Hardy Algebra Research Articles

C⁎-extreme maps and nests

The generalized state space SH(A) of all unital completely positive (UCP) maps on a unital C⁎-algebra A taking values in the algebra B(H) of all bounded operators on a Hilbert space H, is a C⁎-convex set. In this paper, we establish a connection between C⁎-extreme points of SH(A) and a factorization property of certain algebras associated to the UCP maps. In particular, the factorization property of some nest algebras is used to give a complete characterization of those C⁎-extreme maps which are direct sums of pure UCP maps. This significantly extends a result of Farenick and Zhou (1998) [14] from finite to infinite dimensional Hilbert spaces. Also it is shown that normal C⁎-extreme maps on type I factors are direct sums of normal pure UCP maps if and only if an associated algebra is reflexive. Further, a Krein-Milman type theorem is established for C⁎-convexity of the set SH(A) equipped with bounded weak topology, whenever A is a separable C⁎-algebra or it is a type I factor. As an application, we provide a new proof of a classical factorization result on operator-valued Hardy algebras.

Bergman spaces over noncommutative domains and commutant lifting

The goal of the present paper is to provide analogues of Sarason interpolation theorem in the Hardy algebra H∞(D) and Sz.Nagy-Foiaş commutant lifting theorem for contractions on Hilbert spaces in the setting of noncommutative Hardy spaces associated with noncommutative regular domains and varieties. This is accompanied by the study of multi-analytic operators with respect to the universal models associated with the regular domains (resp. varieties) and the study of multipliers of noncommutative Bergman spaces.As applications, we obtain Toeplitz-corona theorems for multi-analytic operators, commutant lifting in several variables where the liftings are in certain Schur classes, factorization of multi-analytic operators, and Nevanlinna-Pick interpolation results for multipliers of Bergman spaces over Reinhardt domains in Cn. In addition, we obtain Andô type dilations and inequalities on noncommutative bi-domains and varieties.

On automorphism groups of Hardy algebras

Let $E$ be a $W^{*}$-correspondence and let $H^{\infty}(E)$ be the associated Hardy algebra. The unit disc of intertwiners $\mathbb{D}((E^{\sigma})^{*})$ plays a central role in the study of $H^{\infty}(E)$. We show a number of results related to the automorphism groups of both $H^{\infty}(E)$ and $\mathbb{D}((E^{\sigma})^{*})$. We find a matrix representation for these groups and describe several features of their algebraic structure. Furthermore, we show an application of $Aut(\mathbb{D}({(E^{\sigma}})^*))$ to the study of Morita equivalence of $W^{*}$-correspondences.

On spaces of vector-valued functions modeled on the Hardy and Bergman algebras

The strong $$N^p(X), \mathcal {N}^p(X)$$ and weak $$wN^p(X), w\mathcal {N}^p(X)$$ spaces of analytic functions with values in a Banach space X, modeled on the large Hardy $$N^p$$ and Bergman $$\mathcal {N}^p$$ algebras respectively, are studied. The Frechet envelopes and the continuous linear functionals on these spaces are described. We show that, although that strong and weak spaces are different, they have the same locally convex structure (the Frechet envelopes are equal).

Bohr inequalities for free holomorphic functions on polyballs

Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball Bn, n=(n1,…,nk)∈Nk, which is a noncommutative analogue of the scalar polyball (Cn1)1×⋯×(Cnk)1. The Bohr radius Kmh(Bn) (resp. Kh(Bn)) associated with the multi-homogeneous (resp. homogeneous) power series expansions of the free holomorphic functions are the main objects of study in this paper. We extend a theorem of Bombieri and Bourgain for the disc D:={z∈C:|z|<1} to the polyball, and obtain the estimations 13k<Kmh(Bn)<2log⁡kk if k>1, extending Boas-Khavinson result for the scalar polydisc Dk.With respect to the homogeneous power series expansion, we prove that Kh(Bn)=1/3, extending the classical result, and obtain analogues of Carathéodory, Fejér, and Egerváry-Szász inequalities for free holomorphic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.When specialized to the regular polydisc Dk (which corresponds to the case n1=⋯=nk=1), we obtain new results concerning Bohr, Landau, Fejér, and Harnack inequalities for operator-valued holomorphic functions and k-pluriharmonic functions on the scalar polydisc Dk. The results of the paper can be used to obtain Bohr type inequalities for the noncommutative ball algebra An, the Hardy algebra Fn∞, and the C⁎-algebra C⁎(S), generated by the universal model S of the polyball Bn.

