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Strongly singular MASAs and mixing actions in finite von Neumann algebras

Published online by Cambridge University Press:  15 September 2008

PAUL JOLISSAINT  and
YVES STALDER
PAUL JOLISSAINT
Affiliation:
Université de Neuchâtel, Institut de Mathémathiques, Emile-Argand 11, Case postale 158, CH-2009 Neuchâtel, Switzerland (email: paul.jolissaint@unine.ch)
YVES STALDER
Affiliation:
Laboratoire de Mathématiques, Université Blaise Pascal, Campus universitaire des Cézeaux, 63177 Aubière Cedex, France (email: yves.stalder@math.univ-bpclermont.fr)
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Abstract

Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A=L(Γ0) is a singular MASA in M=L(Γ) which satisfies a weakly mixing condition. If, moreover, it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, Higman–Neumann–Neumann extensions and semidirect products, and in particular we exhibit examples of singular MASAs that satisfy the weak mixing condition but not the strong mixing one.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1861 - 1878
Copyright
Copyright © 2008 Cambridge University Press

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References

[1] Bass, H.. Group actions on non-Archimedean trees. Arboreal Group Theory (Berkeley, CA, 1988) (Mathematical Sciences Research Institute Publications, 19). Springer, New York, 1991, pp. 69–131.CrossRefGoogle Scholar
[2] Bildea, T.. On the Laplacian subalgebra of some tensors of group von Neumann algebras. Contemp. Math. 414 (2006), 265–271.CrossRefGoogle Scholar
[3] Brown, K. S. and Geoghegan, R.. An infinite-dimensional torsion-free group. Invent. Math. 77 (1984), 367–381.CrossRefGoogle Scholar
[4] Dykema, K. J., Sinclair, A. M. and Smith, R. R.. Values of the Pukánsky invariant in free group factors and the hyperfinite factor. J. Funct. Anal. 240 (2006), 373–398.CrossRefGoogle Scholar
[5] Gromov, M.. Hyperbolic groups. Essays in Group Theory (Mathematical Sciences Research Institute Publications, 8). Springer, New York, 1987, pp. 75–263.CrossRefGoogle Scholar
[6] Jolissaint, P.. Operator algebras related to Thompson’s group F. J. Aust. Math. Soc. 79 (2005), 231–241.CrossRefGoogle Scholar
[7] Neshveyev, S. and Størmer, E.. Dymical Entropy in Operator Algebras. Springer, Berlin, 2006.CrossRefGoogle Scholar
[8] Popa, S.. Cocycle and orbit superrigidity for malleable actions of w-rigid groups. Invent. Math. 170 (2007), 243–295.CrossRefGoogle Scholar
[9] Popa, S.. Strong rigidity of II 1 factors arising from malleable actions of w-rigid groups, I. Invent. Math. 165 (2006), 369–408.CrossRefGoogle Scholar
[10] Popa, S.. Orthogonal pairs of *-subalgebras in finite von Neumann algebras. J. Operator Theory 9 (1983), 253–268.Google Scholar
[11] Popa, S.. Singular maximal abelian *-subalgebras in continuous von Neumann algebras. J. Funct. Anal. 50 (1983), 151–166.CrossRefGoogle Scholar
[12] Radulescu, F.. Singularity of the radial subalgebra of ℒ(F N) and the Pukánsky invariant. Pacific J. Math. 151 (1991), 297–306.CrossRefGoogle Scholar
[13] Robertson, G.. Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces. ArXiv:math.OA/0504362v1, 2005.Google Scholar
[14] Robertson, G., Sinclair, A. M. and Smith, R. R.. Strong singularity for subalgebras of finite factors. Int. J. Math. 14 (2003), 235–258.CrossRefGoogle Scholar
[15] Schmidt, K.. Asymptotic properties of unitary representations and mixing. Proc. London Math. Soc. 48 (1984), 445–460.CrossRefGoogle Scholar
[16] Serre, J.-P.. Arbres, amalgames, SL2 (Astérisque, 46). Société Mathématique de France, Paris, 1977.Google Scholar
[17] Sinclair, A. M. and Smith, R. R.. Strongly singular masas in type II 1 factors. Geom. Funct. Anal. 12 (2002), 199–216.CrossRefGoogle Scholar
[18] Sinclair, A. M. and Smith, R. R.. The Pukánsky invariant for masas in group von Neumann algebras. Illinois J. Math. 49 (2005), 325–343.CrossRefGoogle Scholar
[19] Sinclair, A. M., Smith, R. R., White, S. A. and Wiggins, A.. Strong singularity of singular masas. Illinois J. Math. 51 (2007), 1077–1084.CrossRefGoogle Scholar
[20] Stalder, Y.. Moyennabilité intérieure et extension HNN. Ann. Inst. Fourier (Grenoble) 56 (2006), 309–323.CrossRefGoogle Scholar