Noncommutative $H^p$ spaces associated with type 1 subdiagonal algebras

Abstract

Let $\mathfrak{A}$ be a type 1 subdiagonal algebra in a $\sigma$-finite von Neumann algebra $\mathcal{M}$ with respect to a faithful normal conditional expectation $\Phi$. We consider a Riesz type factorization theorem in noncommutative $H^p$ spaces associated with $\mathfrak{A}$. It is shown that if $1\le r,p,q<\infty$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$, then for any $h\in H^r$, there exist $h_p\in H^p$ and $h_q\in H^q$ such that $h=h_p{h}_q$. Beurling type invariant subspace theorem for noncommutative $L^p(1<p<\infty )$ space is obtained. Furthermore, we show that a $\sigma$-weakly closed subalgebra containing $\mathfrak{A}$ of $\mathcal{M}$ is also a type 1 subdiagonal algebra. As an application, We prove that the relative invariant subspace lattice $La{t}_{\mathcal{M}}\mathfrak{A}$ of $\mathfrak{A}$ in $\mathcal{M}$ is commutative.

arXiv admin note: text overlap with arXiv:2003.12966

References in corpus (1)

• On the maximality of subdiagonal algebras