Analytic Toeplitz Algebra Research Articles

Super-algebras and noncommutative analytic Toeplitz algebras of type 1 subdiagonal algebras

Super-algebras and noncommutative analytic Toeplitz algebras of type 1 subdiagonal algebras

Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem

Given a noncommutative (nc) variety V in the nc unit ball Bd, we consider the algebra H∞(V) of bounded nc holomorphic functions on V. We investigate the problem of when two algebras H∞(V) and H∞(W) are isomorphic. We prove that these algebras are weak-⁎ continuously isomorphic if and only if there is an nc biholomorphism G:W˜→V˜ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form f↦f∘G, where G is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras H∞(Bd) studied by Davidson–Pitts and by Popescu. In particular, we find that Aut(H∞(Bd)) is a proper subgroup of Aut(B˜d).When d<∞ and the varieties are homogeneous, we remove the weak-⁎ continuity assumption, showing that two such algebras are boundedly isomorphic if and only if there is a bi-Lipschitz nc biholomorphism between the similarity envelopes of the nc varieties. We provide two proofs. In the noncommutative setting, our main tool is the noncommutative spectral radius, about which we prove several new results. In the free commutative case, we use a new free commutative Nullstellensatz that allows us to bootstrap techniques from the fully commutative case.

Subdiagonal algebras with Beurling type invariant subspaces

Let A be a maximal subdiagonal algebra in a σ-finite von Neumann algebra M. If every right invariant subspace of A in the noncommutative Hardy space H2 is of Beurling type, then we say A is of type 1. We determine generators of these algebras and consider a Riesz type factorization theorem for the noncommutative H1 space. We show that the right analytic Toeplitz algebra on the noncommutative Hardy space Hp associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive.

Erratum to: The algebraic structure of non-commutative analytic Toeplitz algebras

The algebraic structure of the non-commutative analytic Toeplitz algebra Ln is developed in the original article. Some of the results fail for the case n = ∞, and this implies that certain other results are not established in this case. In Theorem 3.2 of the original article, we showed there is continuous surjection πn,k from Repk(Ln), the space of completely contractive representations of Ln into the k × k matrices Mk , onto the closed unit ball Bn,k ofRn(Mk) by evaluation at the generators. It is further claimed that if T = [T1, . . . , Tn] ∈ Rn(Mk) with ‖T ‖ < 1, then there is a unique representation in π−1 n,k (T ). Further information is obtained for k = 1 in Theorem 3.3 of the original article. Our proof of these results is valid for n < ∞, however, for n = ∞ the uniqueness claim is incorrect. An example due to Michael Hartz (see [2, Example 2.4]) shows that π−1 ∞,1(0) is very large—it contains a copy of the βN\N. The difficulty in the proof of Theorems 3.2 and 3.3 of the original article stems from the use of the factorization A = W X used in Lemma 3.1 of the original article. In the case n = ∞, this factorization comes from Corollary 2.9. The problem is that the infinite sum in Corollary 2.9 converges in the strong topology, not the norm topology, so that when the representation is not strongly continuous (or equivalently,

The Hopf Structure of Some Dual Operator Algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury–Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let \(\mathcal{M}\) be a σ-finite von Neumann algebra and let \(\mathfrak{A} \subseteq \mathcal{M}\) be a maximal subdiagonal algebra with respect to a faithful normal conditional expectation Φ. Based on the Haagerup’s noncommutative Lp space \(L^p \left( \mathcal{M} \right)\) associated with \(\mathcal{M}\), we consider Toeplitz operators and the Hilbert transform associated with \(\mathfrak{A}\). We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space \(H^2 \left( \mathcal{M} \right)\) is just the right analytic Toeplitz algebra. Furthermore, the Hilbert transform on noncommutative \(L^p \left( \mathcal{M} \right)\) is shown to be bounded for 1 < p < ∞. As an application, we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative \(H^1 \left( \mathcal{M} \right)\) as a concrete space of operators.

Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators L e 1 , … , L e n , L f 1 , … , L f m subject to the unitary commutation relations of the form L e i L f j = ∑ k , l u i , j , k , l L f l L e k where u = ( u i , j , k , l ) is an n m × n m unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix u.

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 | &lt; 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 &lt; 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.

Unitary invariants in multivariable operator theory

This paper concerns unitary invariants for n-tuples T:=(Tl,...,Tn) of (not necessarily commuting) bounded linear operators on Hilbert spaces. The author introduces a notion of joint numerical radius and works out its basic properties. Multivariable versions of Berger's dilation theorem, Berger - Kato - Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained. The author studies the joint (spatial) numerical range of T in connection with several unitary invariants for n-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius. He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra Fn.

Multivariable Bohr inequalities

Operator-valued multivariable Bohr type inequalities are obtained for: (i) a class of noncommutative holomorphic functions on the open unit ball of , generalizing the analytic functions on the open unit disc; (ii) the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra ; (iii) a class of noncommutative selfadjoint harmonic functions on the open unit ball of , generalizing the real-valued harmonic functions on the open unit disc; (iv) the Cuntz-Toeplitz algebra , the reduced (resp. full) group -algebra (resp. ) of the free group with generators; (v) a class of analytic functions on the open unit ball of . The classical Bohr inequality is shown to be a consequence of Fejer's inequality for the coefficients of positive trigonometric polynomials and Haager- up-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr's inequality.

