Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Abstract

Let $\mathcal{M}$ be a σ-finite von Neumann algebra and let $\mathfrak{A} \subseteq \mathcal{M}$ be a maximal subdiagonal algebra with respect to a faithful normal conditional expectation Φ. Based on the Haagerup’s noncommutative L ^p space $L^p \left( \mathcal{M} \right)$ associated with $\mathcal{M}$, we consider Toeplitz operators and the Hilbert transform associated with $\mathfrak{A}$. We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space $H^2 \left( \mathcal{M} \right)$ is just the right analytic Toeplitz algebra. Furthermore, the Hilbert transform on noncommutative $L^p \left( \mathcal{M} \right)$ is shown to be bounded for 1 < p < ∞. As an application, we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative $H^1 \left( \mathcal{M} \right)$ as a concrete space of operators.