Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-67gxp Total loading time: 0.281 Render date: 2021-03-05T16:38:44.866Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Asymptotic structure of free product von Neumann algebras

Published online by Cambridge University Press:  20 May 2016

CYRIL HOUDAYER
Affiliation:
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, FRANCE. e-mails: cyril.houdayer@math.u-psud.fr
YOSHIMICHI UEDA
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka, 810-8560, JAPAN. e-mails: ueda@math.kyushu-u.ac.jp

Abstract

Let (M, ϕ) = (M 1, ϕ1) * (M 2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever QM is a von Neumann subalgebra with separable predual such that both Q and QM 1 are the ranges of faithful normal conditional expectations and such that both the intersection QM 1 and the central sequence algebra Q′Mω 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 [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M 1M in arbitrary free product von Neumann algebras.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below.

References

[AH12] Ando, H. and Haagerup, U. Ultraproducts of von Neumann algebras. J. Funct. Anal. 266 (2014), 68426913.CrossRefGoogle Scholar
[BC14] Boutonnet, R. and Carderi, A. Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. Geom. Funct. Anal. 25 (2015), 16881705.CrossRefGoogle Scholar
[BHR12] Boutonnet, R., Houdayer, C. and Raum, S. Amalgamated free product type III factors with at most one Cartan subalgebra. Compositio Math. 150 (2014), 143174.CrossRefGoogle Scholar
[Bl06] Blackadar, B. Operator Algebras. Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry 3. (Springer-Verlag, Berlin, 2006), xx+517 pp.CrossRefGoogle Scholar
[CH08] Chifan, I. and Houdayer, C. Bass–Serre rigidity results in von Neumann algebras. Duke Math. J. 153 (2010), 2354.CrossRefGoogle Scholar
[Co72] Connes, A. Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6 (1973), 133252.CrossRefGoogle Scholar
[Co75] Connes, A. Classification of injective factors. Cases II1, II, IIIλ, λ 1. Ann. of Math. 74 (1976), 73115.CrossRefGoogle Scholar
[Co76] Connes, A. On the cohomology of operator algebras. J. Funct. Anal. 28 (1978), 248253.CrossRefGoogle Scholar
[CS78] Connes, A. and Størmer, E. Homogeneity of the state space of factors of type III1 . J. Funct. Anal. 28 (1978), 187196.CrossRefGoogle Scholar
[FM75] 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, 325359.CrossRefGoogle Scholar
[Ha85] Haagerup, U. Connes' bicentralizer problem and uniqueness of the injective factor of type III1 . Acta Math. 69 (1986), 95148.Google Scholar
[HS90] Haagerup, U. and Størmer, E. Equivalence of normal states on von Neumann algebras and the flow of weights. Adv. Math. 83 (1990), 180262.CrossRefGoogle Scholar
[Ho12a] Houdayer, C. A class of II1 factors with an exotic abelian maximal amenable subalgebra. Trans. Amer. Math. Soc. 366 (2014), 36933707.CrossRefGoogle Scholar
[Ho12b] Houdayer, C. Structure of II1 factors arising from free Bogoljubov actions of arbitrary groups. Adv. Math. 260 (2014), 414457.CrossRefGoogle Scholar
[Ho14] Houdayer, C. Gamma stability in free product von Neumann algebras. Commun. Math. Phys. 336 (2015), 831851.CrossRefGoogle Scholar
[HI15] Houdayer, C. and Isono, Y. Unique prime factorization and bicentralizer problem for a class of type III factors. arXiv:1503.01388 Google Scholar
[HR14] Houdayer, C. and Raum, S. Asymptotic structure of free Araki–Woods factors. Math. Ann. 363 (2015), 237267.CrossRefGoogle Scholar
[HR10] Houdayer, C. and Ricard, É.. Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors. Adv. Math. 228 (2011), 764802.CrossRefGoogle Scholar
[HU15] Houdayer, C. and Ueda, Y.,. Rigidity of free product von Neumann algebra. To appear in Compositio Math. arXiv:1507.02157 Google Scholar
[HV12] Houdayer, C. and Vaes, S. Type III factors with unique Cartan decomposition. J. Math. Pure Appl. 100 (2013), 564590.CrossRefGoogle Scholar
