Noncommutative Ball Research Articles

Bundles of Holomorphic Function Algebras on Subvarieties of the Noncommutative Ball

Bundles of Holomorphic Function Algebras on Subvarieties of the Noncommutative Ball

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.

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.

$$\hbox {H}^2$$ H 2 Spaces of Non-commutative Functions

We define the Hardy spaces of free noncommutative functions on the noncommutative polydisc and the noncommutative ball and study their basic properties. Our technique combines the general methods of noncommutative function theory and asymptotic formulae for integration over the unitary group. The results are the first step in developing the general theory of free noncommutative bounded symmetric domains on the one hand and in studying the asymptotic free noncommutative analogues of classical spaces of analytic functions on the other.

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.

Free holomorphic functions on the unit ball of [formula omitted], II

In this paper we continue the study of free holomorphic functions on the noncommutative ball [ B ( H ) n ] 1 : = { ( X 1 , … , X n ) ∈ B ( H ) n : ‖ X 1 X 1 ∗ + ⋯ + X n X n ∗ ‖ 1 / 2 < 1 } , where B ( H ) is the algebra of all bounded linear operators on a Hilbert space H , and n = 1 , 2 , … or n = ∞ . Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maximum principle, a Naimark type representation theorem, and a Vitali convergence theorem, for free holomorphic functions with operator-valued coefficients. We introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by Schwarz and Harnack. The Borel–Carathéodory theorem is extended to our noncommutative setting. Using a noncommutative generalization of Schwarz's lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball [ B ( H ) n ] 1 , we obtain an analogue of Julia's lemma for free holomorphic functions F : [ B ( H ) n ] 1 → [ B ( H ) m ] 1 . We also obtain Pick–Julia theorems for free holomorphic functions with operator-valued coefficients. We provide a noncommutative generalization of a classical inequality due to Lindelöf, which turns out to be sharper then the noncommutative von Neumann inequality. Finally, we introduce a pseudohyperbolic metric on [ B ( H ) n ] 1 which is invariant under the action of the free holomorphic automorphism group of [ B ( H ) n ] 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on B n , the open unit ball of C n . In this setting, we obtain a Schwarz–Pick type lemma. We also provide commutative versions of these results for operator-valued multipliers of the Drury–Arveson space.

Noncommutative ball maps

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D'Angelo on such analytic maps in C . Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.

Hyperbolic geometry on noncommutative balls

In this paper, we study the noncommutative balls \mathcal C_\rho:={(X_1,\ldots, X_n)\in B(\mathcal H)^n: \ \omega_\rho(X_1,\ldots, X_n)\leq 1},\qquad \rho\in (0,\infty], where \omega_\rho is the joint operator radius for n -tuples of bounded linear operators on a Hilbert space. In particular, \omega_1 is the operator norm, \omega_2 is the joint numerical radius, and \omega_\infty is the joint spectral radius. We introduce a Harnack type equivalence relation on \mathcal C_\rho, \rho&gt;0 , and use it to define a hyperbolic distance \delta_\rho on the Harnack parts (equivalence classes) of \mathcal C_\rho . We prove that the open ball [\mathcal C_\rho]_{&lt;1}:={(X_1,\ldots, X_n)\in B(\mathcal H)^n: \ \omega_\rho(X_1,\ldots, X_n)&lt;1},\qquad \rho&gt;0, is the Harnack part containing 0 and obtain a concrete formula for the hyperbolic distance, in terms of the reconstruction operator associated with the right creation operators on the full Fock space with n generators. Moreover, we show that the \delta_\rho -topology and the usual operator norm topology coincide on [\mathcal C_\rho]_{&lt;1} . While the open ball [\mathcal C_\rho]_{&lt;1} is not a complete metric space with respect to the operator norm topology, we prove that it is a complete metric space with respect to the hyperbolic metric \delta_\rho . In the particular case when \rho=1 and \mathcal H=\mathbb C , the hyperbolic metric \delta_\rho coincides with the Poincaré–Bergman distance on the open unit ball of \mathbb C^n . We introduce a Carathéodory type metric on [\mathcal C_\infty]_{&lt;1} , the set of all n -tuples of operators with joint spectral radius strictly less then 1, by setting d_K(A,B)=\sup_p \|\mathfrak R p(A)-\mathfrak R p(B)\|,\qquad A,B\in [\mathcal C_\infty]_{&lt;1}, where the supremum is taken over all noncommutative polynomials with matrix-valued coefficients p\in \mathbb C[X_1,\ldots, X_n]\otimes M_{m}, m\in \mathbb N , with \mathfrak R p(0)=I and \mathfrak R p(X)\geq 0 for all X\in \mathcal C_1 . We obtain a concrete formula for d_K in terms of the free pluriharmonic kernel on the noncommutative ball [\mathcal C_\infty]_{&lt;1} . We also prove that the metric d_K is complete on [\mathcal C_\infty]_{&lt;1} and its topology coincides with the operator norm topology. We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric \delta_\rho , the Carathéodory metric d_K , and the joint operator radius \omega_\rho, \rho\in (0,\infty] .

