Hardy algebras, W*-correspondences and interpolation theory

Abstract

Given a von Neumann algebra M and a W*-correspondence E over M, we construct an algebra H∞(E) that we call the Hardy algebra of E. When M= Open image in new window =E, H∞(E) is the classical Hardy space H∞Open image in new window of bounded analytic functions on the unit disc. When M= Open image in new window and E= Open image in new windowH∞(E) is the free semigroup algebra studied by Popescu, Davidson and Pitts and many others. We show that given any faithful normal representation σ of M on a Hilbert space H there is a natural correspondence Eσ over the commutant σ(M)′, called the σ-dual of E, and that H∞(E) can be realized in terms of (B(H)-valued) functions on the open unit ball Open image in new window((Eσ)*) in the space of adjoints of elements in Eσ. We prove analogues of the Nevanlinna-Pick theorem in this setting and discover other aspects of the value ‘‘distribution theory’’ for elements in H∞(E). We also analyze the ‘‘boundary behavior’’ of elements in H∞(E) and obtain generalizations of the Sz.-Nagy–Foias functional calculus and the functional calculus of Popescu for c.n.c. row contractions. The correspondence Eσ has a dual that is naturally isomorphic to E and the commutants of certain, so-called induced representations of H∞(E) can be viewed as induced representations of H∞(Eσ). For these induced representations a double commutant theorem is proved.