The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras

Details The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras

2010-05-17

arxiv.org/abs/1005.3049v1

Mathematics

Operator Algebras

22 pages

Scientific paper

A triple of finite von Neumann algebras $B\subseteq N\subseteq M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators $\{u_{\lambda}\}_{\lambda\in \Lambda}$ 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 $x,y\in M$. We prove that a triple of finite von Neumann algebras $B\subseteq N\subseteq M$ has the relative weak asymptotic homomorphism property if and only if $N$ contains the set of all $x\in M$ such that $Bx\subseteq \sum_{i=1}^n x_iB$ 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 $G$.

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