Hardy Algebra Research Articles

Absolute Continuity, Interpolation and the Lyapunov Order

We extend our Nevanlinna–Pick theorem for Hardy algebras and their representations to cover interpolation at the absolutely continuous points of the boundaries of their discs of representations. The Lyapunov order plays a crucial role in our analysis.

Progress in noncommutative function theory

In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. We will show that these spaces of representations can be parameterized as unit balls of certain $W^{*}$-correspondences and the functions can be viewed as Schur class operator functions on these balls. We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory.

Representations of Hardy Algebras: Absolute Continuity, Intertwiners, and Superharmonic Operators

Suppose \({\mathcal{T}_{+}(E)}\) is the tensor algebra of a W*-correspondence E and H ∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of \({\mathcal{T}_{+}(E)}\) on a Hilbert space to ultra-weakly continuous completely contractive representations of H ∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra.

Ergodic actions of convergent Fuchsian groups on quotients of the noncommutative Hardy algebras

We establish that particular quotients of the noncommutative Hardy algebras carry ergodic actions of convergent discrete subgroups of the group SU(n, 1) of automorphisms of the unit ball in ℂ n . To do so, we provide a mean to compute the spectra of quotients of noncommutative Hardy algebra and characterize their automorphisms in terms of biholomorphic maps of the unit ball in ℂ n .

Coherence of the real symmetric Hardy algebra

Let D denote the open unit disk in C centered at 0. Let H∞ R denote the set of all bounded and holomorphic functions defined in D that also satisfy f (z) = f (z) for all z ∈ D. It is shown that H∞ R is a coherent ring.

Free Biholomorphic Classification of Noncommutative Domains

In this paper, we develop a theory of free holomorphic functions on noncommutative Reinhardt domains , generated by positive regular free holomorphic functions f in n noncommuting variables and by positive integers m ≥ 1, where is the algebra of all bounded linear operators on a Hilbert space . Noncommutative Berezin transforms are used to study Hardy algebras H ∞ (D f, rad m ) and domain algebras A(D f, rad m ) associated with and compositions of free holomorphic functions. We obtain noncommutative Cartan type results for formal power series, in several noncommuting indeterminates, which leave invariant the nilpotent parts of the corresponding domains. As a consequence, we characterize the set of all free biholomorphic functions with F(0) = 0. We show that the free biholomorphic classification of the domains is the same as the classification, up to unital completely isometric isomorphisms having completely contractive hereditary extension, of the corresponding noncommutative domain algebras A(D f, rad m ). In particular, we prove that Ψ : A(D f, rad 1 ) → A(D g, rad 1 ) is a unital completely isometric isomorphism if and only if there is a free biholomorphic map φ ∈ Bih(D g 1 , D f 1 ) such that This implies that the noncommutative domains and are free biholomorphic equivalent if and only if the domain algebras A(D f, rad 1 ) and A(D g, rad 1 ) are completely isometrically isomorphic. Using the interaction between the theory of functions in several complex variables and our noncommutative theory, we provide several results concerning the free biholomorphic classification of the noncommutative domains and the classification, up to completely isometric isomorphisms, of the associated noncommutative domain (resp. Hardy) algebras. In particular, we characterize the unit ball of among the noncommutative domains , up to free biholomorphisms. We also obtain characterizations for the unitarily implemented isomorphisms of noncommutative Hardy (resp. domain) algebras in terms of free biholomorphic functions between the corresponding noncommutative domains.

