Factorization for finite subdiagonal algebras of type 1

Abstract

Let ${\mathfrak {A}}$ be a type 1 subdiagonal algebra in a finite von Neumann algebra ${\mathcal {M}}$ with respect to a faithful normal conditional expectation $\Phi$. We consider inner-outer factorization in noncommutative $H^p$($0<p\le \infty$) spaces associated with ${\mathfrak {A}}$. It is shown that for any nonzero $x\in H^p$, there exist a partial isometry $V\in {\mathfrak {A}}$ and an outer $h\in H^p$ such that $x=Vh$. Furthermore, we give a necessary and sufficient condition for a nonzero element in noncommutative $L^p({\mathcal {M}})$ to have a partial BN-factorization associated with ${\mathfrak {A}}$. As an application, we show that for any $0<r,p,q\le \infty$ with $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$, if $h\in H^r$, then there exist $h_p\in H^p$ and $h_q\in H^q$ such that $h=h_ph_q$ and $\left\| h\right\| _r=\left\| h_p\right\| _p\left\| h_q\right\| _q$.