Noncommutative Nest Hardy Spaces and Martingales.

The article focuses on establishing connections between noncommutative Hardy spaces associated with nest algebras and noncommutative martingale Hardy spaces, particularly through the identification of triangular truncation operators as concrete martingale transforms. It develops several foundational results including boundedness properties of triangular truncations on martingale BMO spaces, atomic decompositions of nest Hardy spaces, and duality characterizations linking Hardy spaces and BMO-type spaces defined via projections in finite von Neumann algebras. The work also presents noncommutative maximal inequalities related to families of projections, provides new asymmetric maximal Doob inequalities, and proves complex interpolation results between noncommutative Hardy and BMO spaces. These findings extend classical harmonic analysis and martingale theory concepts to the noncommutative setting of operator algebras equipped with a faithful trace and a nest of projections of order type ℕ.