Abstract
We consider amalgamated free product II1 factors M = M 1*B M 2*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 M i ’s. We apply this to the case where the M i ’s are w-rigid II1 factors, with B equal to either C, to a Cartan subalgebra A in M i , or to a regular hyperfinite II1 subfactor R in M i , to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N 1 * CN2*C …)t, for some t > 0 and some other similar inclusions of algebras C ⊂ N i then, after a permutation of indices, (B ⊂ M i ) is inner conjugate to (C ⊂ N i )t, for all i. Taking B = C and \( M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} \), with {t i }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.
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References
Bekka, M. E. B. & Valette, A., Group cohomology, harmonic functions and the first L 2-Betti number. Potential Anal., 6 (1997), 313–326.
Bisch, D. & Jones, V., Algebras associated to intermediate subfactors. Invent. Math., 128 (1997), 89–157.
Boca, F., Completely positive maps on amalgamated product C*-algebras. Math. Scand., 72 (1993), 212–222.
Burger, M., Kazhdan constants for SL(3, Z). J. Reine Angew. Math., 413 (1991), 36–67.
[Ch] Choda, M., A continuum of nonconjugate property T actions of SL(n, Z) on the hyperfinite II1-factor. Math. Japon., 30 (1985), 133–150.
[C1] Connes, 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.
Connes, A. & Jones, V., Property T for von Neumann algebras. Bull. London Math. Soc., 17 (1985), 57–62.
Dye, H. A., On groups of measure preserving transformations. II. Amer. J. Math., 85 (1963), 551–576.
Dykema, K. J. & Rãdulescu, F., Compressions of free products of von Neumann algebras. Math. Ann., 316 (2000), 61–82.
Feldman, J. & Moore, C. C., Ergodic equivalence relations, cohomology, and von Neumann algebras. I, II. Trans. Amer. Math. Soc., 234 (1977), 289–324, 325–359.
Fernós, T., Relative property (T) and linear groups. Ann. Inst. Fourier (Grenoble), 56 (2006), 1767–1804.
Furman, 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.
Gaboriau, D., Invariants l 2 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.
Gaboriau, D. & Popa, S., An uncountable family of nonorbit equivalent actions of F n . J. Amer. Math. Soc., 18 (2005), 547–559.
Gefter, 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.
Golowin, O. N. & Syadowsky, L. E., Über die Automorphismengruppen der freien Produkte. Rec. Math. [Mat. Sbornik], 4 (46) (1938), 505–514.
Haagerup, U., An example of a nonnuclear C*-algebra, which has the metric approximation property. Invent. Math., 50 (1978), 279–293.
de la Harpe, P. & Valette, A., La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque, 175 (1989).
Hjorth, G., A lemma for cost attained. Ann. Pure Appl. Logic, 143 (2006), 87–102.
Jolissaint, P., Haagerup approximation property for finite von Neumann algebras. J. Operator Theory, 48 (2002), 549–571.
Jones, V. F. R., Index for subfactors. Invent. Math., 72 (1983), 1–25.
Jung, K., A hyperfinite inequality for free entropy dimension. Proc. Amer. Math. Soc., 134:7 (2006), 2099–2108.
Kaniuth, E., Der Typ der regulären Darstellung diskreter Gruppen. Math. Ann., 182 (1969), 334–339.
Kazhdan, 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.
Kosaki, H., Free products of measured equivalence relations. J. Funct. Anal., 207 (2004), 264–299.
Lyndon, R. C. & Schupp, P. E., Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 89. Springer, Berlin–Heidelberg, 1977.
Margulis, G. A., Finitely-additive invariant measures on Euclidean spaces. Ergodic Theory Dynam. Systems, 2 (1982), 383–396.
McDuff, D., Central sequences and the hyperfinite factor. Proc. London Math. Soc., 21 (1970), 443–461.
Monod, N. & Shalom, Y., Orbit equivalence rigidity and bounded cohomology. Ann. of Math., 164 (2006), 825–878.
Nicoara, R., Popa, S. & Sasyk, R., On II1 factors arising from 2-cocycles of w-rigid groups. J. Funct. Anal., 242 (2007), 230–246.
Ozawa, N., A Kurosh-type theorem for type II1 factors. Int. Math. Res. Not., 2006 (2006), Art. ID 97560.
Peterson, J., A 1-cohomology characterization of property (T) in von Neumann algebras. Preprint, 2004. arXiv:math.OA/0409527.
Peterson, J. & Popa, S., On the notion of relative property (T) for inclusions of von Neumann algebras. J. Funct. Anal., 219 (2005), 469–483.
Popa, 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.
Popa, S. & Sasyk, R., On the cohomology of Bernoulli actions. Ergodic Theory Dynam. Systems, 27 (2007), 241–251.
Shalom, Y., Measurable group theory, in European Congress of Mathematics, pp. 391–423. Eur. Math. Soc., Zürich, 2005.
Shlyakhtenko, D., On the classification of full factors of type III. Trans. Amer. Math. Soc., 356 (2004), 4143–4159.
Thoma, E., Eine Charakterisierung diskreter Gruppen vom Typ I. Invent. Math., 6 (1968), 190–196.
Törnquist, A., Orbit equivalence and actions of F n . J. Symbolic Logic, 71 (2006), 265–282.
Ueda, 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.
Valette, A., Group pairs with property (T), from arithmetic lattices. Geom. Dedicata, 112 (2005), 183–196.
Voiculescu, 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.
Zimmer, R. J., Ergodic Theory and Semisimple Groups. Monographs in Mathematics, 81. Birkhäuser, Basel, 1984.
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Ioana, A., Peterson, J. & Popa, S. Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. Acta Math 200, 85–153 (2008). https://doi.org/10.1007/s11511-008-0024-5
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DOI: https://doi.org/10.1007/s11511-008-0024-5