Acta Mathematica

Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

Adrian Ioana, Jesse Peterson, and Sorin Popa

Full-text: Open access

Abstract

We consider amalgamated free product II1 factors M = M1*BM2*B… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitarily conjugated into one of the Mi’s. We apply this to the case where the Mi’s are w-rigid II1 factors, with B equal to either C, to a Cartan subalgebra A in Mi, or to a regular hyperfinite II1 subfactor R in Mi, to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N1 * CN2*C…)t, for some t > 0 and some other similar inclusions of algebras C ⊂ Ni then, after a permutation of indices, (B ⊂ Mi) is inner conjugate to (C ⊂ Ni)t, for all i. Taking B = C and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $, with {ti}i⩾1 = S a given countable subgroup of R+*, we obtain continuously many non-stably isomorphic factors M with fundamental group $ {\user1{\mathcal{F}}}{\left( M \right)} $ equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ and Out(M) abelian and calculable. Taking B = R, we get examples of factors with $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $, Out(M) = K, for any given separable compact abelian group K.

Article information

Source
Acta Math. Volume 200, Number 1 (2008), 85-153.

Dates
Received: 28 February 2006
Revised: 20 November 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891958

Digital Object Identifier
doi:10.1007/s11511-008-0024-5

Zentralblatt MATH identifier
1149.46047

Rights
2008 © Institut Mittag-Leffler

Citation

Ioana, Adrian; Peterson, Jesse; Popa, Sorin. Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. Acta Math. 200 (2008), no. 1, 85--153. doi:10.1007/s11511-008-0024-5. https://projecteuclid.org/euclid.acta/1485891958.


