Nest Hardy and Lipschitz Spaces

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and $\mathbb{A} $ be a nest of projections in $\mathcal{M}$ of order type $\mathbb{N}$. The column nest algebra is defined by $H_\infty ^{c}(\mathbb{A}) =\{x \in \mathcal{M}: ex=exe, \ \forall e \in \mathbb{A}\}$ and for $0<p<\infty $, the associated noncommutative column nest Hardy space $H_{p}^{c}(\mathbb{A})$ is the completion of $H_\infty ^{c}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. We prove that if $(\mathcal{M}_{n})_{n\geq 1}$ is the shell filtration arising from the nest $\mathbb{A}$, then for every $0<p\leq \infty $, the nest Hardy space $H_{p}^{c}(\mathbb{A})$ embeds complementably into the column martingale Hardy space $\mathcal{H}_{p}^{c}(\mathcal{M})$. As an application, we characterize the dual space of $H_{p}^{c}(\mathbb{A})$ for $0<p<1$ as a new type of Lipschitz space associated with the nest $\mathbb{A}$. We also present a martingale proof of the fact that for $0<p<\infty $, the couple $(H_{p}^{c}(\mathbb{A}), H_\infty ^{c}(\mathbb{A}))$ is $K$-closed in the couple $(L_{p}(\mathcal{M}),\mathcal{M})$. This in turn provides a way of extending the study to include nest Hardy spaces associated with symmetric spaces of measurable operators.