THE RELATIVE WEAK ASYMPTOTIC HOMOMORPHISM PROPERTY FOR INCLUSIONS OF FINITE VON NEUMANN ALGEBRAS Academic Article uri icon

abstract

  • 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∥double-struck E signB(xuλ y) - double-struck E signB(double-struck E signN(x) uλ double-struck 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 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) ⊆ 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.

author list (cited authors)

  • FANG, J., GAO, M., & SMITH, R. R.

citation count

  • 4

complete list of authors

  • FANG, JUNSHENG||GAO, MINGCHU||SMITH, ROGER R

publication date

  • July 2011