×

zbMATH — the first resource for mathematics

Asymptotic structure of free product von Neumann algebras. (English) Zbl 1379.46047
Summary: Let \((M,\phi) = (M_1, \phi_1)*(M_2, \phi_2)\) be the free product of any \(\sigma\)-finite von Neumann algebras endowed with any faithful normal states. We show that whenever \(Q\subset M\) is a von Neumann subalgebra with separable predual such that both \(Q\) and \(Q\cap M_1\) are the ranges of faithful normal conditional expectations and such that both the intersection \(Q\cap M_1\) and the central sequence algebra \(Q'\cap M^\omega\) are diffuse (e.g. \(Q\) is amenable), then \(Q\) must sit inside \(M_1\). This result generalizes the previous results of the first named author in [Commun. Math. Phys. 336, No. 2, 831–851 (2015; Zbl 1328.46046)] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion \(M_1\subset M\) in arbitrary free product von Neumann algebras.

MSC:
46L10 General theory of von Neumann algebras
47C15 Linear operators in \(C^*\)- or von Neumann algebras
46L09 Free products of \(C^*\)-algebras
Citations:
Zbl 1328.46046
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ando, H. and Haagerup, U.Ultraproducts of von Neumann algebras. J. Funct. Anal.266 (2014), 6842-6913. doi:10.1016/j.jfa.2014.03.0133198856 · Zbl 1305.46049
[2] Boutonnet, R. and Carderi, A.Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. Geom. Funct. Anal.25 (2015), 1688-1705. doi:10.1007/s00039-015-0348-13432155 · Zbl 1342.46055
[3] Boutonnet, R., Houdayer, C. and Raum, S.Amalgamated free product type III factors with at most one Cartan subalgebra. Compositio Math.150 (2014), 143-174. doi:10.1112/S0010437X13007537 · Zbl 1308.46067
[4] Blackadar, B.Operator Algebras. Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry 3. (Springer-Verlag, Berlin, 2006), xx+517 pp. doi:10.1007/3-540-28517-2 · Zbl 1092.46003
[5] Chifan, I. and Houdayer, C.Bass-Serre rigidity results in von Neumann algebras. Duke Math. J.153 (2010), 23-54. doi:10.1215/00127094-2010-0202641939 · Zbl 1201.46057
[6] Connes, A., Une classification des facteurs de type III, Ann. Sci. École Norm. Sup., 6, 133-252, (1973) · Zbl 0274.46050
[7] Connes, A., Classification of injective factors. Cases II_{1}, II_{∞}, III_{λ}, λ 1, Ann. of Math., 74, 73-115, (1976) · Zbl 0343.46042
[8] Connes, A., On the cohomology of operator algebras, J. Funct. Anal., 28, 248-253, (1978) · Zbl 0408.46042
[9] Connes, A. and Størmer, E.Homogeneity of the state space of factors of type III_{1}. J. Funct. Anal.28 (1978), 187-196. doi:10.1016/0022-1236(78)90085-X · Zbl 0408.46048
[10] Feldman, J. and Moore, C.C.Ergodic equivalence relations, cohomology and von Neumann algebras. I and II. Trans. Amer. Math. Soc.234 (1977), 289-324, 325-359. doi:10.1090/S0002-9947-1977-0578730-20578656 · Zbl 0369.22010
[11] Haagerup, U., Connes’ bicentralizer problem and uniqueness of the injective factor of type III_{1}, Acta Math., 69, 95-148, (1986) · Zbl 0628.46061
[12] Haagerup, U. and Størmer, E.Equivalence of normal states on von Neumann algebras and the flow of weights. Adv. Math.83 (1990), 180-262. doi:10.1016/0001-8708(90)90078-21074023 · Zbl 0717.46054
[13] Houdayer, C., A class of II_{1} factors with an exotic abelian maximal amenable subalgebra, Trans. Amer. Math. Soc., 366, 3693-3707, (2014) · Zbl 1303.46044
[14] Houdayer, C., Structure of II_{1} factors arising from free Bogoljubov actions of arbitrary groups, Adv. Math., 260, 414-457, (2014) · Zbl 1297.46042
[15] Houdayer, C., Gamma stability in free product von Neumann algebras, Commun. Math. Phys., 336, 831-851, (2015) · Zbl 1328.46046
[16] Houdayer, C. and Isono, Y. Unique prime factorization and bicentralizer problem for a class of type III factors. arXiv:1503.01388 · Zbl 1371.46050
[17] Houdayer, C. and Raum, S.Asymptotic structure of free Araki-Woods factors. Math. Ann.363 (2015), 237-267. doi:10.1007/s00208-015-1168-13394379 · Zbl 1339.46057
[18] Houdayer, C. and Ricard, É.. Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors. Adv. Math.228 (2011), 764-802. doi:10.1016/j.aim.2011.06.0102822210 · Zbl 1267.46071
[19] Houdayer, C. and Ueda, Y.,. Rigidity of free product von Neumann algebra. To appear in Compositio Math.arXiv:1507.02157 · Zbl 1379.46046
