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Proceedings of the American Mathematical Society

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Toeplitz and Hankel operators associated with subdiagonal algebras

Author: Bebe Prunaru
Journal: Proc. Amer. Math. Soc. 139 (2011), 1387-1396
MSC (2010): Primary 46L51, 47B35; Secondary 47L25, 47L30
Published electronically: August 31, 2010
MathSciNet review: 2748431
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Abstract: Let $ \mathcal M$ be a $ \sigma$-finite von Neumann algebra and let $ \mathcal A\subset\mathcal M$ be a maximal subdiagonal algebra with respect to some faithful normal expectation $ \mathcal E$ on $ \mathcal M.$ Let $ \phi$ be a normal faithful $ \mathcal E$-invariant state on $ \mathcal M$, let $ L^2(\mathcal M,\phi)$ be the non-commutative Lebesgue space in the sense of U. Haagerup, and consider the Hardy space $ H^2(\mathcal A,\phi)\subset L^2(\mathcal M,\phi)$ associated with the pair $ (\mathcal A,\phi).$ For each $ x\in\mathcal M$, the Toeplitz operator $ T_x\in B(H^2(\mathcal A,\phi))$ and the Hankel operator $ H_x\in B(H^2(\mathcal A,\phi),H^2(\mathcal A,\phi)^\perp)$ are defined as in the classical case of the unit circle. We show that the mapping $ x\mapsto T_x$ is completely isometric on $ \mathcal M$ and therefore $ \sigma(x)\subset\sigma(T_x)$ for all $ x\in\mathcal M.$ We also show that $ \Vert H_x\Vert=dist(x,\mathcal A)$ for every $ x\in\mathcal M.$

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Additional Information

Bebe Prunaru
Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania

Keywords: Subdiagonal algebras, Toeplitz operators, Hankel operators, non-commutative Hardy spaces.
Received by editor(s): September 10, 2009
Received by editor(s) in revised form: April 26, 2010
Published electronically: August 31, 2010
Additional Notes: This research was partially supported by Grant PNII - Programme “Idei” (code 1194).
Communicated by: Mario Bonk
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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