Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.