THE RELATIVE WEAK ASYMPTOTIC HOMOMORPHISM PROPERTY FOR INCLUSIONS OF FINITE VON NEUMANN ALGEBRAS
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A triple of finite von Neumann algebras B ⊆ N ⊆ M is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators {uλ}λ∈Λ in B such that limλ∥doublestruck E signB(xuλ y)  doublestruck E signB(doublestruck E signN(x) uλ doublestruck E signN(y))∥2 = 0 for all x,y ∈ M. We prove that a triple of finite von Neumann algebras B ⊆ N ⊆ M has the relative weak asymptotic homomorphism property if and only if N contains the set of all x ∈ M such that Bx ⊆ ∑i=1n xiB for a finite number of elements x1, ..., xn in M. Such an x is called a onesided quasinormalizer of B, and the von Neumann algebra generated by all onesided quasinormalizers of B is called the onesided quasinormalizer algebra of B. We characterize onesided quasinormalizer algebras for inclusions of group von Neumann algebras and use this to show that onesided quasinormalizer algebras and quasinormalizer algebras are not equal in general. We also give some applications to inclusions L(H) ⊆ 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. © 2011 World Scientific Publishing Company.
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FANG, J., GAO, M., & SMITH, R. R.
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FANG, JUNSHENGGAO, MINGCHUSMITH, ROGER R
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Discrete Group

Homomorphism

Normalizer

Quasinormalizer

Von Neumann Algebra
Identity
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