Subdiagonal algebras with Beurling type invariant subspaces

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

“…We give a Riesz type factorization theorem for non-commutative H P (1 ≤ p < ∞) which says that every element in H r is a product of two elements in H p and H q respectively with 1 r = 1 p + 1 q for any 1 ≤ r, p, q < ∞ in Section 2. This gives an answer of a problem in [15]. Moreover, we consider a Beurling type invariant subspace theorem in L p (M) when 1 < p < ∞ in Section 3.…”

“…Recently, Labuschagne in [23] extended their results to noncommutative H 2 for maximal subdiagonal algebras in a σ-finite von Neumann algebra based on noncommutative H p spaces in [13]. We also discussed Riesz type factorization theorem in noncommutative H 1 and Beurling type invariant subspace theorem for L 1 (M) associated with a type 1 subdiagonal algebra in the sense that every right invariant subspace of a maximal subdiagonal algebra in noncommutative H 2 is of Beurling type in [15,16]. Do hold for type 1 subdiagonal algebras?…”

“…It is worth noting that there were several successful examples showing that some famous mature theorems about the classical Hardy spaces have been extended to the noncommutative Hardy spaces based on subdiagonal algebras(cf. [3,4,5,6,13,14,15,17,18]). Marsalli and West [24] gave a Riesz factorization theorem for finite noncommutative H p spaces.…”

“…[16]). In this paper, we continuously consider type 1 subdiagonal algebras introduced in [15]. Let A be a type 1 subdiagonal algebra with respect to Φ with diagonal D, that is, every invariant right invariant subspace M ⊆ H 2 of A is of type 1.…”

“…Note that there is a family of column orthogonal partial isometries U = {U n : n ≥ 1} such that H 2 0 = ⊕ col n U n H 2 and those partial isometries together with the diagonal D consist of generators of A(cf. [15]). We will fixed those partial isometries throughout this paper.…”