# zbMATH — the first resource for mathematics

Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. (English) Zbl 1342.46055
The authors develop a new approach, via the centralizers of certain states on the reduced $$C^*$$-algebras, to study maximal amenable von Neumann subalgebras coming from maximal amenable subgroups. Let $$\Lambda\subset \Gamma$$ be a countable discrete group. $$\Lambda$$ is said to be singular in $$\Gamma$$ if there is a continuous action of $$\Gamma$$ on some compact Hausdorff space $$X$$ such that, for any $$\mu \in \mathrm{Prob}_{\Lambda}(X)$$ and $$g\in \Gamma \backslash \Lambda$$, we have $$g\cdot \mu \perp \mu$$. The authors show that, if $$\Lambda$$ is singular in $$\Gamma$$, then $$L\Lambda$$ is maximal amenable inside $$L\Gamma$$. Many examples are given as applications of this result. In particular, for $$n\geq 2$$, if $$\Gamma=SL_n(\mathbb{Z})$$ and $$\Lambda\subset \Gamma$$ is the subgroup of upper triangular matrices, then $$\Lambda$$ is singular in $$\Gamma$$.
The paper also gives many equivalent characterizations of singularity, and a simple proof for showing strong solidity for some group von Neumann algebras.

##### MSC:
 46L10 General theory of von Neumann algebras 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations
Full Text:
##### References:
 [1] Adams, S., Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology, 33, 765-783, (1994) · Zbl 0838.20042 [2] Anantharaman-Delaroche, C., Systèmes dynamiques non-commutatifs et moyennabilité, Mathematische Annalen, 279, 297-315, (1987) · Zbl 0608.46036 [3] Borel, A.; Tits, J., Groupes réductifs, Institut des Hautes Études Scientifiques Publications Mathématiques, 27, 55-150, (1965) · Zbl 0145.17402 [4] R. Boutonnet and A. Carderi. Maximal amenable subalgebras in von Neumann algebras associated with hyperbolic groups (preprint 2013). arXiv:1310.5864 · Zbl 1369.46052 [5] Bowditch, B.H., Relatively hyperbolic groups, International Journal of Algebra and Computation, 22, 1250016, 66, (2012) · Zbl 1259.20052 [6] N. Brown and N. Ozawa. $$C$$\^{}{*}-algebras and finite-dimensional approximations. In: Graduate Studies in Mathematics, Vol. 88. American Mathematical Society, Providence (2008) · Zbl 1160.46001 [7] Cameron, J.; Fang, J.; Ravichandran, M.; White, S., The radial masa in a free group factor is maximal injective, Journal of the London Mathematical Society, 82, 787-809, (2010) · Zbl 1237.46043 [8] Chifan, I.; Sinclair, T., On the structural theory of II_{1} factors of negatively curved groups, Annales scientifiques de École normale supérieure, 46, 1-33, (2013) · Zbl 1290.46053 [9] Connes, A., Classification of injective factors. cases II_{1}, II_{∞}, III_{$$λ$$}, $$λ$$ ≠ 1, Annals of Mathematics, 104, 73-115, (1976) · Zbl 0343.46042 [10] E. Ghys and P. de la Harpe (eds.). Sur les groupes hyperboliques d’après Mikhael Gromov. In: Progress in Mathematics, Vol. 83. Birkhäuser, Boston (1990) · Zbl 1213.46053 [11] Houdayer, C., A class of II_{1} factors with an exotic abelian maximal amenable subalgebra, Transactions of the American Mathematical Society, 366, 3693-3707, (2014) · Zbl 1303.46044 [12] Knapp A.W.: Lie groups beyond an introduction. In: Progress in Mathematics, Vol. 140, Birkhauser, Boston (1996) · Zbl 0862.22006 [13] B. Leary. On maximal amenable subalgebras in amalgamated free products. PhD thesis, UCLA (in preparation) · Zbl 1227.22003 [14] Moore, C.C., Amenable subgroups of semi-simple groups and proximal flows, Israel Journal of Mathematics, 34, 121-138, (1979) · Zbl 0431.22014 [15] Ozawa, N., Solid von Neumann algebras, Acta Mathematics, 192, 111-117, (2004) · Zbl 1072.46040 [16] Ozawa, N., Boundary amenability of relatively hyperbolic groups, Topology and its Applications, 153, 2624-2630, (2006) · Zbl 1109.20037 [17] Ozawa, N., Weak amenability of hyperbolic groups, Groups, Geometry, and Dynamics, 2, 271-280, (2008) · Zbl 1147.43003 [18] Ozawa, N., A comment on free group factors, Noncommutative harmonic analysis with applications to probability II. Banach Center Publications, 89, 241-245, (2010) · Zbl 1214.46038 [19] Ozawa, N., Examples of groups which are not weakly amenable, Kyoto Journal of Mathematics, 52, 333-344, (2012) · Zbl 1242.43007 [20] Ozawa, N.; Popa, S., On a class of II_{1} factors with at most one Cartan subalgebra, Annals of Mathematics, 172, 713-749, (2010) · Zbl 1201.46054 [21] Ozawa, N.; Popa, S., On a class of II_{1} factors with at most one Cartan subalgebra II, American Journal of Mathematics, 132, 841-866, (2010) · Zbl 1213.46053 [22] Peterson, J.; Thom, A., Group cocycles and the ring of affiliated operators, Inventiones Mathematicae, 185, 561-592, (2011) · Zbl 1227.22003 [23] Popa, S., Maximal injective subalgebras in factors associated with free groups, Advances in Mathematics, 50, 27-48, (1983) · Zbl 0545.46041 [24] Popa, S.; Vaes, S., Unique Cartan decomposition for II_{1} factors arising from arbitrary actions of free groups, Acta Mathematica, 212, 141-198, (2014) · Zbl 1307.46047 [25] S. Popa and S. Vaes. Unique Cartan decomposition for II_{1} factors arising from arbitrary actions of hyperbolic groups. Journal für die reine und angewandte Mathematik (to appear) · Zbl 1307.46047 [26] Prasad, G.; Raghunathan, M.S., Cartan subgroups and lattices in semi-simple groups, Annals of Mathematics, 96, 296-317, (1972) · Zbl 0245.22013 [27] Prasad, G.; Rapinchuk, A.S., Existence of irreducible R-regular elements in Zariski-dense subgroups, Mathematical Research Letters, 10, 21-32, (2003) · Zbl 1029.22020 [28] Robertson, G.; Steger, T., Malnormal subgroups of lattices and the pukánszky invariant in group factors, Journal of Functional Analysis, 258, 2708-2713, (2010) · Zbl 1204.22005 [29] Shen, J., Maximal injective subalgebras of tensor products of free group factors, Journal of Functional Analysis, 240, 334-348, (2006) · Zbl 1113.46061 [30] Zimmer R.J.: Ergodic theory and semisimple groups. In: Monographs in Mathematics, Vol. 81, Birkhauser, Basel (1984) · Zbl 0571.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.