Multivariable Nehari problem and interpolation

In this paper we obtain a multivariable Nehari type theorem for Hankel operators, extending the classical result as well as Treil–Volberg generalization. The result is based on a new generalization of the noncommutative commutant lifting theorem which extends to several variables the commutant lifting result obtained by Treil and Volberg, and recently generalized by Biswas, Foiaş, and Frazho. An extension of I.S. Iokhvidov–Ky Fan theorem is obtained and used to prove a multivariable Adamian–Arov–Krein type theorem for generalized Hankel operators. As consequences, we obtain Carathéodory and Nevanlinna–Pick type interpolation results for “meromorphic” operators on Fock spaces (resp. operator-valued meromorphic functions on the unit ball of C n ). The Nehari type results are used to obtain descriptions of Hankel operators acting on Fock spaces or weighted Fock spaces (multivariable Dirichlet type spaces) and to solve new “weighted” interpolation problems (e.g. Sarason) for noncommutative analytic Toeplitz algebras, which have consequences to the operator-valued analytic interpolation in the unit ball of C n . Moreover, using central intertwining liftings, we obtain explicit formulas for certain solutions of the weighted interpolation problems. As another application of the generalized noncommutative commutant lifting theorem, we derive an abstract interpolation problem for multipliers. This is used to solve a left tangential Nevanlinna–Pick type interpolation problem with operatorial argument for noncommutative analytic Toeplitz algebras.

Central Intertwining Lifting, Suboptimization, and Interpolation in Several Variables

In this paper we obtain an explicit central intertwining lifting for row contractions. This is used to prove Kaftal–Larson–Weiss and Foiaş–Frazho H∞−H2 suboptimization type results for the noncommutative (resp. commutative) analytic Toeplitz algebra F∞n (resp. W∞n). The algebra F∞n (resp. W∞n) can be viewed as a multivariable noncommutative (resp. commutative) analogue of the Hardy space H∞. Similar results are provided for F∞n⊗B(K, K′) and W∞n⊗B(K, K′), where B(K, K′) is the set of all bounded linear operators acting on Hilbert spaces. New extensions of the Sarason, Carathéodory, and Nevanlinna–Pick type interpolation results are obtained for F∞n⊗B(K, K′) and some consequences to the operator-valued analytic interpolation in the unit ball of Cn are considered.

!COMMUTANT LIFTING, TENSOR ALGEBRAS, AND FUNCTIONAL CALCULUS

AbstractA non-commutative multivariable analogue of Parrott’s generalization of the Sz.-Nagy–Foia\c{s} commutant lifting theorem is obtained. This yields Tomita-type commutant results and interpolation theorems (e.g. Sarason, Nevanlinna–Pick, Carathéodory) for $F_n^\infty\,\bar{\otimes}\,\M$, the weakly-closed algebra generated by the spatial tensor product of the non-commutative analytic Toeplitz algebra $F_n^\infty$ and an arbitrary von Neumann algebra $\M$. In particular, we obtain interpolation theorems for bounded analytic functions from the open unit ball of $\mathbb{C}^n$ into a von Neumann algebra.A variant of the non-commutative Poisson transform is used to extend the von Neumann inequality to tensor algebras, and to provide a generalization of the functional calculus for contractive sequences of operators on Hilbert spaces. Commutative versions of these results are also considered.AMS 2000 Mathematics subject classification: Primary 47L25; 47A57; 47A60. Secondary 30E05

Invariant Subspaces and Hyper-Reflexivity for Free Semigroup Algebras

A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup algebra generated by the left regular representation of the free semigroup on n generators. This algebra is the appropriate non-commutative n-dimensional analogue of the analytic Toeplitz algebra. We develop a detailed picture of the invariant subspace structure analogous to Beurling's theorem and show that this algebra is hyper-reflexive with distance constant at most 51. The free semigroup algebras, known as atomic, for which the range projections of words in the generators lie in an atomic masa are completely classified. This provides a complete classification for a large class of representations of the Cuntz C*-algebras On. This allows us to describe completely the invariant subspace structure of these algebras, and thereby show that these algebras are all hyper-reflexive. 1991 Mathematics Subject Classification: 47D25.

Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras

The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup onn generators. We obtain a distance formula to an arbitrary WOT-closed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna-Pick type interpolation theorems

The algebraic structure of non-commutative analytic Toeplitz algebras

The non-commutative analytic Toeplitz algebra is the wot– closed algebra generated by the left regular representation of the free semigroup on n generators. We develop a detailed picture of the algebraic structure of this algebra. In particular, we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n-ball. The k-dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k × kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of H∞ over the unit disk. In [6, 17, 18, 20], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in n non-commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators. The papers cited obtain a compelling analogue of Beurling’s theorem and inner–outer factorization. In this paper, we add further evidence. The main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of Cn. The kernel is the subgroup of quasi-inner automorphisms, which are trivial modulo the wot-closed commutator ideal. Additional evidence of analytic properties comes from the structure of k-dimensional (completely contractive) representations, which have a structure very similar to the fibration of the maximal ideal space of H∞ over the unit disk. An important tool in our analysis is a detailed structure theory for wot-closed right ideals. Curiously, left ideals remain more obscure. The non-commutative analytic Toeplitz algebra Ln is determined by the left regular representation of the free semigroup Fn on n generators z1, . . . , zn which acts on `2(Fn) by λ(w)ξv = ξwv for v, w in Fn. In particular, the algebra Ln is the unital, wot-closed algebra generated by the isometries Li = λ(zi) for 1 ≤ i ≤ n. This algebra and its norm-closed version (the noncommutative disk algebra) were introduced by Popescu [19] in an abstract sense in connection with a non-commutative von Neumann inequality and 1991 Mathematics Subject Classification. 47D25. March 9, 1997; October 9, 1997 final draft. First author partially supported by an NSERC grant and a Killam Research Fellowship. Second author partially supported by an NSF grant.

Analytic Toeplitz algebras and intertwining operators

Analytic Toeplitz algebras and intertwining operators