We investigate the structure of crossed product von Neumann algebras arising from Bogoljubov actions of countable groups on Shlyakhtenko’s free Araki–Woods factors. Among other results, we settle the questions of factoriality and Connes’ type classification. We moreover provide general criteria regarding fullness and strong solidity. As an application of our main results, we obtain examples of type factors that are prime, have no Cartan subalgebra and possess a maximal amenable abelian subalgebra. We also obtain a new class of strongly solid type factors with prescribed Connes’ invariants that are not isomorphic to any free Araki–Woods factors.
Nous étudions la structure de produits croisés d’algèbres de von Neumann provenant des actions Bogoljubov de groupes dénombrables sur les facteurs d’Araki–Woods libres. Nous résolvons notamment les questions concernant la factorialité et la classification du type de Connes. Nous donnons également des critères généraux concernant le caractère plein et la solidité forte. Comme application de nos résultats, nous obtenons des exemples de facteurs de type qui sont premiers, sans sous-algèbre de Cartan et qui possèdent une sous-algèbre maximale moyennable abélienne. Nous obtenons aussi une nouvelle classe de facteurs fortement solides de type avec des invariants de Connes prescrits et qui ne sont pas isomorphes à des facteurs d’Araki–Woods libres.
Revised:
Accepted:
Published online:
Keywords: Free Araki–Woods factors; Fullness; Maximal amenable subalgebras; Popa’s deformation/rigidity theory; Type $\mathrm{III}$ factors; Ultraproduct von Neumann algebras
Mot clés : Facteurs d’Araki–Woods libres ; Facteurs pleins ; Sous-algèbres maximales moyennables ; Théorie déformation/rigidité de Popa ; Facteurs de type $\mathrm{III}$ ; Ultraproduits d’algèbres de von Neumann
@article{AIF_2021__71_2_473_0, author = {Houdayer, Cyril and Trom, Benjamin}, title = {Structure of extensions of free {Araki{\textendash}Woods} factors}, journal = {Annales de l'Institut Fourier}, pages = {473--514}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {71}, number = {2}, year = {2021}, doi = {10.5802/aif.3391}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3391/} }
TY - JOUR AU - Houdayer, Cyril AU - Trom, Benjamin TI - Structure of extensions of free Araki–Woods factors JO - Annales de l'Institut Fourier PY - 2021 SP - 473 EP - 514 VL - 71 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3391/ DO - 10.5802/aif.3391 LA - en ID - AIF_2021__71_2_473_0 ER -
%0 Journal Article %A Houdayer, Cyril %A Trom, Benjamin %T Structure of extensions of free Araki–Woods factors %J Annales de l'Institut Fourier %D 2021 %P 473-514 %V 71 %N 2 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3391/ %R 10.5802/aif.3391 %G en %F AIF_2021__71_2_473_0
Houdayer, Cyril; Trom, Benjamin. Structure of extensions of free Araki–Woods factors. Annales de l'Institut Fourier, Volume 71 (2021) no. 2, pp. 473-514. doi : 10.5802/aif.3391. http://archive.numdam.org/articles/10.5802/aif.3391/
[1] Amenable correspondences and approximation properties for von Neumann algebras, Pac. J. Math., Volume 171 (1995) no. 2, pp. 309-341 | DOI | MR | Zbl
[2] Ultraproducts of von Neumann algebras, J. Funct. Anal., Volume 266 (2014) no. 12, pp. 6842-6913 | DOI | MR | Zbl
[3] Crossed-products by locally compact groups: intermediate subfactors, J. Oper. Theory, Volume 79 (2018) no. 1, pp. 101-137 | MR | Zbl
[4] Amenable absorption in amalgamated free product von Neumann algebras, Kyoto J. Math., Volume 58 (2018) no. 3, pp. 583-593 | MR | Zbl
[5] Amalgamated free product type factors with at most one Cartan subalgebra, Compos. Math., Volume 150 (2014) no. 1, pp. 143-174 | DOI | MR | Zbl
[6] Strong solidity of free Araki–Woods factors, Am. J. Math., Volume 140 (2018) no. 5, pp. 1231-1252 | DOI | MR | Zbl
