Hardy algebras, W*-correspondences and interpolation theory

By P. Muhly, B. Solel. 2003

112 citations recorded by the source corpus.

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= =E, H∞(E) is the classical Hardy space H∞ of bounded analytic functions on the unit disc. When M= and E= H∞(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 ((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–Foiaş 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e63ad54edf283afc1cc07ccf7d00aeb2a5581682

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Citations in the corpus

Noncommutative reproducing kernel Hilbert spaces (2016, background): “nd the BBLS-completely positive kernels appear prominently in the characterization of the generalized Schur classes and associated generalized Nevanlinna-Pick interpolation theory of Muhly-Solel (see [34, 35, 36]). NC REPRODUCING KERNEL HILBERT SPACES 49 3.4. Smoothness properties. We nowexplore the”
Free holomorphic functions on the unit ball of B(H)n (2006, unknown): “onnection with a noncommutative von Neumann inequality for row contractions. This class was extended to more general settings by Ball-Groenewald-Malakorn (see [5], [6]), and by Muhly-Solel (see [26], [27], [28]). The Muhly-Solel paper [28] gives an intrinsic characterization for the Schur class S ba”
Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “mials 11 of norm one. However, it was shown that this is the same as the closure of alg(L) in the weak-operator topology. This algebra is also a particular instance of Muhly and Solel’s Hardy algebra [50], and would be denoted in their context as H1(E). The algebra of bounded holomorphic functions o”
Operator Theory on Noncommutative Domains (2007, unknown): “We should also remark that, in recent years, many results concerning the theory of row contractions were extended by Muhly and Solel ([54], [ 55 ], [56], [57]) to representations of tensor algebras over C ∗ -correspondences and Hardy algebras.”
Schur class operator functions and automorphisms of Hardy algebras (2006, background): “O A ] 1 3 Ju n 20 07 S hur Class Operator Fun tions and”
Free holomorphic functions and interpolation (2007, methodology): “In related areas of research, we remark the work of Helton, McCullough, Putinar, and Vinnikov, on symmetric noncommutative polynomials ([13], [14], [15], [16]), and the work of Muhly and Solel on representations of tensor algebras over C ∗ correspondences (see [19], [ 20 ]).”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “er; this point is explored in more detail in [24]. The recent paper of Norton [67] explores similar connections between the interpolation theory of Constantinescu-Johnson [32] and that of Muhly-Solel [62]. Acknowledgement: The research of the ﬁrst and third authors was partially supported by the US-”
Free holomorphic automorphisms of the unit ball of B(ℋ) n (2008, background): “... noncommutative Hardy algebra H ∞ (B(H) n) which, due to Davidson-Pitts results on the automorphisms of F ∞ n (see [9]), implies that Aut(B(H) n ) is isomorphic to the group of contractive automorphisms of H ∞ (B(H) n ). We remark here that the conformal automorphisms of Bn also occur in the work”
Isomorphisms of algebras associated with directed graphs (2003, background): “(Our result can be thought as an extension of Solel’s result to Hardy algebras [ 20 ] associated with finite graphs.) In particular, we show that spatially isomorphic free semigroupoid algebras are unitarily isomorphic.”