Toeplitz and Hankel operators associated with subdiagonal algebras

Abstract

Let M be a σ-finite von Neumann algebra and let A ⊂ M be a maximal subdiagonal algebra with respect to some faithful normal expectation E on M. Let φ be a normal faithful E-invariant state on M, let L 2 (M, φ) be the non-commutative Lebesgue space in the sense of U. Haagerup, and consider the Hardy space H 2 (A, φ) C L 2 (M, φ) associated with the pair (A, φ). For each x ∈ M, the Toeplitz operator T x ∈ B(H 2 (A, φ)) and the Hankel operator H x ∈ B ( H 2 (A, φ), H 2 (A, φ)⊥) are defined as in the classical case of the unit circle. We show that the mapping x ↦ T x is completely isometric on M and therefore σ(x) C σ(T x ) for all x ∈ M. We also show that ∥H x ∥ = dist(x, A) for every x ∈ M.