We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras 
${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $
over an amenable von Neumann subalgebra 
$B$
. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra 
$({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$
with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product 
${\mathrm{II} }_{1} $
factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation 
$ \mathcal{R} $
defined on a standard measure space and which splits as the free product 
$ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $
of recurrent subequivalence relations gives rise to a nonamenable factor 
$\mathrm{L} ( \mathcal{R} )$
with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors 
${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $
arising from nonsingular free ergodic actions 
$\Gamma \curvearrowright (X, \mu )$
on standard measure spaces of amalgamated groups 
$\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $
over a finite subgroup 
$\Sigma $
.
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.
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.
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.
${C}^{\ast } $-algebras and finite-dimensional approximations, in Graduate Studies in Mathematics, Vol. 88 (American Mathematical Society, Providence, RI, 2008).
${\mathrm{II} }_{1} $ factors of negatively curved groups, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 1–33.
${\mathrm{II} }_{1} $ factors of negatively curved groups, II. Actions by product groups, Adv. Math. 245 (2013), 208–236.
$\mathrm{III} $, Ann. Sci. Éc. Norm. Supér. (4) 6 (1973), 133–252.
${\mathrm{II} }_{1} $ factor with two non-conjugate Cartan subalgebras, Bull. Amer. Math. Soc. 6 (1982), 211–212.
$\mathrm{I} $ and 
$\mathrm{II} $, Trans. Amer. Math. Soc. 234 (1977), 289–324; 325–359.
$\mathrm{III} $ factors with unique Cartan decomposition, J. Math. Pures Appl. (9) 100 (2013), 564–590.
${\mathrm{II} }_{1} $factors, arXiv:1207.0054.
$w$-rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008), 85–153.
$1$-bounded von Neumann algebras, Geom. Funct. Anal. 17 (2007), 1180–1200.
${\mathrm{II} }_{1} $ factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (2010), 713–749.
${\mathrm{II} }_{1} $ factors with at most one Cartan subalgebra 
$\mathrm{II} $, Amer. J. Math. 132 (2010), 841–866.
${\mathrm{II} }_{1} $ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), 809–899.
${\mathrm{II} }_{1} $ factors arising from malleable actions of w-rigid groups 
$\mathrm{I} $, 
$\mathrm{II} $, Invent. Math. 165 (2006), 369–408; 409–451.
${\mathrm{II} }_{1} $factors arising from arbitrary actions of free groups, Acta Math., to appear, arXiv:1111.6951.
${\mathrm{II} }_{1} $factors arising from arbitrary actions of hyperbolic groups, J. Reine Angew. Math., to appear, arXiv:1201.2824.
$\mathrm{III} $ factors, Proc. Natl. Acad. Sci. USA 97 (2000), 12439–12441.
${\mathrm{II} }_{1} $ factors with many non conjugate Cartan subalgebras, Adv. Math. 231 (2012), 2224–2251.
$\mathrm{I} $, in Encyclopaedia of mathematical sciences, Vol. 124, Operator Algebras and Non-commutative Geometry, vol. 5 (Springer, Berlin, 2002).
$\mathrm{II} $, in Encyclopaedia of mathematical sciences, Vol. 125, Operator Algebras and Non-commutative Geometry, vol. 6 (Springer, Berlin, 2003).
${\mathrm{II} }_{1} $ factors, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 743–788.
${\mathrm{II} }_{1} $ factor, Math. Ann. 355 (2013), 661–696.
${C}^{\ast } $-algebras., in Operator algebras and their connections with topology and ergodic theory, Lecture Notes in Mathematics, vol. 1132 (Springer, Heidelberg, 1985), 556–588.
$\mathrm{III} $. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), 172–199.Email your librarian or administrator to recommend adding this journal to your organisation's collection.
* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.
Usage data cannot currently be displayed