[20] Houdayer, C. and Vaes, S.Type III factors with unique Cartan decomposition. J. Math. Pure Appl.100 (2013), 564-590. doi:10.1016/j.matpur.2013.01.013 · Zbl 1291.46052
[21] Ioana, A., Cartan subalgebras of amalgamated free product II_{1} factors, Ann. Sci. École Norm. Sup., 48, 71-130, (2015) · Zbl 1351.46058
[22] Ioana, A., Peterson, J. and Popa, S.Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math.200 (2008), 85-153. doi:10.1007/s11511-008-0024-52386109 · Zbl 1149.46047
[23] Jones, V. F.R., Index for subfactors, Invent. Math., 72, 1-25, (1983) · Zbl 0508.46040
[24] Kadison, R. V., Diagonalizing matrices, Amer. J. Math., 106, 1451-1468, (1984) · Zbl 0585.46048
[25] Kosaki, H., Characterization of crossed product (properly infinite case), Pacific J. Math., 137, 159-167, (1989) · Zbl 0693.46058
[26] Krieger, W., On ergodic flows and the isomorphism of factors, Math. Ann., 223, 19-70, (1976) · Zbl 0332.46045
[27] Martín, M. and Ueda, Y.On the geometry of von Neumann algebra preduals. Positivity18 (2014), 519-530. doi:10.1007/s11117-013-0259-z3249917 · Zbl 1314.46014
[28] Masuda, T. and Tomatsu, R. Classification of actions of discrete Kac algebras on injective factors. To appear in Mem. Amer. Math. Soc.arXiv:1306.5046
[29] Ocneanu, A.Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics, 1138 (Springer-Verlag, Berlin, 1985), iv+115 pp. doi:10.1007/BFb0098579 · Zbl 0608.46035
[30] Ozawa, N.A remark on amenable von Neumann subalgebras in a tracial free product. Proc. Japan Acad. Ser. A Math. Sci.91 (2015), 104. doi:10.3792/pjaa.91.1043365404 · Zbl 1351.46059
[31] Peterson, J., L2-rigidity in von Neumann algebras, Invent. Math., 175, 417-433, (2009) · Zbl 1170.46053
[32] Pimsner, M. and Popa, S.Entropy and index for subfactors. Ann. Sci. École Norm. Sup.19 (1986), 57-106. · Zbl 0646.46057
[33] Popa, S., Maximal injective subalgebras in factors associated with free groups, Adv. Math., 50, 27-48, (1983) · Zbl 0545.46041
[34] Popa, S., Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math., 111, 375-405, (1993) · Zbl 0787.46047
[35] Popa, S., On a class of type II_{1} factors with Betti numbers invariants, Ann. of Math., 163, 809-899, (2006) · Zbl 1120.46045
[36] Popa, S., Strong rigidity of II_{1} factors arising from malleable actions of w-rigid groups I, Invent. Math., 165, 369-408, (2006) · Zbl 1120.46043
[37] Popa, S., On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc., 21, 981-1000, (2008) · Zbl 1222.46048
[38] Takesaki, M.Theory of Operator Algebras. I. Encyclopedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5 (Springer, Berlin, 2002), xx+415 pp. · Zbl 0990.46034
[39] Takesaki, M.Theory of operator algebras. II. Encyclopaedia of Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry, 6 (Springer-Verlag, Berlin, 2003), xxii+518 pp. doi:10.1007/978-3-662-10451-4 · Zbl 1059.46031
[40] Ueda, Y., Amalgamated free products over Cartan subalgebra, Pacific J. Math., 191, 359-392, (1999) · Zbl 1030.46085
[41] Ueda, Y., Remarks on free products with respect to non-tracial states, Math. Scand., 88, 111-125, (2001) · Zbl 1026.46048
[42] Ueda, Y., Fullness, Connes’ χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras, Trans. Amer. Math. Soc., 355, 349-371, (2003) · Zbl 1028.46097
[43] Ueda, Y., Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math., 228, 2647-2671, (2011) · Zbl 1252.46059
[44] Ueda, Y., On type III_{1} factors arising as free products, Math. Res. Lett., 18, 909-920, (2011) · Zbl 1243.46053
[45] Ueda, Y., Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting, J. London Math. Soc., 88, 25-48, (2013) · Zbl 1285.46048
[46] Vaes, S., Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa), Astérisque, 311, 237-294, (2007) · Zbl 1194.46085
[47] Vaes, S., Explicit computations of all finite index bimodules for a family of II_{1} factors, Ann. Sci. École Norm. Sup., 41, 743-788, (2008) · Zbl 1194.46086
[48] Voiculescu, D.-V.. Symmetries of some reduced free product C*-algebras. Operator algebras and their Connections with Topology and Ergodic Theory. Lecture Notes in Mathematics 1132 (Springer-Verlag, 1985), 556-588.
[49] Voiculescu, D.-V., Dykema, K.J. and Nica, A.. Free random variables. CRM Monograph Series 1 (American Mathematical Society, Providence, RI, 1992).
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.