_1 \ast_B \mathcal{M}_2$ over an abelian von Neumann algebra $B$ is prime, that is, cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\rm II}_1$ and in the type ${\rm III}$ cases, new examples of prime factors.

Article information

Source
Duke Math. J. Volume 153, Number 1 (2010), 23-54.

Dates
First available in Project Euclid: 28 April 2010

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1272480931

Digital Object Identifier
doi:10.1215/00127094-2010-020

Mathematical Reviews number (MathSciNet)
MR2641939

Zentralblatt MATH identifier
1201.46057

Subjects
Primary: 46L10: General theory of von Neumann algebras
Secondary: 46L54: Free probability and free operator algebras 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

Citation

Chifan, Ionut; Houdayer, Cyril. Bass-Serre rigidity results in von Neumann algebras. Duke Math. J. 153 (2010), no. 1, 23--54. doi:10.1215/00127094-2010-020. http://projecteuclid.org/euclid.dmj/1272480931.


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References

  • S. Adams, Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), 765–783.
  • A. Alvarez and D. Gaboriau, Free products, orbit equivalence and measure equivalence rigidity, preprint.
  • C. Anantharaman-Delaroche, Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. Math. 171 (1995), 309–341.
  • L. Barnett, Free product von Neumann algebras of type ${\rm III}$, Proc. Amer. Math. Soc. 123 (1995), 543–553.
  • M. E. B. Bekka and A. Valette, Group cohomology, harmonic functions and the first $L^2$-Betti number, Potential Anal. 6 (1997), 313–326.
  • I. Chifan and A. Ioana, Ergodic subequivalence relations induced by a Bernoulli action, preprint.
  • A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252.
  • —, Almost periodic states and factors of type ${\rm III_1}$, J. Funct. Anal. 16 (1974), 415–445.
  • —, Classification of injective factors, Ann. of Math. (2) 104 (1976), 73–115.
  • —, Noncommutative Geometry, Academic Press, San Diego, 1994.
  • A. Connes and E. Størmer, Homogeneity of the state space of factors of type ${\rm III_1}$, J. Funct. Anal. 28 (1978), 187–196.
  • A. Connes and M. Takesaki, The flow of weights on factors of type III, Tôhoku Math. J. (2) 29 (1977), 473–575.
  • M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507–549.
  • E. G. Effros and E. C. Lance, Tensor products of operator algebras, Adv. Math. 25 (1977), 1–34.
  • T. Falcone and M. Takesaki, The non-commutative flow of weights on a von Neumann algebra, J. Funct. Anal. 182 (2001), 170–206.
  • D. Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math. 139 (2000), 41–98.
  • M. Gao and M. Junge, Examples of prime von Neumann algebras, Int. Math. Res. Not. IMRN 2007, no. 15, art. ID rnm042.
  • L. Ge, Applications of free entropy to finite von Neumann algebras, ${\it II}$, Ann. of Math. (2) 147 (1998), 143–157.
  • R. H. Hermann and V. F. R. Jones, “Central sequences in crossed products” in Operator Algebras and Mathematical Physics (Iowa City, Ia., 1985), Contemp. Amer. Math. Soc., Providence Math. 62 1987, 539–544.
  • C. Houdayer, Construction of type ${\rm II_1}$ factors with prescribed countable fundamental group, J. Reine Angew. Math. 634 (2009), 169–207.
  • 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.
  • G. Kuhn and T. Steger, More irreducible boundary representations of free groups, Duke Math. J. 82 (1996), 381–436.
  • N. Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), 111–117.
  • —, A Kurosh-type theorem for type ${\rm II_1}$ factors, Int. Math. Res. Not. IMRN 2006, art. ID 97560.
  • —, An example of a solid von Neumann algebra, Hokkaido Math. J. 38 (2009), 557–561.
  • N. Ozawa and S. Popa, On a class of ${\it II}_1$ factors with at most one Cartan subalgebra, to appear in Ann. of Math. (2), preprint.
  • C. Pensavalle and T. Steger, Tensor products with anisotropic principal series representations of free groups, Pacific J. Math. 173 (1996), 181–202.
  • J. Peterson, $L^2$-rigidity in von Neumann algebras, Invent. Math. 175 (2009), 417–433.
  • J. Peterson and A. Thom, Group cocycles and the ring of affiliated operators, preprint.
  • S. Popa, On a class of type ${\rm II_1}$ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), 809–899.
  • —, Some rigidity results for non-commutative Bernoulli Shifts, J. Funct. Anal. 230 (2006), 273–328.
  • —, Strong rigidity of ${\rm II_1}$ factors arising from malleable actions of w-rigid groups, ${\it I}$, Invent. Math. 165 (2006), 369–408.
  • —, On Ozawa's property for free group factors, Int. Math. Res. Not. IMRN 2007 art. ID rnm036.
  • —, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc. 21 (2008), 981–1000.
  • J. Ramagge and G. Robertson, Factors from trees, Proc. Amer. Math. Soc. 125 (1997), 2051–2055.
  • é. Ricard and Q. Xu, Khintchine type inequalities for reduced free products and applications, J. Reine Angew. Math. 599 (2006), 27–59.
  • D. Shlyakhtenko, Prime type ${\rm III}$ factors, Proc. Nat. Acad. Sci. USA 97 (2000), 12439–12441.
  • —, Theory of Operator Algebras, ${\it II}$, Encyclopaedia Math. Sci. 125, Oper. Alg. Non-commut. Geom. 6, Springer, Berlin, 2003.
  • M. Takesaki, Duality for crossed products and structure of von Neumann algebras of type ${\rm III}$, Acta Math. 131 (1973), 249–310.
  • A. Törnquist, Orbit equivalence and actions of, $\mathbb{F}_n$, J. Symbolic Logic 71 (2006), 265–282.
  • Y. Ueda, Amalgamated free product over Cartan subalgebra, Pacific J. Math. 191 (1999), 359–392.
  • —, Remarks on free products with respect to non-tracial states, Math. Scand. 88 (2001), 111–125.
  • —, Fullness, Connes' $\chi$-groups, and ultra-products of amalgamated free products over Cartan subalgebras, Trans. Amer. Math. Soc. 355 (2003), 349–371.
  • —, “Amalgamated free products over Cartan subalgebra, ${\rm II}$: Supplementary results and examples” in Operator Algebras and Applications, Adv. Stud. Pure Math. 38, Math. Soc. Japan, Tokyo, 2004, 239–265.
  • S. Vaes, Rigidity results for Bernoulli actions and their von Neumann algebras (after S. Popa), Astérisque 311 (2007), 237–294., Séminaire Bourbaki 2005/2006, no. 961.
  • —, Explicit computations of all finite index bimodules for a family of ${\rm II_1}$ factors, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 743–788.
  • D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, CRM Monog. Ser. 1, Amer. Math. Soc., Providence, $1992$.