Nonlinear partial differential equations on noncommutative Euclidean spaces

$L^p$-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 potential theory

We propose to view hermitian metrics on trivial holomorphic vector bundles $E\toΩ$ as noncommutative analogs of functions defined on the base $Ω$, and curvature as the notion corresponding to the Laplace operator or $\partial\overline\partial$. We discuss noncommutative generalizations of basic results of ordinary potential theory, mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.

Noncommutative Harmonic and Subharmonic Polynomials and other Noncommutative Partial Differential Equations

Solutions to Laplace's equation are called harmonic functions. Harmonic functions arise in many applications, such as physics and the theory of stochastic processes. Of interest classically are harmonic polynomials, which have a simple classification. Further, the work of Reznick, building on the work of others, namely Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason, has led to a classification of all polynomial solutions to a differential equation of arising from a homogeneous polynomial over an algebraically closed field. The definition of harmonicity can be extended t…

A noncommutative version of the Banach-Stone theorem (II)

In this paper, we extend the Banach-Stone theorem to the non commutative case, i.e, we give a partial answere to the question 2.1 of [13], and we prove that the structure of the postliminal C*-algebras A determines the topology of its primitive ideals space.

量子欧氏空间上量化导数的谱渐近性

Spectral Asymptotics for Quantized Derivatives on Quantum Euclidean Spaces

该研究利用非交换Wiener-Ikehara Tauberian定理与C*代数版伪微分算子理论，推导量子欧氏空间一阶齐次Sobolev空间元素量化导数的谱渐近性，推广了已有结果。

Noncommutative variations on Laplace's equation

As a first step at developing a theory of noncommutative nonlinear elliptic partial differential equations, we analyze noncommutative analogues of Laplace's equation and its variants (some of the them nonlinear) over noncommutative tori. Along the way we prove noncommutative analogues of many results in classical analysis, such as Wiener's Theorem on functions with absolutely convergent Fourier series, and standard existence and non-existence theorems on elliptic functions. We show that many many classical methods, including the Maximum Principle, the direct method of the calculus of variation…

The noncommutative Choquet boundary

Let S be an operator system -- a self-adjoint linear subspace of a unital C*-algebra A such that contains 1 and A=C*(S) is generated by S. A boundary representation for S is an irreducible representation πof C*(S) on a Hilbert space with the property that $π\restriction_S$ has a unique completely positive extension to C*(S). The set $\partial_S$ of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system $S\subseteq C(X)$ that separates points of X. It is known that the closure of the Choquet boundary of a func…

Singular integrals in quantum Euclidean spaces

In this paper, we establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes' pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce $L_p$-boundedness and Sobolev $p$-estimates for regular, exotic and forbidden symbols in the expected ranks. In the $L_2$ level both Calderón-Vaillancourt and Bourdaud theorems for exotic and …

Cauchy-Riemann equations for free noncommutative functions

In classical complex analysis analyticity of a complex function $f$ is equivalent to differentiability of its real and imaginary parts $u$ and $v$, respectively, together with the Cauchy-Riemann equations for the partial derivatives of $u$ and $v$. We extend this result to the context of free noncommutative functions on tuples of matrices of arbitrary size. In this context, the real and imaginary parts become so called real noncommutative functions, as appeared recently in the context of Löwner's theorem in several noncommutative variables. Additionally, as part of our investigation of real no…

Duality in Scalar Field Theory on Noncommutative Phase Spaces

We describe a novel duality symmetry of Phi(4)-theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates. This duality acts on the fields by Fourier transformation and scaling. It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of non-interacting scalar particles coupled to a constant background electromagnetic field. The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged. Using a parti…