The noncommutative \(H^{(r,s)}_{p}({\mathcal A};\ell_{\infty})\) and \(H_{p}({\mathcal A};\ell_{1})\) spaces
type: Item
instance of: scholarly article
title: The noncommutative \(H^{(r,s)}_{p}({\mathcal A};\ell_{\infty})\) and \(H_{p}({\mathcal A};\ell_{1})\) spaces
zbMATH Open document ID: 1356.46050
author: N. E. Zubov
published in: Positivity
publication date: 2015-11-18T00:00:00Z
review text: Let \(\mathcal{M}\) be a finite von Neumann algebra equipped with a normal faithful tracial state \(\tau\), and let \(\mathcal{D}\) be a von Neumann subalgebra. A finite subdiagonal algebra \(\mathcal{A}\) for the pair is defined in the usual way, using the unique conditional expectation \(\Phi: \mathcal{M}\to\mathcal{D}\). The authors define noncommutative vector-valued Hardy spaces \(H^{(r,s)}_p(\mathcal{A};\ell_\infty)\) and \(H_p(\mathcal{A};\ell_1)\) in a way analogous to that of the noncommutative vector-valued \(L_p\) spaces of \textit{M. Junge} [J. Reine Angew. Math. 549, 149--190 (2002; Zbl 1004.46043)], with \(H_p(\mathcal{A};\ell_\infty)\) being the symmetric version \(H^{(2p,2p)}_p(\mathcal{A};\ell_\infty)\). Basic properties of the spaces together with proper duality results are established, and contractivity of the extensions of the conditional expectation \(\phi\) to the spaces is proved. Complex interpolation functors are shown to yield the intermediate vector-valued Hardy spaces: \(H_p=(H_{p_0},H_{p_1})_\theta\) for \(1/p=(1-\theta)/p_0+\theta/p_1\), for the symmetric versions of \(\ell_\infty\)-valued spaces as well as for the \(\ell_1\)-valued ones.
reviewed by: Stanisław Goldstein
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 46L51
zbMATH DE Number: 6510850
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: subdiagonal algebras
zbMATH Keywords: vector-valued noncommutative Hardy spaces
zbMATH Keywords: Riesz factorization
zbMATH Keywords: complex interpolation
MaRDI profile type: Publication
full work available at URL: s11117-015-0332-x
OpenAlex ID: W2460667639
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DOI: 10.1007/S11117-015-0332-X
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description: scientific article; zbMATH DE number 6510850