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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the relative weak asymptotic homomorphism property for triples of group von Neumann algebras
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by Paul Jolissaint PDF
Proc. Amer. Math. Soc. 140 (2012), 1393-1396 Request permission

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 \[ \lim _{i\in I}\Vert \mathbb {E}_B(xu_iy)-\mathbb {E}_B(\mathbb {E}_N(x)u_i\mathbb {E}_N(y))\Vert _2=0 \] 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
  • Vitaly Bergelson and Joseph Rosenblatt, Mixing actions of groups, Illinois J. Math. 32 (1988), no. 1, 65–80. MR 921351
  • I. Chifan, On the normalizing algebra of a MASA in a II$_1$ factor, arXiv:math.OA/0606225, 2006.
  • 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
  • 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
  • © Copyright 2011 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1393-1396
  • MSC (2010): Primary 46L10; Secondary 22D25
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10990-0
  • MathSciNet review: 2869123