[7] Gradient forms and strong solidity of free quantum groups (2018) (https://arxiv.org/abs/1802.01968)
[8] Bass–Serre rigidity results in von Neumann algebras, Duke Math. J., Volume 153 (2010) no. 1, pp. 23-54 | MR | Zbl
[9] Une classification des facteurs de type , Ann. Sci. Éc. Norm. Supér., Volume 6 (1973), pp. 133-252 | DOI | Numdam | MR | Zbl
[10] Almost periodic states and factors of type , J. Funct. Anal., Volume 16 (1974), pp. 415-445 | DOI | MR | Zbl
[11] Classification of injective factors. Cases , , , , Ann. Math., Volume 104 (1976), pp. 73-115 | DOI | MR
[12] Factors of type , property , and closure of inner automorphisms, J. Oper. Theory, Volume 14 (1985), pp. 189-211 | MR | Zbl
[13] Interpolated free group factors, Pac. J. Math., Volume 163 (1994) no. 1, pp. 123-135 | DOI | MR | Zbl
[14] The standard form of von Neumann algebras, Math. Scand., Volume 37 (1975), pp. 271-283 | DOI | MR
[15] Connes’ bicentralizer problem and uniqueness of the injective factor of type , Acta Math., Volume 69 (1986), pp. 95-148 | Zbl
[16] Pointwise inner automorphisms of von Neumann algebras, J. Funct. Anal., Volume 92 (1990) no. 1, pp. 177-201 | DOI | MR | Zbl
[17] A class of factors with an exotic abelian maximal amenable subalgebra, Trans. Am. Math. Soc., Volume 366 (2014) no. 7, pp. 3693-3707 | DOI | MR | Zbl
[18] Structure of factors arising from free Bogoljubov actions of arbitrary groups, Adv. Math., Volume 260 (2014), pp. 414-457 | DOI | MR | Zbl
[19] Gamma stability in free product von Neumann algebras, Commun. Math. Phys., Volume 336 (2015) no. 2, pp. 831-851 | DOI | MR | Zbl
[20] Unique prime factorization and bicentralizer problem for a class of type factors, Adv. Math., Volume 305 (2017), pp. 402-455 | DOI | MR | Zbl
[21] Factoriality, Connes’ type invariants and fullness of amalgamated free product von Neumann algebras, Proc. R. Soc. Edinb., Sect. A, Math., Volume 150 (2020) no. 3, pp. 1495-1532 | DOI | MR | Zbl
[22] Fullness and Connes’ invariant of type tensor product factors, J. Math. Pures Appl., Volume 121 (2019), pp. 113-134 | DOI | MR | Zbl
[23] Asymptotic structure of free Araki–Woods factors, Math. Ann., Volume 363 (2015) no. 1-2, pp. 237-267 | DOI | MR | Zbl
[24] Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors, Adv. Math., Volume 228 (2011) no. 2, pp. 764-802 | DOI | MR | Zbl
[25] Strongly solid factors with an exotic MASA, Int. Math. Res. Not., Volume 2011 (2011) no. 6, pp. 1352-1380 | MR | Zbl
[26] Classification of a family of non almost periodic free Araki–Woods factors, J. Eur. Math. Soc., Volume 21 (2019) no. 10, pp. 3113-3142 | DOI | MR | Zbl
[27] Asymptotic structure of free product von Neumann algebras, Math. Proc. Camb. Philos. Soc., Volume 161 (2016) no. 3, pp. 489-516 | DOI | MR | Zbl
[28] Rigidity of free product von Neumann algebras, Compos. Math., Volume 152 (2016) no. 12, pp. 2461-2492 | DOI | MR | Zbl
[29] Type factors with unique Cartan decomposition, J. Math. Pures Appl., Volume 100 (2013) no. 4, pp. 564-590 | DOI | MR | Zbl
[30] Cartan subalgebras of amalgamated free product factors, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 1, pp. 71-130 | DOI | MR | Zbl
[31] Unique prime factorization for infinite tensor product factors, J. Funct. Anal., Volume 276 (2019) no. 7, pp. 2245-2278 | DOI | MR | Zbl
[32] Unitary conjugacy for type subfactors and W-superrigidity (2019) (https://arxiv.org/abs/1902.01049)
[33] Central sequences in crossed products of full factors, Duke Math. J., Volume 49 (1982), pp. 29-33 | MR | Zbl
[34] Thin factors with no Cartan subalgebras, Kyoto J. Math., Volume 59 (2019) no. 4, pp. 815-867 | MR | Zbl
[35] Fullness of crossed products of factors by discrete groups (2018) (https://arxiv.org/abs/1811.08274, to appear in Proc. R. Soc. Edinb., Sect. A, Math.)
