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Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

Published online by Cambridge University Press:  29 January 2019

Cyril Houdayer
Affiliation:
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS Université Paris-Saclay, 91405 Orsay, FRANCE (cyril.houdayer@math.u-psud.fr)
Yusuke Isono
Affiliation:
RIMS, Kyoto University, 606-8502Kyoto, JAPAN (isono@kurims.kyoto-u.ac.jp)

Abstract

We investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.

Type
Research Article
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Ando, H. and Haagerup, U.. Ultraproducts of von Neumann algebras. J. Funct. Anal. 266 (2014), 68426913.CrossRefGoogle Scholar
2Boutonnet, R. and Houdayer, C.. Amenable absorption in amalgamated free product von Neumann algebras. Kyoto J. Math. 58 (2018), 583593.CrossRefGoogle Scholar
3Boutonnet, R., Houdayer, C. and Raum, S.. Amalgamated free product type III factors with at most one Cartan subalgebra. Compos. Math. 150 (2014), 143174.CrossRefGoogle Scholar
4Boutonnet, R., Houdayer, C. and Vaes, S.. Strong solidity of free Araki–Woods factors. Amer. J. Math. 140 (2018), 12311252.CrossRefGoogle Scholar
5Connes, A.. Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6 (1973), 133252.CrossRefGoogle Scholar
6Connes, A.. Almost periodic states and factors of type III1. J. Funct. Anal. 16 (1974), 415445.CrossRefGoogle Scholar
7Connes, A.. Outer conjugacy classes of automorphisms of factors. Ann. Sci. École Norm. Sup. 8 (1975), 383419.CrossRefGoogle Scholar
8Connes, A.. Classification of injective factors. Cases II1, II, IIIλ, λ ≠ 1. Ann. Math. 74 (1976), 73115.CrossRefGoogle Scholar
9Connes, A.. On the spatial theory of von Neumann algebras. J. Funct. Anal. 35 (1980), 153164.CrossRefGoogle Scholar
10Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I and II. Trans. Amer. Math. Soc. 234 (1977), 289324, 325–359.CrossRefGoogle Scholar
11Gaboriau, D.. Coût des relations d'équivalence et des groupes. Invent. Math. 139 (2000), 4198.CrossRefGoogle Scholar
12Haagerup, U.. The standard form of von Neumann algebras. Math. Scand. 37 (1975), 271283.CrossRefGoogle Scholar
13Haagerup, U.. Operator valued weights in von Neumann algebras, I. J. Funct. Anal. 32 (1979), 175206.CrossRefGoogle Scholar
14Haagerup, U.. Operator valued weights in von Neumann algebras, II. J. Funct. Anal. 33 (1979), 339361.CrossRefGoogle Scholar
15Houdayer, C. and Isono, Y.. Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence. Comm. Math. Phys. 348 (2016), 9911015.CrossRefGoogle Scholar
16Houdayer, C. and Isono, Y.. Unique prime factorization and bicentralizer problem for a class of type III factors. Adv. Math. 305 (2017), 402455.CrossRefGoogle Scholar
17Houdayer, C. and Ueda, Y.. Asymptotic structure of free product von Neumann algebras. Math. Proc. Cambridge Philos. Soc. 161 (2016), 489516.CrossRefGoogle Scholar
18Houdayer, C. and Ueda, Y.. Rigidity of free product von Neumann algebras. Compos. Math. 152 (2016), 24612492.CrossRefGoogle Scholar
19Houdayer, C. and Vaes, S.. Type III factors with unique Cartan decomposition. J. Math. Pures Appl. 100 (2013), 564590.CrossRefGoogle Scholar
20Houdayer, C., Marrakchi, A. and Verraedt, P.. Fullness and Connes' τ invariant of type III tensor product factors. J. Math. Pures Appl. 121 (2019), 113134.CrossRefGoogle Scholar
21Houdayer, C., Marrakchi, A. and Verraedt, P.. Strongly ergodic equivalence relations: spectral gap and type III invariants. To appear in Ergodic Theory Dynam. Systems. arXiv:1704.07326.Google Scholar
22Houdayer, C., Shlyakhtenko, D. and Vaes, S.. Classification of a family of non almost periodic free Araki-Woods factors. To appear in J. Eur. Math. Soc. arXiv:1605.06057.Google Scholar
23Ioana, A.. Cartan subalgebras of amalgamated free product II1 factors. With an appendix joint with Stefaan Vaes. Ann. Sci. École Norm. Sup. 48 (2015), 71130.CrossRefGoogle Scholar