Noncommutative transforms and free pluriharmonic functions

In this paper, we study free pluriharmonic functions on noncommutative balls [B(H)n]γ, γ>0, and their boundary behavior. These functions have the formf(X1,…,Xn)=∑k=1∞∑|α|=kbαXα⁎+a0I+∑k=1∞∑|α|=kaαXα,aα,bα∈C, where the convergence of the series is in the operator norm topology for any (X1,…,Xn)∈[B(H)n]γ, and B(H) denotes the algebra of all bounded linear operators on a Hilbert space H. The main tools used in this study are certain noncommutative transforms which are introduced in the present paper and which generalize the classical transforms of Berezin, Poisson, Fantappiè, Herglotz, and Cayley. Several classical results from complex analysis have free analogues in our noncommutative multivariable setting.

Free pluriharmonic majorants and commutant lifting

In this paper we initiate the study of sub-pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball [ B ( H ) n ] 1 : = { ( X 1 , … , X n ) ∈ B ( H ) n : ‖ X 1 X 1 ∗ + ⋯ + X n X n ∗ ‖ 1 / 2 < 1 } , where B ( H ) is the algebra of all bounded linear operators on a Hilbert space H . Several classical results from complex analysis have analogues in this noncommutative multivariable setting. We present basic properties for sub-pluriharmonic curves, characterize the class of sub-pluriharmonic curves that admit free pluriharmonic majorants and find, in this case, the least free pluriharmonic majorants. We show that, for any free holomorphic function Θ on [ B ( H ) n ] 1 , the map φ : [ 0 , 1 ) → C ∗ ( R 1 , … , R n ) , φ ( r ) : = Θ ( r R 1 , … , r R n ) ∗ Θ ( r R 1 , … , r R n ) , is a sub-pluriharmonic curve in the Cuntz–Toeplitz algebra generated by the right creation operators R 1 , … , R n on the full Fock space with n generators. We prove that Θ is in the noncommutative Hardy space H ball 2 if and only if φ has a free pluriharmonic majorant. In this case, we find Herglotz–Riesz and Poisson type representations for the least pluriharmonic majorant of φ. Moreover, we obtain a characterization of the unit ball of H ball 2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of φ, when Θ is in the unit ball of H ball 2 . In the second part of this paper, we introduce a generalized noncommutative commutant lifting (GNCL) problem which extends, to our noncommutative multivariable setting, several lifting problems including the classical Sz.-Nagy–Foiaş commutant lifting problem and the extensions obtained by Treil–Volberg, Foiaş–Frazho–Kaashoek, and Biswas–Foiaş–Frazho, as well as their multivariable noncommutative versions. We solve the GNCL problem and, using the results regarding sub-pluriharmonic curves and free pluriharmonic majorants on noncommutative balls, we provide a complete description of all solutions. In particular, we obtain a concrete Schur type description of all solutions in the noncommutative commutant lifting theorem.

Noncommutative balls and mirror quantum spheres

Noncommutative analogues of n-dimensional balls are defined by repeated application of the quantum double suspension to the classical low-dimensional spaces. In the `even-dimensional' case they correspond to the Twisted Canonical Commutation Relations of Pusz and Woronowicz. Then quantum spheres are constructed as double manifolds of noncommutative balls. Both C*-algebras and polynomial algebras of the objects in question are defined and analyzed, and their relations with previously known examples are presented. Our construction generalizes that of Hajac, Matthes and Szymanski for `dimension 2', and leads to a new class of quantum spheres (already on the C*-algebra level) in all `even-dimensions'.

Hyperbolic geometry on the unit ball of $B(\mathcal{H})^{n}$ and dilation theory

In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball $[B(H)^n]_1^-$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, and its implications to noncommutative function theory. The central object is an intertwining operator $L_{B,A}$ of the minimal isometric dilations of $A, B\in [B(H)^n]_1^-$, which establishes a strong connection between noncommutative hyperbolic geometry on $[B(H)^n]_1^-$ and multivariable dilation theory. The goal of this paper is to study the operator $L_{B,A}$ and its connections to the hyperbolic metric $\delta$ on the Harnack parts of $[B(H)^n]_1^-$. We study the geometric structure of the operator $L_{B,A}$ and obtain new characterizations for the Harnack domination (resp. equivalence) in $[B(H)^n]_1^-$. We express $\|L_{B,A}\|$ in terms of the reconstruction operators $R_A$ and $R_B$, and obtain a Schwartz-Pick lemma for contractive free holomorphic functions on $[B(H)^n]_1$ with respect to the intertwining operator $L_{B,A}$. As a consequence, we deduce a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric.

Noncommutative balls and their doubles

Quantum analogues ofn-dimensional balls are defined via suitable generators and relations. In the even case they correspond to the Twisted Canonical Commutation Relations of Pusz and Woronowicz. Then we construct quantum spheres as double manifolds of the noncommutative balls.