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The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras | | Junsheng Fang
; Mingchu Gao
; Roger R. Smith
; | | Date: |
17 May 2010 | | Abstract: | A triple of finite von Neumann algebras $Bsubseteq Nsubseteq M$ is said to
have the relative weak asymptotic homomorphism property if there exists a net
of unitary operators ${u_{lambda}}_{lambdain Lambda}$ in $B$ such that
$$lim_{lambda}|mathbb{E}}_B(xu_{lambda}y)-{mathbb{E}}_B({mathbb{E}}_N(x)u_{lambda}{mathbb{E}}_N(y))|_2=0$$
for all $x,yin M$. We prove that a triple of finite von Neumann algebras
$Bsubseteq Nsubseteq M$ has the relative weak asymptotic homomorphism
property if and only if $N$ contains the set of all $xin M$ such that
$Bxsubseteq sum_{i=1}^n x_iB$ for a finite number of elements $x_1,...,x_n$
in $M$. Such an $x$ is called a one sided quasi-normalizer of $B$, and the von
Neumann algebra generated by all one sided quasi-normalizers of $B$ is called
the one sided quasi-normalizer algebra of $B$.
We characterize one sided quasi-normalizer algebras for inclusions of group
von Neumann algebras and use this to show that one sided quasi-normalizer
algebras and quasi-normalizer algebras are not equal in general. We also give
some applications to inclusions $L(H)subseteq L(G)$ arising from containments
of groups. For example, when $L(H)$ is a masa we determine the unitary
normalizer algebra as the von Neumann algebra generated by the normalizers of
$H$ in $G$. | | Source: | arXiv, 1005.3049 | | Services: | Forum | Review | PDF | Favorites | |
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