Free holomorphic automorphisms of the unit ball of B(ℋ) n

In this paper we continue the study of free holomorphic functions on the noncommutative ball where B (ℋ) is the algebra of all bounded linear operators on a Hilbert space ℋ, and n = 1, 2, . . . or n = ∞. These are noncommutative multivariable analogues of the analytic functions on the open unit disc 𝔻 ≔ { z ∈ ℂ : | z | < 1}. The theory of characteristic functions for row contractions (elements in ) is used to determine the group of all free holomorphic automorphisms of [ B (ℋ) n ] 1 . It is shown that , the Moebius group of the open unit ball 𝔹 n ≔ { λ ∈ ℂ n : ∥ λ ∥ 2 < 1}. We show that the noncommutative Poisson transform commutes with the action of the automorphism group . This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra C *( S 1 , . . . , S n ), which leave invariant the noncommutative disc algebra . This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals new connections with noncommutative multivariable operator theory, especially, the theory of characteristic functions for row contractions and the noncommutative Poisson transforms. We show that the unitarily implemented automorphisms of the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra , respectively, are determined by the free holomorphic automorphisms of [ B (ℋ) n ] 1 , via the noncommutative Poisson transform. Moreover, we prove that . We also prove that any completely isometric automorphism of the noncommutative disc algebra has the form where . We deduce a similar result for the noncommutative Hardy algebra , which, due to Davidson-Pitts results on the automorphisms of , implies that is isomorphic to the group of contractive automorphisms of . We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group . This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under . Finally, in the last section, we prove a maximum principle for free holomorphic functions on the noncommutative ball [ B (ℋ) n ] 1 and provide some extensions of the classical Schwarz lemma to our noncommutative setting.

Sufficient conditions for the projective freeness of Banach algebras

Let R be a unital semi-simple commutative complex Banach algebra, and let M ( R ) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M ( R ) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H ∞ ( X ) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.

On the Krull Dimension of Rings of Transfer Functions

In this article, we prove that the Krull dimension of several commonly used classes of transfer functions of infinite dimensional linear control systems is infinite. On the other hand, we also show that the weak Krull dimension of the Hardy algebra \(H^{\infty}(\mathbb{D})\) , the disk algebra \(A(\mathbb{D})\) and the Wiener algebra \(W_{+}(\mathbb{D})\) is equal to 1.

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras H∞(E), which we call Hardy algebras, and which are noncommutative generalizations of classical H∞, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which “reproduces” the values, on $${\mathbb{D}}((E^{\sigma})^*)$$ , of the “functions” coming from H∞(E). We present results that are natural generalizations of the Poisson integral formula. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the “model space” for the canonical model that can be attached to a point in the disc $${\mathbb{D}}((E^{\sigma})^*)$$ . We also connect our Poisson kernel to various “point evaluations” and to the idea of curvature.

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let H(D; (E)) denote the algebra of all functions f:D (E) that are holomorphic. If is a subalgebra of H(D; (E)) , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra H (D; (E)), the disk algebra A(D; (E)) and the Wiener algebra W+(D; (E)) are all infinite.

Schur class operator functions and automorphisms of Hardy algebras

Let E be a W^{\ast} -correspondence over a von Neumann algebra M and let H^{\infty}(E) be the associated Hardy algebra. If \sigma is a faithful normal representation of M on a Hilbert space H , then one may form the dual correspondence E^{\sigma} and represent elements in H^{\infty}(E) as B(H) -valued functions on the unit ball \mathbb{D}(E^{\sigma})^{\ast} . The functions that one obtains are called Schur class functions and may be characterized in terms of certain Pick-like kernels. We study these functions and relate them to system matrices and transfer functions from systems theory. We use the information gained to describe the automorphism group of H^{\infty}(E) in terms of special Möbius transformations on \mathbb{D}(E^{\sigma}) . Particular attention is devoted to the H^{\infty} -algebras that are associated to graphs.