Morita Equivalence of $$W^{*}$$ W ∗ -Correspondences and Their Hardy Algebras

Muhly and Solel developed a notion of Morita equivalence for $C^{*}$- correspondences, which they used to show that if two $C^{*}$-correspondences $E$ and $F$ are Morita equivalent then their tensor algebras $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$ are (strongly) Morita equivalent operator algebras. We give the weak$^{*}$ version of this result by considering (weak) Morita equivalence of $W^{*}$-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of $W^{*}$-correspondences $E$ and $F$ implies weak Morita equivalence of their Hardy algebras $H^{\infty}(E)$ and $H^{\infty}(F)$. We give special attention to $W^{*}$-graph correspondences and show a number of results related to their Morita equivalence.

Andô dilations and inequalities on non-commutative domains

We obtain intertwining dilation theorems for non-commutative regular domains 𝒟f and non-commutative varieties 𝒱J in B(𝓗)n, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain 𝒟f (respectively, variety 𝒱J ) as well as a Schur-type representation for the unit ball of the Hardy algebra associated with the variety 𝒱J. We provide Andô-type dilations and inequalities for bi-domains 𝒟f ×c 𝒟g consisting of all pairs (X,Y ) of tuples X := (X1,…, Xn1) ∊ 𝒟f and Y := (Y1,…, Yn2) ∊ 𝒟g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n1 = n2 = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties 𝒱J1×c 𝒱J2, where 𝒱J1 and 𝒱J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n1 = n2 = 1 are also discussed.

Interpolation and Commutant Lifting with Weights

Our two principle goals are generalizations of the commutant lifting theorem and the Nevanlinna-Pick interpolation theorem to the context of Hardy algebras built from \(W^*\)-correspondences endowed with a sequence of weights. These theorems generalize theorems of Muhly and Solel from 1998 and 2004, respectively, which were proved in settings without weights. Of special interest is the fact that commutant lifting in our setting is a consequence of Parrott’s Lemma; it is inspired by work of Arias.

Reflexivity of non-commutative Hardy algebras

Let H∞(E) be a non-commutative Hardy algebra associated with a W⁎-correspondence E. These algebras were introduced in 2004 by Muhly and Solel, and generalize the classical Hardy algebra of the unit disc H∞(D). As a special case one obtains also the algebra Fd∞ of Popescu, which is H∞(Cd) in our setting.In this paper we view the algebra H∞(E) as acting on a Hilbert space via an induced representation. We write it ρπ(H∞(E)) and we study the reflexivity of ρπ(H∞(E)). This question was studied by Arias and Popescu in the context of the algebra Fd∞, and by other authors in several other special cases. As it will be clear from our work, the extension to the case of a general W⁎-correspondence E over a general W⁎-algebra M requires new techniques and approach.We obtain some partial results in the general case and we turn to the case of a correspondence over a factor. Under some additional assumptions on the representation π:M→B(H) we show that ρπ(H∞(E)) is reflexive. Then we apply these results to analytic crossed products ρπ(H∞(Mα)) and obtain their reflexivity for any automorphism α∈Aut(M) whenever M is a factor. Finally, we show also the reflexivity of the compression of the Hardy algebra to a suitable coinvariant subspace M, which may be thought of as a generalized symmetric Fock space.

