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

Abstract

It is well-known that the equalityLG⊖LH=span{Lg:g∈G−H}SOT¯ holds for G an i.c.c. group and H a subgroup in G, where LG and LH are the corresponding group von Neumann algebras and LG⊖LH is the set {x∈LG:ELH(x)=0} with ELH the conditional expectation defined from LG onto LH. Inspired by this, it is natural to ask whether the equalityN⊖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.