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 {u} in B such that [Formula: see text] 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 [Formula: see text] 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.

published proceedings

  • INTERNATIONAL JOURNAL OF MATHEMATICS

author list (cited authors)

  • Fang, J., Gao, M., & Smith, R. R.

citation count

  • 6

complete list of authors

  • Fang, Junsheng||Gao, Mingchu||Smith, Roger R

publication date

  • July 2011