Generalized interpolation for type 1 subdiagonal algebras

Abstract

Let \({\mathfrak {A}}\) be a maximal subdiagonal algebra with diagonal \({\mathfrak {D}}\) in a \(\sigma \)-finite von Neumann algebra \({\mathcal {M}}\) with respect to a faithful normal conditional expectation \(\Phi \). We firstly give a type decomposition of an invariant subspace \({\mathfrak {M}}\) of \({\mathfrak {A}}\) in the acting Hilbert space. We then revisit certain useful properties of type 1 subdiagonal algebras. It is shown that a two-sided invariant subspace \({\mathfrak {M}}\) in the noncommutative \(H^2\) space has the form \({\mathfrak {M}}=\oplus _{n\ge 1}^{col}W_nH^2\) for a family of partial isometries \(\{W_n:n\ge 1\}\) satisfying \( W_n^*W_m=0\) when \(n\not =m\), \(W_n^*W_n\in {\mathfrak {D}}\) and \(\sum _{n\ge 1} W_nW_n^*=I\) if \({\mathfrak {D}}\) is a factor. Furthermore, we give a noncommutative version of the Sarason’s generalized interpolation theorem for such a two-sided invariant subspace of a type 1 subdiagonal algebra.