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

TL;DR

This paper investigates the structure of the orthogonal complement of a type I von Neumann algebra within a type II_1 factor, establishing a span equality for unitary operators in this setting.

Contribution

It proves that the orthogonal complement of a type I von Neumann algebra in a type II_1 factor is spanned by its unitary operators, confirming a natural generalization of a known group algebra result.

Findings

The orthogonal complement is spanned by its unitaries.

Affirmative answer for the span equality in this setting.

Extension of known group algebra results to von Neumann algebras.

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.