Export citation

References

  • B ekka, M. E. B. & V alette, A., Group cohomology, harmonic functions and the first L2-Betti number. Potential Anal., 6 (1997), 313–326.
  • B isch, D. & J ones, V., Algebras associated to intermediate subfactors. Invent. Math., 128 (1997), 89–157.
  • B oca, F., Completely positive maps on amalgamated product C*-algebras. Math. Scand., 72 (1993), 212–222.
  • B urger, M., Kazhdan constants for SL(3, Z). J. Reine Angew. Math., 413 (1991), 36–67.
  • [Ch] C hoda, M., A continuum of nonconjugate property T actions of SL(n, Z) on the hyperfinite II1-factor. Math. Japon., 30 (1985), 133–150.
  • [C1] C onnes, A., Outer conjugacy classes of automorphisms of factors. Ann. Sci. École Norm. Sup., 8 (1975), 383–419.
  • — Sur la classification des facteurs de type II. C. R. Acad. Sci. Paris Sér. A-B, 281 (1975), 13–15.
  • — Classification of injective factors. Cases II1, II, IIIλ, λ ≠ 1. Ann. of Math., 104 (1976), 73–115.
  • — A factor of type II1 with countable fundamental group. J. Operator Theory, 4 (1980), 151–153.
  • C onnes, A. & J ones, V., Property T for von Neumann algebras. Bull. London Math. Soc., 17 (1985), 57–62.
  • D ye, H. A., On groups of measure preserving transformations. II. Amer. J. Math., 85 (1963), 551–576.
  • D ykema, K. J. & Rãdulescu, F., Compressions of free products of von Neumann algebras. Math. Ann., 316 (2000), 61–82.
  • F eldman, J. & M oore, C. C., Ergodic equivalence relations, cohomology, and von Neumann algebras. I, II. Trans. Amer. Math. Soc., 234 (1977), 289–324, 325–359.
  • F ernós, T., Relative property (T) and linear groups. Ann. Inst. Fourier (Grenoble), 56 (2006), 1767–1804.
  • F urman, A., Orbit equivalence rigidity. Ann. of Math., 150 (1999), 1083–1108.
  • — Outer automorphism groups of some ergodic equivalence relations. Comment. Math. Helv., 80 (2005), 157–196.
  • G aboriau, D., Invariants l2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci., 95 (2002), 93–150.
  • — Examples of groups that are measure equivalent to the free group. Ergodic Theory Dynam. Systems, 25 (2005), 1809–1827.
  • G aboriau, D. & P opa, S., An uncountable family of nonorbit equivalent actions of Fn. J. Amer. Math. Soc., 18 (2005), 547–559.
  • G efter, S. L., Cohomology of the ergodic action of a T-group on the homogeneous space of a compact Lie group, in Operators in Function Spaces and Problems in Function Theory, pp. 77–83 (Russian). Naukova Dumka, Kiev, 1987.
  • — Outer automorphism group of the ergodic equivalence relation generated by translations of dense subgroup of compact group on its homogeneous space. Publ. Res. Inst. Math. Sci., 32 (1996), 517–538.
  • G olowin, O. N. & S yadowsky, L. E., Über die Automorphismengruppen der freien Produkte. Rec. Math. [Mat. Sbornik], 4 (46) (1938), 505–514.
  • H aagerup, U., An example of a nonnuclear C*-algebra, which has the metric approximation property. Invent. Math., 50 (1978), 279–293.
  • de la H arpe, P. & V alette, A., La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque, 175 (1989).
  • H jorth, G., A lemma for cost attained. Ann. Pure Appl. Logic, 143 (2006), 87–102.
  • J olissaint, P., Haagerup approximation property for finite von Neumann algebras. J. Operator Theory, 48 (2002), 549–571.
  • J ones, V. F. R., Index for subfactors. Invent. Math., 72 (1983), 1–25.
  • J ung, K., A hyperfinite inequality for free entropy dimension. Proc. Amer. Math. Soc., 134:7 (2006), 2099–2108.
  • K aniuth, E., Der Typ der regulären Darstellung diskreter Gruppen. Math. Ann., 182 (1969), 334–339.
  • K azhdan, D. A., On the connection of the dual space of a group with the structure of its closed subgroups. Funktsional. Anal. i Prilozhen., 1 (1967), 71–74 (Russian); English translation in Funct. Anal. Appl., 1 (1967), 63–65.
  • K osaki, H., Free products of measured equivalence relations. J. Funct. Anal., 207 (2004), 264–299.
  • L yndon, R. C. & S chupp, P. E., Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 89. Springer, Berlin–Heidelberg, 1977.
  • M argulis, G. A., Finitely-additive invariant measures on Euclidean spaces. Ergodic Theory Dynam. Systems, 2 (1982), 383–396.
  • M cD uff, D., Central sequences and the hyperfinite factor. Proc. London Math. Soc., 21 (1970), 443–461.
  • M onod, N. & S halom, Y., Orbit equivalence rigidity and bounded cohomology. Ann. of Math., 164 (2006), 825–878.
  • N icoara, R., P opa, S. & S asyk, R., On II1 factors arising from 2-cocycles of w-rigid groups. J. Funct. Anal., 242 (2007), 230–246.
  • O zawa, N., A Kurosh-type theorem for type II1 factors. Int. Math. Res. Not., 2006 (2006), Art. ID 97560.
  • P eterson, J., A 1-cohomology characterization of property (T) in von Neumann algebras. Preprint, 2004. arXiv:math.OA/0409527.
  • P eterson, J. & P opa, S., On the notion of relative property (T) for inclusions of von Neumann algebras. J. Funct. Anal., 219 (2005), 469–483.
  • P opa, S., Orthogonal pairs of *-subalgebras in finite von Neumann algebras. J. Operator Theory, 9 (1983), 253–268.
  • — Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math., 111 (1993), 375–405.
  • — Free-independent sequences in type II1 factors and related problems, in Recent Advances in Operator Algebras (Orléans, 1992). Astérisque, 232 (1995), 187–202.
  • — Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property T. Doc. Math., 4 (1999), 665–744.
  • — On a class of type II1 factors with Betti numbers invariants. Ann. of Math., 163 (2006), 809–899.
  • — Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions. J. Inst. Math. Jussieu, 5 (2006), 309–332.
  • — Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal., 230 (2006), 273–328.
  • — Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. I, II. Invent. Math., 165 (2006), 369–408, 409–451.
  • — A unique decomposition result for HT factors with torsion free core. J. Funct. Anal., 242 (2007), 519–525.
  • P opa, S. & S asyk, R., On the cohomology of Bernoulli actions. Ergodic Theory Dynam. Systems, 27 (2007), 241–251.
  • S halom, Y., Measurable group theory, in European Congress of Mathematics, pp. 391–423. Eur. Math. Soc., Zürich, 2005.
  • S hlyakhtenko, D., On the classification of full factors of type III. Trans. Amer. Math. Soc., 356 (2004), 4143–4159.
  • T homa, E., Eine Charakterisierung diskreter Gruppen vom Typ I. Invent. Math., 6 (1968), 190–196.
  • T örnquist, A., Orbit equivalence and actions of Fn. J. Symbolic Logic, 71 (2006), 265–282.
  • U eda, Y., Amalgamated free product over Cartan subalgebra. Pacific J. Math., 191 (1999), 359–392.
  • — Notes on treeability and costs for discrete groupoids in operator algebra framework, in Operator Algebras (Abel Symposium 2004), Abel Symp., 1, pp. 259–279. Springer, Berlin–Heidelberg, 2006.
  • V alette, A., Group pairs with property (T), from arithmetic lattices. Geom. Dedicata, 112 (2005), 183–196.
  • V oiculescu, D., Symmetries of some reduced free product C*-algebras, in Operator Algebras and their Connections with Topology and Ergodic Theory (Bu°teni, 1983), Lecture Notes in Math., 1132, pp. 556–588. Springer, Berlin–Heidelberg, 1985.
  • — The analogues of entropy and of Fisher’s information measure in free probability theory. II. Invent. Math., 118 (1994), 411–440.
  • Z immer, R. J., Ergodic Theory and Semisimple Groups. Monographs in Mathematics, 81. Birkhäuser, Basel, 1984.