24Ioana, A., Peterson, J. and Popa, S.. Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math. 200 (2008), 85153.CrossRefGoogle Scholar
25Isono, Y.. Unique prime factorization for infinite tensor product factors. arXiv:1712.00925.Google Scholar
26Jones, V. F. R.. Index for subfactors. Invent. Math. 72 (1983), 125.CrossRefGoogle Scholar
27Jones, V. F. R. and Schmidt, K.. Asymptotically invariant sequences and approximate finiteness. Amer. J. Math. 109 (1987), 91114.CrossRefGoogle Scholar
28Kadison, R. V.. Diagonalizing matrices. Amer. J. Math. 106 (1984), 14511468.CrossRefGoogle Scholar
29Kosaki, H.. Extension of Jones' theory on index to arbitrary factors. J. Funct. Anal. 66 (1986), 123140.CrossRefGoogle Scholar
30Marrakchi, A.. Spectral gap characterization of full type III factors. To appear in J. Reine Angew. Math. arXiv:1605.09613.Google Scholar
31Masuda, T. and Tomatsu, R.. Classification of actions of discrete Kac algebras on injective factors. Mem. Amer. Math. Soc. 245 (2017), no. 1160, ix+118 pp.Google Scholar
32McDuff, D.. Central sequences and the hyperfinite factor. Proc. London Math. Soc. 21 (1970), 443461.CrossRefGoogle Scholar
33Murray, F. and von Neumann, J.. Rings of operators. IV. Ann. Math. 44 (1943), 716808.CrossRefGoogle Scholar
34Ocneanu, A.. Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics, 1138. Springer-Verlag, Berlin, 1985. iv+115 pp.CrossRefGoogle Scholar
35Ozawa, N. and Popa, S.. On a class of II1 factors with at most one Cartan subalgebra. Ann. Math. 172 (2010), 713749.CrossRefGoogle Scholar
36Peterson, J.. L2-rigidity in von Neumann algebras. Invent. Math. 175 (2009), 417433.CrossRefGoogle Scholar
37Pimsner, M. and Popa, S.. Entropy and index for subfactors. Ann. Sci. École Norm. Sup. 19 (1986), 57106.CrossRefGoogle Scholar
38Popa, S.. On a problem of R.V. Kadison on maximal abelian *-subalgebras in factors. Invent. Math. 65 (1981), 269281.CrossRefGoogle Scholar
39Popa, S.. Maximal injective subalgebras in factors associated with free groups. Adv. Math. 50 (1983), 2748.CrossRefGoogle Scholar
40Popa, S.. Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111 (1993), 375405.CrossRefGoogle Scholar
41Popa, S.. Classification of subfactors and their endomorphisms. CBMS Regional Conference Series in Mathematics, 86. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1995. x+110 pp.CrossRefGoogle Scholar
42Popa, S.. On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163 (2006), 809899.CrossRefGoogle Scholar
43Popa, S.. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I. Invent. Math. 165 (2006), 369408.CrossRefGoogle Scholar
44Popa, S.. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21 (2008), 9811000.CrossRefGoogle Scholar
45Takesaki, 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.CrossRefGoogle Scholar
46Ueda, Y.. Amalgamated free products over Cartan subalgebra. Pacific J. Math. 191 (1999), 359392.CrossRefGoogle Scholar
47Ueda, Y.. Fullness, Connes' χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras. Trans. Amer. Math. Soc. 355 (2003), 349371.CrossRefGoogle Scholar
48Ueda, Y.. Factoriality, type classification and fullness for free product von Neumann algebras. Adv. Math. 228 (2011), 26472671.CrossRefGoogle Scholar
49Ueda, Y.. On type III1 factors arising as free products. Math. Res. Lett. 18 (2011), 909920.CrossRefGoogle Scholar
50Ueda, Y.. Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting. J. London Math. Soc. 88 (2013), 2548.CrossRefGoogle Scholar
51Voiculescu, 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.CrossRefGoogle Scholar
52Voiculescu, D.-V., Dykema, K.J. and Nica, A.. Free random variables. CRM Monograph Series 1. (Providence, RI: American Mathematical Society, 1992).CrossRefGoogle Scholar

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