Hardy Algebras, W*-Correspondences and Interpolation Theory
| dc.creator | Muhly, Paul S. | |
| dc.creator | Solel, Baruch | |
| dc.date | 2003-08-10 | |
| dc.date.accessioned | 2026-07-07T05:00:17Z | |
| dc.date.available | 2026-07-07T05:00:17Z | |
| dc.description | Given a von Neumann algebra $M$ and a $W^{\ast}$-correspondence $E$ over $M$, we construct an algebra $H^{\infty}(E)$ that we call the Hardy algebra of $E$. When $M=\mathbb{C}=E$, then $H^{\infty}(E)$ is the classical Hardy space $H^{\infty}(\mathbb{T})$ of bounded analytic functions on the unit disc. We show that given any faithful normal representation $σ$ of $M$ on a Hilbert space $H$ there is a natural correspondence $E^σ$ over the commutant $σ(M)^{\prime}$, called the $σ$-dual of $E$, and that $H^{\infty}(E)$ can be realized in terms of ($B(H)$-valued) functions on the open unit ball $\mathbb{D}((E^σ)^{\ast})$ in the space of adjoints of elements in $E^σ$. We prove analogues of the Nevanlinna-Pick theorem in this setting and discover other aspects of the value ``distribution theory'' for elements in $H^{\infty}(E)$. We also analyze the ``boundary behavior'' of elements in $H^{\infty}(E)$ and obtain generalizations of the Sz.-Nagy--Foia\c {s} functional calculus. The correspondence $E^σ$ has a dual that is naturally isomorphic to $E$ and the commutants of certain, so-called induced representations of $H^{\infty}(E)$ can be viewed as induced representations of $H^{\infty}(E^σ)$. For these induced representations a double commutant theorem is proved. | |
| dc.description | 74 pages, Latex file | |
| dc.identifier | https://arxiv.org/abs/math/0308088 | |
| dc.identifier | http://arxiv.org/abs/math/0308088 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68284 | |
| dc.subject | Operator Algebras | |
| dc.subject | Functional Analysis | |
| dc.subject | 46L08, 46L89, 46L52, 47L30, 47L55, 47L65, 47L75 | |
| dc.title | Hardy Algebras, W*-Correspondences and Interpolation Theory | |
| dc.type | text |