Skip to main content Accesibility Help

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

  • Cyril Houdayer (a1) and Yusuke Isono (a2)

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.

Hide All
1Ando, H. and Haagerup, U.. Ultraproducts of von Neumann algebras. J. Funct. Anal. 266 (2014), 68426913.
2Boutonnet, R. and Houdayer, C.. Amenable absorption in amalgamated free product von Neumann algebras. Kyoto J. Math. 58 (2018), 583593.
3Boutonnet, R., Houdayer, C. and Raum, S.. Amalgamated free product type III factors with at most one Cartan subalgebra. Compos. Math. 150 (2014), 143174.
4Boutonnet, R., Houdayer, C. and Vaes, S.. Strong solidity of free Araki–Woods factors. Amer. J. Math. 140 (2018), 12311252.
5Connes, A.. Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6 (1973), 133252.
6Connes, A.. Almost periodic states and factors of type III1. J. Funct. Anal. 16 (1974), 415445.
7Connes, A.. Outer conjugacy classes of automorphisms of factors. Ann. Sci. École Norm. Sup. 8 (1975), 383419.
8Connes, A.. Classification of injective factors. Cases II1, II, IIIλ, λ ≠ 1. Ann. Math. 74 (1976), 73115.
9Connes, A.. On the spatial theory of von Neumann algebras. J. Funct. Anal. 35 (1980), 153164.
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.
11Gaboriau, D.. Coût des relations d'équivalence et des groupes. Invent. Math. 139 (2000), 4198.
12Haagerup, U.. The standard form of von Neumann algebras. Math. Scand. 37 (1975), 271283.
13Haagerup, U.. Operator valued weights in von Neumann algebras, I. J. Funct. Anal. 32 (1979), 175206.
14Haagerup, U.. Operator valued weights in von Neumann algebras, II. J. Funct. Anal. 33 (1979), 339361.
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.
16Houdayer, C. and Isono, Y.. Unique prime factorization and bicentralizer problem for a class of type III factors. Adv. Math. 305 (2017), 402455.
17Houdayer, C. and Ueda, Y.. Asymptotic structure of free product von Neumann algebras. Math. Proc. Cambridge Philos. Soc. 161 (2016), 489516.
18Houdayer, C. and Ueda, Y.. Rigidity of free product von Neumann algebras. Compos. Math. 152 (2016), 24612492.
19Houdayer, C. and Vaes, S.. Type III factors with unique Cartan decomposition. J. Math. Pures Appl. 100 (2013), 564590.
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.
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.
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.
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.
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.
25Isono, Y.. Unique prime factorization for infinite tensor product factors. arXiv:1712.00925.
26Jones, V. F. R.. Index for subfactors. Invent. Math. 72 (1983), 125.
27Jones, V. F. R. and Schmidt, K.. Asymptotically invariant sequences and approximate finiteness. Amer. J. Math. 109 (1987), 91114.
28Kadison, R. V.. Diagonalizing matrices. Amer. J. Math. 106 (1984), 14511468.
29Kosaki, H.. Extension of Jones' theory on index to arbitrary factors. J. Funct. Anal. 66 (1986), 123140.
30Marrakchi, A.. Spectral gap characterization of full type III factors. To appear in J. Reine Angew. Math. arXiv:1605.09613.
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.
32McDuff, D.. Central sequences and the hyperfinite factor. Proc. London Math. Soc. 21 (1970), 443461.
33Murray, F. and von Neumann, J.. Rings of operators. IV. Ann. Math. 44 (1943), 716808.
34Ocneanu, A.. Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics, 1138. Springer-Verlag, Berlin, 1985. iv+115 pp.
35Ozawa, N. and Popa, S.. On a class of II1 factors with at most one Cartan subalgebra. Ann. Math. 172 (2010), 713749.
36Peterson, J.. L2-rigidity in von Neumann algebras. Invent. Math. 175 (2009), 417433.
37Pimsner, M. and Popa, S.. Entropy and index for subfactors. Ann. Sci. École Norm. Sup. 19 (1986), 57106.
38Popa, S.. On a problem of R.V. Kadison on maximal abelian *-subalgebras in factors. Invent. Math. 65 (1981), 269281.
39Popa, S.. Maximal injective subalgebras in factors associated with free groups. Adv. Math. 50 (1983), 2748.
40Popa, S.. Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111 (1993), 375405.
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.
42Popa, S.. On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163 (2006), 809899.
43Popa, S.. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I. Invent. Math. 165 (2006), 369408.
44Popa, S.. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21 (2008), 9811000.
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.
46Ueda, Y.. Amalgamated free products over Cartan subalgebra. Pacific J. Math. 191 (1999), 359392.
47Ueda, Y.. Fullness, Connes' χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras. Trans. Amer. Math. Soc. 355 (2003), 349371.
48Ueda, Y.. Factoriality, type classification and fullness for free product von Neumann algebras. Adv. Math. 228 (2011), 26472671.
49Ueda, Y.. On type III1 factors arising as free products. Math. Res. Lett. 18 (2011), 909920.
50Ueda, Y.. Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting. J. London Math. Soc. 88 (2013), 2548.
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.
52Voiculescu, D.-V., Dykema, K.J. and Nica, A.. Free random variables. CRM Monograph Series 1. (Providence, RI: American Mathematical Society, 1992).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed