2003/08/10 by Paul S. Muhly, Muhly, Paul S., Baruch Solel +1
Mathematics · #Advanced Operator Algebra Research #Algebraic and Geometric Analysis #Holomorphic and Operator Theory #math.FA #math.OA #msc:46L08 #msc:46L52 #msc:46L89 #msc:47L30 #msc:47L55 #msc:47L65 #msc:47L75
paper · pdf · doi:10.48550/arxiv.math/0308088
74 pages, Latex file
arxiv created 2003/08/10 · arxiv updated 2009/12/01
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=ℂ=E, then H∞(E) is the classical Hardy space H∞(\mathbbT) 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 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 \mathbbD((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--Foia\c s functional calculus. 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.