Noncommutative Berezin transforms and multivariable operator model theory

In this paper, we initiate the study of a class D p m ( H ) of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space H , where m ⩾ 2 , n ⩾ 2 , and p is a positive regular polynomial in n noncommutative indeterminates. These domains are defined by certain positivity conditions on p, i.e., D p m ( H ) : = { X : = ( X 1 , … , X n ) : ( 1 − p ) k ( X , X ∗ ) ⩾ 0 for 1 ⩽ k ⩽ m } . Each such a domain has a universal model ( W 1 , … , W n ) of weighted shifts acting on the full Fock space F 2 ( H n ) with n generators. The study of D p m ( H ) is close related to the study of the weighted shifts W 1 , … , W n , their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra A n ( D p m ) , the Hardy algebra F n ∞ ( D p m ) , and the C ∗ -algebra C ∗ ( W 1 , … , W n ) . A good part of this paper deals with these issues. The main tool, which we introduce here, is a noncommutative Berezin type transform associated with each n-tuple of operators in D p m ( H ) . The study of this transform and its boundary behavior leads to Fatou type results, functional calculi, and a model theory for n-tuples of operators in D p m ( H ) . These results extend to noncommutative varieties V p , Q m ( H ) ⊂ D p m ( H ) generated by classes Q of noncommutative polynomials. When m ⩾ 2 , n ⩾ 2 , p = Z 1 + ⋯ + Z n , and Q = 0 , the elements of the corresponding variety V p , Q m ( H ) can be seen as multivariable noncommutative analogues of Agler's m-hypercontractions. Our results apply, in particular, when Q consists of the noncommutative polynomials Z i Z j − Z j Z i , i , j = 1 , … , n . In this case, the model space is a symmetric weighted Fock space F s 2 ( D p m ) , which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in C n , and the universal model is the n-tuple ( M λ 1 , … , M λ n ) of multipliers by the coordinate functions. In this particular case, we obtain a model theory for commuting n-tuples of operators in D p m ( H ) , recovering several results already existent in the literature.

The Euler–Lagrange Theory for Schur's algorithm: Wall pairs

This paper develops a techniques of Wall pairs for the study of periodic exposed quadratic irrationalities in the unit ball of the Hardy algebra.

Canonical Models for Representations of Hardy Algebras

We develop a model theory for completely non coisometric representations of the Hardy algebra of a W*-correspondence defined over a von Neumann algebra. It follows very closely the model theory developed by Sz.-Nagy and Foias for studying single contraction operators on Hilbert space and it extends work of Popescu for row contractions.

Hardy algebras, W*-correspondences and interpolation theory

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.

Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in L2(T)

A function f in the unit ball B of the Hardy algebra H∞ on the unit disc D={z∈C : |z|<1} is a non-exposed point of B (|f|<1 a.e. on T={ζ∈C : |ζ|=1}) iff limn ∫T |fn|2 dm=0, where m is the Lebesgue measure on T and (fn)n⩾0 are the Schur functions of f. This result easily implies Rakhmanov's well-known theorem which states that limn an=0 if σ′>0 a.e. on T, (an)n⩾0 being the parameters of the orthogonal polynomials (ϕn)n⩾0 in L2(dσ). We prove that fnbn is the Schur function of the probability measure |ϕn|2 dσ, which leads to an important formula relating |ϕn|2 σ′ to fn and bn=ϕn/ϕ*n. A probability measure σ is called a Rakhmanov measure if (*)−limn |ϕn|2 dσ=dm. We show that a probability measure σ with parameters (an)n⩾0 is a Rakhmanov measure iff the an's satisfy the Máté–Nevai condition limn anan+κ=0 for every κ=1, 2, … . Next, we prove that even approximants An/Bn of the Wall continued fraction for f converge in L2(T) iff either f is an inner function or limn an=0. This implies that measures satisfying limn anan+κ=0, κ=1, 2, …, and limn |an|>0 are all singular.

Gleason-type decompositions for H^∞(B_n) and Lumer's Hardy algebra of the ball

Gleason-type decompositions for H^∞(B_n) and Lumer's Hardy algebra of the ball

Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein type-theorem

A class of commutative Banach algebras which satisfy a BochnerSchoenberg-Eberlein-type inequality is introduced. Commutative ${C^ * }$-algebras, the disk algebra and the Hardy algebra on the open disk are examples.

On some basic relations in the abstract Hardy algebra theory

On some basic relations in the abstract Hardy algebra theory