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 $$L_G^{}\ominus L_H^{}=\overline{\mathrm{span}\{L_g:g\in G-H{\}}^{\mathrm{SOT}}}$$ 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.

11 pages