[Io12] Ioana, A. Cartan subalgebras of amalgamated free product II1 factors. Ann. Sci. École Norm. Sup. 48 (2015), 71130.CrossRefGoogle Scholar
[IPP05] Ioana, 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
[Jo82] Jones, V.F.R. Index for subfactors. Invent. Math. 72 (1983), 125.CrossRefGoogle Scholar
[Ka82] Kadison, R.V. Diagonalizing matrices. Amer. J. Math. 106 (1984), 14511468.CrossRefGoogle Scholar
[Ko88] Kosaki, H. Characterization of crossed product (properly infinite case). Pacific J. Math. 137 (1989), 159167.CrossRefGoogle Scholar
[Kr75] Krieger, W. On ergodic flows and the isomorphism of factors. Math. Ann. 223 (1976),1970.CrossRefGoogle Scholar
[MU12] Martín, M. and Ueda, Y. On the geometry of von Neumann algebra preduals. Positivity 18 (2014), 519530.CrossRefGoogle Scholar
[MT13] 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 Google Scholar
[Oc85] Ocneanu, A. Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics, 1138 (Springer-Verlag, Berlin, 1985), iv+115 pp.CrossRefGoogle Scholar
[Oz15] Ozawa, N. A remark on amenable von Neumann subalgebras in a tracial free product. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 104.CrossRefGoogle Scholar
[Pe06] Peterson, J. L2-rigidity in von Neumann algebras. Invent. Math. 175 (2009), 417433.CrossRefGoogle Scholar
[PP84] Pimsner, M. and Popa, S. Entropy and index for subfactors. Ann. Sci. École Norm. Sup. 19 (1986), 57106.CrossRefGoogle Scholar
[Po83] Popa, S. Maximal injective subalgebras in factors associated with free groups. Adv. Math. 50 (1983), 2748.CrossRefGoogle Scholar
[Po90] Popa, S. Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111 (1993), 375405.CrossRefGoogle Scholar
[Po01] Popa, S. On a class of type II1 factors with Betti numbers invariants. Ann. of Math. 163 (2006), 809899.CrossRefGoogle Scholar
[Po03] Popa, S. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I. Invent. Math. 165 (2006), 369408.CrossRefGoogle Scholar
[Po06] Popa, S. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21 (2008), 9811000.CrossRefGoogle Scholar
[Ta02] 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.Google Scholar
[Ta03] 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.CrossRefGoogle Scholar
[Ue98a] Ueda, Y. Amalgamated free products over Cartan subalgebra. Pacific J. Math. 191 (1999), 359392.CrossRefGoogle Scholar
[Ue98b] Ueda, Y. Remarks on free products with respect to non-tracial states. Math. Scand. 88 (2001), 111125.CrossRefGoogle Scholar
[Ue00] Ueda, Y. Fullness, Connes' χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras. Trans. Amer. Math. Soc. 355 (2003), 349371.CrossRefGoogle Scholar
[Ue10] Ueda, Y. Factoriality, type classification and fullness for free product von Neumann algebras. Adv. Math. 228 (2011), 26472671.CrossRefGoogle Scholar
[Ue11] Ueda, Y. On type III1 factors arising as free products. Math. Res. Lett. 18 (2011), 909920.CrossRefGoogle Scholar
[Ue12] Ueda, Y. Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting. J. London Math. Soc. 88 (2013), 2548.CrossRefGoogle Scholar
[Va06] Vaes, S. Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa). Astérisque 311 (2007), 237294.Google Scholar
[Va07] Vaes, S. Explicit computations of all finite index bimodules for a family of II1 factors. Ann. Sci. École Norm. Sup. 41 (2008), 743788.CrossRefGoogle Scholar
[Vo85] 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), 556588.Google Scholar
[VDN92] Voiculescu, D.-V., Dykema, K.J. and Nica, A.. Free random variables. CRM Monograph Series 1 (American Mathematical Society, Providence, RI, 1992).CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 67 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 5th March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Asymptotic structure of free product von Neumann algebras
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Asymptotic structure of free product von Neumann algebras
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Asymptotic structure of free product von Neumann algebras
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *