Hardy Algebras, W*-Correspondences and Interpolation Theory

Abstract

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 $\sigma$ of $M$ on a Hilbert space $H$ there is a natural correspondence $E^{\sigma }$ over the commutant $\sigma (M{)}^{\prime }$, called the $\sigma$-dual of $E$, and that $H^{\infty }(E)$ can be realized in terms of ($B(H)$-valued) functions on the open unit ball $\mathbb{D}((E^{\sigma }{)}^{\ast })$ in the space of adjoints of elements in $E^{\sigma }$. 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 $E^{\sigma }$ 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 $H^{\infty }(E^{\sigma })$. For these induced representations a double commutant theorem is proved.

74 pages, Latex file

References in corpus (1)

• Free Semigroupoid Algebras