[36] Spectral gap characterization of full type factors, J. Reine Angew. Math., Volume 753 (2019), pp. 193-210 | DOI | MR | Zbl
[37] Rings of operators. IV, Ann. Math., Volume 44 (1943), pp. 716-808 | DOI | MR | Zbl
[38] Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, 1138, Springer, 1985, iv+115 pages | DOI | MR | Zbl
[39] Haagerup approximation property for arbitrary von Neumann algebras, Publ. Res. Inst. Math. Sci., Volume 51 (2015) no. 3, pp. 567-603 | DOI | MR | Zbl
[40] A Kurosh type theorem for type factors, Int. Math. Res. Not., Volume 2006 (2006) no. 1, 97560, 21 pages | Zbl
[41] On a class of factors with at most one Cartan subalgebra, Ann. Math., Volume 172 (2010) no. 1, pp. 713-749 | DOI | MR | Zbl
[42] On a problem of R.V. Kadison on maximal abelian -subalgebras in factors, Invent. Math., Volume 65 (1981), pp. 269-281 | DOI | MR | Zbl
[43] Maximal injective subalgebras in factors associated with free groups, Adv. Math., Volume 50 (1983), pp. 27-48 | DOI | MR | Zbl
[44] On a class of type factors with Betti numbers invariants, Ann. Math., Volume 163 (2006) no. 3, pp. 809-899 | DOI | MR | Zbl
[45] Strong rigidity of factors arising from malleable actions of w-rigid groups , Invent. Math., Volume 165 (2006) no. 2, pp. 369-408 | DOI | Zbl
[46] On the superrigidity of malleable actions with spectral gap, J. Am. Math. Soc., Volume 21 (2008) no. 4, pp. 981-1000 | DOI | MR | Zbl
[47] Unique Cartan decomposition for factors arising from arbitrary actions of free groups, Acta Math., Volume 212 (2014) no. 1, pp. 141-198 | DOI | Zbl
[48] Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math., Volume 115 (1994) no. 2, pp. 347-389 | DOI | MR | Zbl
[49] Free quasi-free states, Pac. J. Math., Volume 177 (1997) no. 2, pp. 329-368 | DOI | MR | Zbl
[50] Some applications of freeness with amalgamation, J. Reine Angew. Math., Volume 500 (1998), pp. 191-212 | DOI | MR | Zbl
[51] -valued semicircular systems, J. Funct. Anal., Volume 166 (1999) no. 1, pp. 1-47 | DOI | MR | Zbl
[52] On the classification of full factors of type , Trans. Am. Math. Soc., Volume 356 (2004) no. 10, pp. 4143-4159 | DOI | MR | Zbl
[53] Conditional expectations in von Neumann algebras, J. Funct. Anal., Volume 9 (1972), pp. 306-321 | DOI | MR | Zbl
[54] Theory of Operator Algebras. , Encyclopaedia of Mathematical Sciences, 124, Springer, 2002, xx+415 pages
[55] Theory of operator algebras. , Encyclopaedia of Mathematical Sciences, 125, Springer, 2003, xxii+518 pages
[56] Amalgamated free product over Cartan subalgebra, Pac. J. Math., Volume 191 (1999) no. 2, pp. 359-392 | DOI | MR | Zbl
[57] Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math., Volume 228 (2011) no. 5, pp. 2647-2671 | DOI | MR | Zbl
[58] États quasi-libres libres et facteurs de type (d’après D. Shlyakhtenko), Séminaire Bourbaki. Volume 2003/2004 (Astérisque), Volume 299, Société Mathématique de France, 2005, pp. 329-350 | Zbl
[59] Rigidity results for Bernoulli actions and their von Neumann algebras (after S. Popa), Séminaire Bourbaki. Volume 2005/2006 (Astérisque), Volume 311, Société Mathématique de France, 2007, pp. 237-294 | Numdam | Zbl
[60] Symmetries of some reduced free product -algebras, Operator algebras and their connections with topology and ergodic theory (Lecture Notes in Mathematics), Volume 1132, Springer, 1985, pp. 556-588 | DOI | MR | Zbl
[61] The analogues of entropy and of Fisher’s information measure in free probability theory. . The absence of Cartan subalgebras, Geom. Funct. Anal., Volume 6 (1996) no. 1, pp. 172-199 | DOI | MR | Zbl
[62] Free random variables, CRM Monograph Series, 1, American Mathematical Society, 1992 | DOI | Zbl
Cited by Sources: