A note on noncommutative holomorphic and harmonic functions on the unit disk

Free holomorphic functions on the unit ball of B(H)^n

We develop a theory of holomorphic functions in several noncommuting (free) variables and thus provide a framework for the study of arbitrary n-tuples of operators. The main topics are the following: Free holomorphic functions and Hausdorff derivations; Cauchy, Liouville, and Schwartz type results for free holomorphic functions; Algebras of free holomorphic functions; Free analytic functional calculus and noncommutative Cauchy transforms; Weierstrass and Montel theorems for free holomorphic functions; Free pluriharmonic functions and noncommutative Poisson transforms; Hardy spaces of free holo…

Harmonic analysis approach to Gromov--Hausdorff convergence for noncommutative tori

We show that the rotation algebras are limit of matrix algebras in a very strong sense of convergence for algebras with additional Lipschitz structure. Our results generalize to higher dimensional noncommutative tori and operator valued coefficients. In contrast to previous results by Rieffel, Li, Kerr, and Latrémolière we use Lipschitz norms induced by the `carré du champ' of certain natural dynamical systems, including the heat semigroup.

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs B_n of bounded linear operators on Hilbert spaces. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on B_n, study their properties, and obtain hyperbolic versions of Schwarz-Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the operatorial polydisk D_k, our hyperbolic metric is compl…

$L^p$-理论与量子调和分析中的非交换几何

$L^p$-Theory and Noncommutative Geometry in Quantum Harmonic Analysis

本文提出量子调和分析中非交换$L^p$理论及扩展至李群，揭示其与Connes非交换几何的联系，并研究量子Segal代数性质，为统一框架奠定基础。

Noncommutative reproducing kernel Hilbert spaces

The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with motivation and applications from a variety of areas (e.g., noncommutative functional calculus, fr…

On Harmonic Noncommutative $L^p$-Operators on Locally Compact Quantum Groups

For a locally compact quantum group $\G$ with tracial Haar weight $φ$, and a quantum measure $μ$ on $\G$, we study the space ${H}_μ^p$ of $μ$-harmonic operators in the non-commutative $L^p$-space ${L}^p(\G)$ associated to the Haar weight $φ$. The main result states that if $μ$ is non-degenerate, then ${H}_μ^p$ is trivial for all $1 \leq p < \infty$.

Noncommutative analogues of Stein spaces of finite embedding dimension

We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions…

Operator theory on noncommutative domains

In this paper we study noncommutative domains D_f in B(H)^n, generated by positive regular free holomorphic functions f, where B(H) is the algebra of all bounded linear operators on a Hilbert space H.

Noncommutative hyperbolic metrics

We characterize certain noncommutative domains in terms of noncommutative holomorphic equivalence via a pseudometric that we define in purely algebraic terms. We prove some properties of this pseudometric and provide an application to free probability.

Noncommutative harmonic analysis on semigroup and ultracontractivity

We extend some classical results of Cowling and Meda to the noncommutative setting. Let $(T_t)_{t>0}$ be a symmetric contraction semigroup on a noncommutative space $L_p(\mathcal{M}),$ and let the functions $ϕ$ and $ψ$ be regularly related. We prove that the semigroup $(T_t)_{t>0}$ is $ϕ$-ultracontractive, i.e. $\|T_t x\|_\infty \leq C ϕ(t)^{-1} \|x\|_1$ for all $x\in L_1(\mathcal{M})$ and $ t>0$ if and only if its infinitesimal generator $L$ has the Sobolev embedding properties: $\|ψ(L)^{-α} x\|_q \leq C'\|x\|_p$ for all $x\in L_p(\mathcal{M}),$ where $1<p<q<\infty$ and $α=\frac{1}{p}-\frac{1…

Noncommutative hyperbolic geometry on the unit ball of $B(H)^n$

In this paper we introduce a hyperbolic distance $δ$ on the noncommutative open ball $[B(H)^n]_1$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, which is a noncommutative extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We prove that $δ$ is invariant under the action of the group $Aut([B(H)^n]_1)$ of all free holomorphic automorphisms of $[B(\cH)^n]_1$, and show that the $δ$-topology and the usual operator norm topology coincide on $[B(H)^n]_1$. Moreover, we prove that $[B(H)^n]_1$ is a complete metric space with respect to the hype…

Free holomorphic functions and interpolation

In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna-Pick interpolation problem for analytic functions with positive real parts on the open unit disc. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball on C^n.

Free holomorphic functions on the unit ball of B(H)^n. II

In this paper we continue the study of free holomorphic functions on the unit ball of B(H)^n, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. Several classical results from complex analysis have free analogues in our noncommutative setting.

通过非交换几何棱镜观察经典调和分析

Classical harmonic analysis viewed through the prism of noncommutative geometry

本文通过非交换几何框架，将经典调和分析与L^p空间分析结合，提出新的K-理论群和索引定理，揭示Hilbert变换和Riesz变换的作用。

Free Holomorphic Functions on Polydomains

In this paper, we continue to develop the theory of free holomorphic functions on noncommutative regular polydomains. We find analogues of several classical results from complex analysis such as Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem for entire functions, in our multivariable setting. We also provide a maximum principle and a Schwarz type lemma. These results are used to prove analogues of Weierstrass, Montel, and Vitali theorems for the algebra of free holomorphic functions on the regular polydomain, which turns out to be a complete metric space.

算子空间上全纯函数的线性化

Linearizing holomorphic functions on operator spaces

该研究提出算子空间开单位球上完全有界全纯函数的概念，赋予其算子空间结构，找到标量值情形下的非交换预对偶并证明其线性化性质与逼近性质传递性。

紧量子齐性凯勒空间

Compact Quantum Homogeneous Kähler Spaces

本文研究紧量子齐性凯勒空间，探讨其与Woronowicz量子群理论的兼容性，分析Dolbeault-Dirac算子的分析行为及扭后的Fredholm性与欧拉特征数的关系。

非交换单位球子簇上的有界非交换解析函数代数

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

该研究确定了非交换单位球子簇上有界非交换解析函数代数的结构，探讨了其等距同构条件，并在齐次和交换情况下得到完美零点定理，为非交换单位球函数论提供新结果。

An invitation to harmonic analysis associated with semigroups of operators

This article is an introduction to our recent work in harmonic analysis associated with semigroups of operators, in the effort of finding a noncommutative Calderón-Zygmund theory for von Neumann algebras. The classical CZ theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of such metrics -or with very little information on the metric- Markov semigroups of operators appear to be the right substitutes of classical metric/geometric tools in harmonic analysis. Our approach is particularly useful in the noncommutative setting bu…