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Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra. (English) Zbl 1314.46075

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

MSC:

46L52 Noncommutative function spaces
47L75 Other nonselfadjoint operator algebras
46K50 Nonselfadjoint (sub)algebras in algebras with involution
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
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