Holomorphic automorphisms of noncommutative polyballs

In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of automorphisms of the Cuntz-Toeplitz algebra C*(S) which leaves invariant the noncommutative polyball algebra A_n; the group of unitarily implemented automorphisms of the polyball algebra A_n and the noncommutative Hardy algebra F_n^\infty, respectively. We prove that the free holomorphic automorphism group Aut(B_n) is a sigma-compact, locally compact topological group with respect to the topology induced by an appropriate metric. Finally, we obtain a concrete unitary projective representation of the topological group Aut(B_n)in terms of noncommutative Berezin kernels associated with regular polyballs.

Comparing Two Generalized Noncommutative Nevanlinna–Pick Theorems

We explore the relationship between two noncommutative generalizations of the classical Nevanlinna–Pick theorem: one proved by Constantinescu and Johnson in 2003 and the other proved by Muhly and Solel in 2004. To make the comparison, we generalize Constantinescu and Johnson’s theorem to the context of \(W^*\)-correspondences and Hardy algebras. After formulating the so-called displacement equation in this context, we are able to follow Constantinescu and Johnson’s line of reasoning in our proof. Though our result is similar in appearance to Muhly and Solel’s, closer inspection reveals differences. Nevertheless, when the given data lie in the center of the dual correspondence, the theorems are essentially the same.

Generalized Inner-Outer Factorizations in Non Commutative Hardy Algebras

Let \({H^{\infty}(E)}\) be a non commutative Hardy algebra, associated with a \({W^*}\)-correspondence E. In this paper we construct factorizations of inner-outer type of the elements of \({H^{\infty}(E)}\) represented via the induced representation, and of the elements of its commutant. These factorizations generalize the classical inner-outer factorization of elements of \({H^\infty(\mathbb{D})}\). Our results also generalize some results that were obtained by several authors in some special cases.

Berezin transforms on noncommutative varieties in polydomains

Let Q be a set of polynomials in noncommutative indeterminates Zi,j, i∈{1,…,k}, j∈{1,…,ni}. In this paper, we study noncommutative varietiesVQ(H):={X={Xi,j}∈D(H):g(X)=0 for all g∈Q}, where D(H) is a regular polydomain in B(H)n1+⋯+nk and B(H) is the algebra of bounded linear operators on a Hilbert space H. Under natural conditions on Q, we show that there is a universal modelS={Si,j} such that g(S)=0, g∈Q, acting on a subspace of a tensor product of full Fock spaces. We characterize the variety VQ(H) and its pure part in terms of the universal model and a class of completely positive linear maps. We obtain a characterization of those elements in VQ(H) which admit characteristic functions and prove that the characteristic function is a complete unitary invariant for the class of completely non-coisometric elements. We study the universal model S, its joint invariant subspaces and the representations of the universal operator algebras it generates: the variety algebraA(VQ), the Hardy algebra F∞(VQ), and the C⁎-algebra C⁎(VQ). Using noncommutative Berezin transforms associated with each variety, we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains. This includes various commutative cases which are closely connected to the theory of holomorphic functions in several complex variables and algebraic geometry.

Free biholomorphic functions and operator model theory, II

In a companion to this paper, we introduced the class of n-tuples f=(f1,…,fn) of formal power series in noncommutative indeterminates Z1,…,Zn with the model property and developed an operator model theory for pure n-tuples of operators in noncommutative domains Bf(H)⊂B(H)n, where the associated universal model is an n-tuple (MZ1,…,MZn) of multiplication operators on a Hilbert space H2(f) of formal power series. In the present paper, we continue this work by considering the completely non-coisometric (c.n.c) case. In the second part of the paper, several results concerning the noncommutative multivariable operator theory on the unit ball [B(H)n]1− are extended to noncommutative varieties Vf,J(H)⊆Bf(H) defined byVf,J(H):={(T1,…,Tn)∈Bf(H):ψ(T1,…,Tn)=0 for any ψ∈J} for an appropriate evaluation ψ(T1,…,Tn) and associated with WOT-closed two-sided ideals J of the Hardy algebra H∞(Bf). In particular, we obtain commutative versions for all the results.

