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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the relative weak asymptotic homomorphism property for triples of group von Neumann algebras


Author: Paul Jolissaint
Journal: Proc. Amer. Math. Soc. 140 (2012), 1393-1396
MSC (2010): Primary 46L10; Secondary 22D25
Published electronically: August 5, 2011
MathSciNet review: 2869123
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Abstract: A triple of finite von Neumann algebras $ B\subset N\subset M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitaries $ (u_i)_{i\in I}\subset U(B)$ such that

$\displaystyle \lim_{i\in I}\Vert \mathbb{E}_B(xu_iy)-\mathbb{E}_B(\mathbb{E}_N(x)u_i\mathbb{E}_N(y))\Vert_2=0 $

for all $ x,y\in M$. Recently, J. Fang, M. Gao and R. Smith proved that the triple $ B\subset N\subset M$ has the relative weak asymptotic homomorphism property if and only if $ N$ contains the set of all $ x\in M$ such that $ Bx\subset\sum_{i=1}^n x_iB$ for finitely many elements $ x_1,\ldots,x_n\in M$. Furthermore, if $ H<G$ is a pair of groups, they get a purely algebraic characterization of the weak asymptotic homomorphism property for the pair of von Neumann algebras $ L(H)\subset L(G)$, but their proof requires a result which is very general and whose proof is rather long. We extend the result to the case of a triple of groups $ H<K<G$, we present a direct and elementary proof of the above-mentioned characterization, and we introduce three more equivalent conditions on the triple $ H<K<G$, one of them stating that the subspace of $ H$-compact vectors of the quasi-regular representation of $ H$ on $ \ell^2(G/H)$ is contained in $ \ell^2(K/H)$.


References [Enhancements On Off] (What's this?)

  • 1. Vitaly Bergelson and Joseph Rosenblatt, Mixing actions of groups, Illinois J. Math. 32 (1988), no. 1, 65–80. MR 921351 (89g:28029)
  • 2. I. Chifan, On the normalizing algebra of a MASA in a II$ _1$ factor, arXiv:math.OA/0606225, 2006.
  • 3. J. Fang, M. Gao and R. R. Smith, The relative weak asymptotic homomorphism property for inclusions of finite von Neumann algebras, arXiv:math.OA/1005.3049 v1, 2010.

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Additional Information

Paul Jolissaint
Affiliation: Université de Neuchâtel, Institut de Mathémathiques, Emile-Argand 11, 2000 Neuchâtel, Switzerland
Email: paul.jolissaint@unine.ch

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10990-0
PII: S 0002-9939(2011)10990-0
Keywords: von Neumann algebra, one-sided quasi-normalizer, discrete group, quasi-regular representation, asymptotic homomorphism
Received by editor(s): November 8, 2010
Received by editor(s) in revised form: November 18, 2010, and January 5, 2011
Published electronically: August 5, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society