Noncommutative Nest Hardy Spaces and Martingales

Let (M,τ ) be a finite von Neumann algebra equipped with a normalized faithful trace and let A be a nest of order type N. The nest algebra is defined by H∞ʳ(A)=∈ M:ex=exe,e∈ A\, and the related noncommutative Hardy space Hₚʳ(A) is the completion of H∞ʳ(A) in Lₚ(M). In the present paper, we make a connection between Hₚʳ(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.