The noncommutative \(H^{(r,s)}_{p}({\mathcal A};\ell_{\infty})\) and \(H_{p}({\mathcal A};\ell_{1})\) spaces
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.
vector-valued noncommutative Hardy spaces
scientific article; zbMATH DE number 6510850
scientific article