Unitary operators in the orthogonal complement of a type $\mathrm{I}$ von Neumann algebra in a type $\mathrm{II}^{}_{1}$ factor

Abstract

It is well-known that the equality $$L^{}_{G}\ominus L^{}_{H}=\bar{\mathrm{span}\{L_{g}:g\in G-H\}^{\mathrm{SOT}}}$$ 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 $L^{}_{G}\ominus L^{}_{H}$ is the set $\{x\in L^{}_{G}:E^{}_{L^{}_{H}}(x)=0\}$ 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 $$N\ominus A=\bar{\mathrm{span}\{u: u\mbox{is unitary in}N\ominus A\}^{\mathrm{SOT}}}$$ holds for $N$ a type $\mbox{II}^{}_{1}$ 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.