On the relative weak asymptotic homomorphism property for triples of group von Neumann algebras
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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 \[ \lim _{i\in I}\Vert \mathbb {E}_B(xu_iy)-\mathbb {E}_B(\mathbb {E}_N(x)u_i\mathbb {E}_N(y))\Vert _2=0 \] 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
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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