Unitary operators in the orthogonal complement of a type I von Neumann subalgebra in a type II_1 factor?

摘要

It is well-known that the equality L_G Θ L_H= span{L_g: g ∈ G - H}~(SOT) holds for G an i.c.c. group and H a subgroup in G, where L_G and L_H are the corresponding group von Neumann algebras and L_G Θ L_H is the set {x ∈ L_G: E_(LH)(x) = 0} with E_(LH) the conditional expectation defined from L_G onto L_H. Inspired by this, it is natural to ask whether the equality N Θ A = span{u: u is unitary in N ? A}~(SOT) holds for N a type II_1 factor and A a von Neumann subalgebra of N. In this paper, we give an affirmative answer to this question for the case A a type I von Neumann algebra.