×

zbMATH — the first resource for mathematics

Strong rigidity of \(\text{II}_1\) factors arising from malleable actions of \(w\)-rigid groups. I. (English) Zbl 1120.46043
The paper under review gives a remarkable answer to a challenging problem concerning type II\(_1\) factors, originating in the classical work of F. J. Murray and J. von Neumann [Ann. Math. 44, 716–808 (1943; Zbl 0060.26903)]. For any subgroup \(S\subset {\mathbb R}_+^*\) the existence of a II\(_1\) factor \(M\) with fundamental group \({\mathcal F}(M)=S\) is proved. When \(S\) is countable \(M\) can be chosen to be a separable factor (i.e., \(M_*\) is separable). Previous progress was made in 1943 by Murray and von Neumann (proving \({\mathcal F}(R)={\mathbb R}_+^*\) for the hyperfinite II\(_1\) factor \(R\)), in 1980 by A. Connes [J. Oper. Theory 4, 151–153 (1980; Zbl 0455.46056)] (proving that \({\mathcal F}(M)\) is countable if \(M\) is the von Neumann algebra of a discrete countable ICC group with Kazhdan’s property T), in 1987 by V. Ya. Golodets and N. I. Nessonov [J. Funct. Anal. 70, 80–89 (1987; Zbl 0614.46053)] (constructing, for each countable set \(\Sigma\subset {\mathbb R}_+^*\), a type II\(_1\) factor \(M\) with \({\mathcal F}(M)\) countable and \(\Sigma\subset {\mathcal F}(M)\)), and very recently by the author [Ann. Math. (2) 163, 809–899 (2006; Zbl 1120.46045)] (for any II\(_1\) factor \(M\) with a Cartan subalgebra satisfying simultaneously a relative Haagerup type compact approximation property and a relative property T, e.g., \(M={\mathcal L}({\mathbb Z^2} \rtimes SL_2({\mathbb Z}))\), one has \({\mathcal F}(M)=\{ 1\}\)).
The class of II\(_1\) factors considered here consist of crossed products \(M=N \rtimes_\sigma G\) with two properties. The first one requires the discrete ICC group \(G\) to be { \(w\)-rigid} (i.e., to contain an infinite normal subgroup with the relative property T). The second one requires the action \(\sigma\) to be { malleable} and { mixing}. Malleability, as defined by the author [J. Funct. Anal. 230, 273–328 (2006; Zbl 1097.46045)], is related with realizability of the finite von Neumann algebra \((N,\tau)\) as the centralizer \({\mathcal N}_\varphi\) of a von Neumann algebra with discrete decomposition \(({\mathcal N},\varphi)\) and with the existence of a { gauged extension} \(\tilde{\sigma}\) of \(\sigma\) to \((\tilde{{\mathcal N}},\tilde{\varphi})=({\mathcal N}\otimes {\mathcal N},\varphi\otimes \varphi)\) which leaves \({\mathcal N}={\mathcal N}\otimes 1\subset \tilde{{\mathcal N}}\) invariant and commutes with a certain action \(\alpha\) of \({\mathbb R}\) on \(\tilde{{\mathcal N}}\) for which \(\alpha_1 ({\mathcal N}\otimes 1)=1\otimes {\mathcal N}\). The almost periodic spectrum \(S({\mathcal N},\varphi)\), given by the pure point spectrum of the modular group \(\Delta_\varphi\), is a subgroup \(S(\tilde{\sigma})\subset {\mathcal F} (N\rtimes_\sigma G)\subset {\mathbb R}_+^*\). Moreover, the inclusion \(M=N\rtimes_\sigma G \subset {\mathcal N}\rtimes_{\tilde{\sigma}} G\) generates a family \(\text{Aut}_t (M;\tilde{\sigma})\) of \(t\)-scaling automorphisms, \(t\in S(\tilde{\sigma})\), such that any two of them differ by an inner automorphism of \(M\). The embeddings of \({\mathcal L}(G)\) in \(N\rtimes_\sigma G\) are very rigid, as shown by the following central result in the paper.
Theorem. Let \(G_i\) be \(w\)-rigid ICC groups, \(\sigma_i :G_i \rightarrow \text{ Aut} (N_i,\tau_i)\) malleable mixing actions with gauged extensions \(\tilde{\sigma}_i\) and \(M_i=N_i \rtimes_{\sigma_i} G_i\), \(i=0,1\). Let \(\theta:M_0 \cong M_1^s\) be an isomorphism for some \(s>0\). Then there exist unique \(\beta_i \in S(\tilde{\sigma}_i)\) and unique \(\theta^i_{\beta_i} \in \text{ Aut}_{\beta_i} (M_i; \tilde{\sigma}_i)\) such that \(\theta^1_{\beta_1} (\theta ({\mathcal L}(G_0)))={\mathcal L}(G_1)^{s\beta_1}\) and \(\theta (\theta^0_{\beta_0} ({\mathcal L}(G_0)))={\mathcal L}(G_1)^{s\beta_0}\). Moreover, \(\beta_0=\beta_1\).
In particular, taking \(G_i=G\), \(\sigma_i=\sigma\), \(M_i=N\rtimes _\sigma G\) with \(G\) a \(w\)-rigid group and \(\sigma\) a malleable mixing action, one has
\[ \forall s\in {\mathcal F}(N\rtimes _\sigma G),\;\exists t\in S(\tilde{\sigma}),\;st\in {\mathcal F}({\mathcal L} (G)). \]
Corollary. Let \(G\) be a discrete ICC \(w\)-rigid group for which \({\mathcal F}({\mathcal L}(G))=\{ 1\}\), for instance \(G={\mathbb Z}^2 \rtimes SL_2({\mathbb Z})\), and \(\sigma\) be a malleable mixing action on \(N\). Then \({\mathcal F}(N\rtimes _\sigma G)=S(\tilde{\sigma})\).
Bernoulli \(G\)-shifts (both of classical and Connes-Størmer type) and Bogoliubov shifts provide prominent examples of malleable mixing actions. Classical (commutative) Bernoulli \(G\)-shifts \(\sigma\) on \(\bigotimes_{g\in G} {\mathbb T}\) only give \(S(\tilde{\sigma})=\{ 1\}\) and \(\text{Aut}(M;\tilde{\sigma})=\text{ Inn} (M)\). Myriads of examples are however produced by Connes-Størmer \(G\)-shifts \(\sigma\) on ITPFI factors \(({\mathcal N},\varphi)=\bigotimes_{g\in G} (B({\mathcal H}),\varphi_0)\). When the state \(\varphi_0\) is defined by the list of eigenvalues \(\{ t_i \}_i\), the group \({\mathcal F}(N\rtimes _\sigma G)=S(\tilde{\sigma})\) coincides with the multiplicative subgroup in \({\mathbb R}_+^*\) generated by the ratios \(\{ t_i / t_j\}_{i,j}\); thus any subgroup of \({\mathbb R}_+^*\) is effectively realized as the fundamental group of a type II\(_1\) factor.

MSC:
46L35 Classifications of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
22F10 Measurable group actions
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
46L10 General theory of von Neumann algebras
46L40 Automorphisms of selfadjoint operator algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Araki, H., Woods, J.: A classification of factors. Publ. Res. Inst. Math. Sci. 6, 51–130 (1968) · Zbl 0206.12901
[2] Burger, M.: Kazhdan constants for SL(3,\(\mathbb{Z}\)). J. Reine Angew. Math. 413, 36–67 (1991) · Zbl 0704.22009
[3] Christensen, E.: Subalgebras of a finite algebra. Math. Ann. 243, 17–29 (1979) · Zbl 0406.46052
[4] Choda, M.: Group factors of the Haagerup type. Proc. Japan Acad., Ser. A 59, 174–177 (1983) · Zbl 0523.46038
[5] Connes, A.: A type II1 factor with countable fundamental group. J. Oper. Theory 4, 151–153 (1980) · Zbl 0455.46056
[6] Connes, A.: Une classification des facteurs de type III. Ann. Éc. Norm. Sup. 6, 133–252 (1973) · Zbl 0274.46050
[7] Connes, A.: Almost periodic states and factors of type III1. J. Funct. Anal. 16, 415–455 (1974) · Zbl 0302.46050
[8] Connes, A., Jones, V.F.R.: Property T for von Neumann algebras. Bull. Lond. Math. Soc. 17, 57–62 (1985), · Zbl 1190.46047
[9] Connes, A., Størmer, E.: Entropy for automorphisms of II1 von Neumann algebras. Acta Math. 134, 289–306 (1974) · Zbl 0326.46032
[10] Delaroche, C., Kirilov, A.: Sur les relations entre l’espace dual d’un groupe et la structure de ses sous-groupes fermes. Sem. Bourbaki, 20’eme année, 1967–1968, no. 343, juin 1968
[11] Dixmier, J.: Les algèbres d’opérateurs sur l’espace Hilbertien (Algèbres de von Neumann). Paris: Gauthier-Villars 1957, 1969 · Zbl 0088.32304
[12] Feldman, J., Moore, C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras I, II. Trans. Am. Math. Soc. 234, 289–324, 325–359 (1977) · Zbl 0369.22009
[13] Furman, A.: Orbit equivalence rigidity. Ann. Math. 150, 1083–1108 (1999) · Zbl 0943.22012
[14] Gaboriau, D.: Invariants 2 de rélations d’équivalence et de groupes. Publ. Math. Inst., Hautes Étud. Sci. 95, 93–150 (2002) · Zbl 1022.37002
[15] Gaboriau, D., Popa, S.: An uncountable family of non orbit equivalent actions of \(\mathbb{F}_n\) . J. Am. Math. Soc. 18, 547–559 (2005) (math.GR/0306011) · Zbl 1155.37302
[16] Gefter, S., Golodets, V.: Fundamental groups for ergodic actions and actions with unit fundamental groups. Publ. Res. Inst. Math. Sci. 24, 821–847 (1988) · Zbl 0684.22003
[17] Golodets, V.Y., Nesonov, N.I.: T-property and nonisomorphic factors of type II and III. J. Funct. Anal. 70, 80–89 (1987) · Zbl 0614.46053
[18] Haagerup, U.: An example of non-nuclear C*-algebra which has the metric approximation property. Invent. Math. 50, 279–293 (1979) · Zbl 0408.46046
[19] de la Harpe, A., Valette, A.: La propriété T de Kazhdan pour les groupes localement compacts. Astérisque 175, 1–156 (1989)
[20] Hjort, G.: A converse to Dye’s theorem. UCLA preprint, September 2002. To appear in Trans. Am. Math. Soc.
[21] Jolissaint, P.: On the property (T) for pairs of topological groups. Enseign. Math., II. Sér. 51, 31–45 (2005) · Zbl 1106.22006
[22] Jones, V.F.R.: A converse to Ocneanu’s theorem. J. Oper. Theory 4, 21–23 (1982)
[23] Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983) · Zbl 0508.46040
[24] Jones, V.F.R.: Ten problems. In: Mathematics: perspectives and frontieres, pp. 79–91, ed. by V. Arnold, M. Atiyah, P. Lax, B. Mazur. AMS 2000 · Zbl 0969.57001
[25] Jones, V.F.R., Popa, S.: Some properties of MASA’s in factors. In: Invariant subspaces and other topics, pp. 89–102, Operator Theory: Adv. Appl. Boston: Birkhäuser 1982 · Zbl 0508.46041
[26] Kadison, R.V.: Problems on von Neumann algebras. Baton Rouge Conference 1967, unpublished
[27] Kazhdan, D.: Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. Appl. 1, 63–65 (1967) · Zbl 0168.27602
[28] Margulis, G.: Finitely-additive invariant measures on Euclidian spaces. Ergodic Theory Dyn. Syst. 2, 383–396 (1982) · Zbl 0532.28012
[29] Monod, N., Shalom, Y.: Orbit equivalence rigidity and bounded cohomology. Preprint 2002. To appear in Ann. Math. · Zbl 1129.37003
[30] Murray, F., von Neumann. J.: Rings of operators. Ann. Math. 37, 116–229 (1936) · Zbl 0014.16101
[31] Murray, F., von Neumann, J.: On rings of operators IV. Ann. Math. 44, 716–808 (1943) · Zbl 0060.26903
[32] Popa, S.: Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. 230, 273–328 (2006) · Zbl 1097.46045
[33] Popa, S.: Correspondences. INCREST preprint 1986, unpublished
[34] Popa, S.: On a class of type II1 factors with Betti numbers invariants. MSRI preprint no 2001–024, revised math.OA/0209130. To appear in Ann. Math.
[35] Popa, S.: On the fundametal group of type II1 factors. Proc. Natl. Acad. Sci. 101, 723–726 (2004) (math.OA/0210467) · Zbl 1064.46048
[36] Popa, S.: Free independent sequences in type II1 factors and related problems. Astérisque 232, 187–202 (1995) · Zbl 0840.46039
[37] Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups II. Invent. Math., DOI 10.1007/s00222-006-0502-3 (math.OA/0407137) · Zbl 1120.46044
[38] Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups III. In preparation · Zbl 1120.46044
[39] Powers, R.: Representation of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. Math. 86, 138–171 (1967) · Zbl 0157.20605
[40] Powers, R., Størmer, E.: Free states of the canonical anticommutation relations. Comm. Math. Phys. 16, 1–33 (1970) · Zbl 0186.28301
[41] Sakai, S.: C*-algebras and W*-algebras. Berlin, Heidelberg, New York: Springer 1971 · Zbl 0219.46042
[42] Takesaki, M.: The structure of a von Neumann algebra with a homogeneous periodic state. Acta Math. 131, 281–308 (1973) · Zbl 0267.46047
[43] Takesaki, M.: Conditional expectation in von Neumann algebra. J. Funct. Anal. 9, 306–321 (1972) · Zbl 0245.46089
[44] Takesaki, M.: Theory of Operator Algebras II. Encyclopedia of Mathematical Sciences 125. Berlin, Heidelberg, New York: Springer 2002 · Zbl 0990.46034
[45] Valette, A.: Group pairs with relative property (T) from arithmetic lattices. Geom. Dedicata 112, 183–196 (2005) · Zbl 1076.22012
[46] Zimmer, R.: Ergodic theory and semisimple groups. Boston: Birkhäuser 1984 · Zbl 0571.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.