Unitary operators in the orthogonal complement of a type I von Neumann subalgebra in a type II1 factor

It is well-known that the equality L G ⊖ 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 ⊖ L H is the set { x ∈ L G : E L H ( x ) = 0 } with E L H the conditional expectation defined from L G onto L H . Inspired by this, it is natural to ask whether the equality N ⊖ A = span { u : u is unitary in N ⊖ A } SOT ¯ holds for N a type II1 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.