Tensorial Function Theory: From Berezin Transforms to Taylor’s Taylor Series and Back

Let H ∞(E) be the Hardy algebra of a W*-correspondence E over a W*-algebra M. Then the ultraweakly continuous completely contractive representations of H ∞(E) are parametrized by certain sets $${{\mathcal{AC}}(\sigma)}$$ indexed by NRep(M)—the normal *-representations σ of M. Each set $${{\mathcal{AC}}(\sigma)}$$ has analytic structure, and each element $${F \in H^{\infty}(E)}$$ gives rise to an analytic operator-valued function $${\widehat{F}_{\sigma}}$$ on $${{\mathcal{AC}}(\sigma)}$$ that we call the σ-Berezin transform of F. The sets $${\{{\mathcal{AC}}(\sigma)\}_{\sigma\in\Sigma}}$$ and the family of functions $${\{\widehat{F}_{\sigma}\}_{\sigma\in\Sigma}}$$ exhibit “matricial structure” that was introduced by Joeseph Taylor in his work on noncommutative spectral theory in the early 1970s. Such structure has been exploited more recently in other areas of free analysis and in the theory of linear matrix inequalities. Our objective here is to determine the extent to which the matricial structure characterizes the Berezin transforms.

Reformulation of the extension of the [formula omitted]-metric for [formula omitted

The classical ν-metric introduced by Vinnicombe in robust control theory for rational plants was extended to classes of nonrational transfer functions in Ball (2012) [1]. In Sasane (2012) [11], an extension of the classical ν-metric was given when the underlying ring of stable transfer functions is the Hardy algebra, H∞. However, this particular extension to H∞ did not directly fit in the abstract framework given in Ball (2012) [1]. In this paper we show that the case of H∞ also fits into the general abstract framework in Ball (2012) [1] and that the ν-metric defined in this setting is identical to the extension of the ν-metric defined in Sasane (2012) [11]. This is done by introducing a particular Banach algebra, which is the inductive limit of certain C∗-algebras.

Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains $${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$$ generated by positive regular free holomorphic functions f and certain classes of n-tuples $${\varphi = (\varphi_1, \ldots, \varphi_n)}$$ of formal power series in noncommutative indeterminates Z 1, . . . , Z n . Noncommutative Poisson transforms are employed to show that each abstract domain $${\mathbb{D}_f^\varphi}$$ has a universal model consisting of multiplication operators (M Z1, . . . , M Z n ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M Z1, . . . , M Z n and show that all pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ are compressions of $${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$$ to their coinvariant subspaces. We show that the eigenvectors of $${M_{Z_1}^*, \ldots, M_{Z_n}^*}$$ are precisely the noncommutative Poisson kernels $${\Gamma_\lambda}$$ associated with the elements $${\lambda}$$ of the scalar domain $${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra $${H^\infty(\mathbb{D}_f^\varphi)}$$ . We introduce the characteristic function of an n-tuple $${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$$ , present a model for pure n-tuples of operators in the noncommutative domain $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ .

Classification of noncommutative domain algebras

Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of Cn defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete classification of these algebras based upon techniques inspired by multivariate complex analysis, and more specifically the classification of domains in hermitian spaces up to biholomorphic equivalence.

Test Functions, Schur–Agler Classes and Transfer-Function Realizations: The Matrix-Valued Setting

Given a collection of test functions, one defines the associated Schur–Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur–Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with $${{\mathbb C}^{N}}$$ of the scalar-valued version.

Extension of the $\nu$-metric for stabilizable plants over $H^\infty$

An abstract $\nu$-metric was introduced in [1], with a view towards extending the classical $\nu$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. Here we give an important concrete special instance of the abstract $\nu$-metric, namely the case when the ring of stable transfer functions is the Hardy algebra $H^\infty$, by verifying that all the assumptions demanded in the abstract set-up are satisfied. This settles the open question implicit in [2].