One noncommutative differential calculus coming from the inner derivation

Nonpure (Non)Commutative Analysis, Geometry and Mechanics, part 1: differential and integral calculus

We construct and study differential and integral calculus on the space of states of a C*-algebra by equipping it with a formal smooth structure. To achieve this goal we first concentrate on the space of nonpure states of a commutative C*-algebra as a guideline for the noncommutative case. In particular, we prove Stokes' theorem over the both commutative and noncommutative smooth Wasserstein space.

R-transform Theory On a Tower of Amalgamated Noncommutative Probability Spaces

In [9], we observed Amalgamated R-transform Theory. Different from the original definition of Voiculescu and Speicher, we define R-transforms of operator-valued random variable(s) by operator-valued formal series. By doing that we can establish Amalgamated R-transform calculus. In this paper, we will concentrate on observing R-transforms on a tower of amalgamated noncommutative probability spaces. We will consider the R-transform calculus and relation between R-transforms on the tower and distributions with respect to the tower. As application, we will observe the compatibility of the given to…

Functional calculus and multi-analytic models on regular $Λ$-polyballs

The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular $Λ$-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.) $k$-tuples in ${\bf B}_Λ(H)$, and to study the characteristic functions and the associated multi-analytic models for the c.n.c. elements in the regular $Λ$-polyball. In addition, we show that the characteristic function is a complete unitary invariant for the class of c.n.c. $k$-tuples in ${\bf B}_Λ(H)$. These results extend the corresponding classical results of …

The Theory of Pseudo-differential Operators on the Noncommutative n-Torus

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on $\mathbb R^n$, one may derive a similar pseudo-differential calculus on noncommutative $n$ tori $\mathbb T_θ^n$, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for $n=2,4$, as shown in the groundbreaking paper in which t…

源于群在 C*-代数上作用的非交换可数主丛

Noncommutative numerable principal bundles from group actions on C*-algebras

本文引入局部平凡 $G$-C*-代数定义，利用 Pedersen 理想多重子推广 Gelfand-Naimark 对偶性，研究非单 C*-代数上群作用产生的非交换主丛局部平凡性，并证明其 $G$-作用的自由性。

非交换几何中的除法差：重新排列引理、函数计算与展开公式

Divided Differences in Noncommutative Geometry: Rearrangement Lemma, Functional Calculus and Expansional Formula

本文推广了Connes-Tretkoff-Moscovici重新排列引理并给出简洁证明，同时建立多变量函数计算基础，并展示模函数递推公式与除法差的联系，还揭示了Magnus展开式相关公式的概念解释。

Fractional powers on noncommutative $L_p$ for $p<1$

We prove that the homogeneous functional calculus associated to $x\mapsto |x|^θ$ or $x\mapsto {\rm sgn}\, (x) |x|^θ$ for $0<θ<1$ is $θ$-Hölder on selfadjoint elements of noncommutative $L_p$-spaces for $0<p\leq\infty$ with values in $L_{p/θ}$. This extends an inequality of Birman, Koplienko and Solomjak also obtained by Ando.

On some topics of analysis on noncommutative spaces

We consider a conservative Markov semigroup on a semi-finite $W^*$-algebra. It is known that under some reasonable assumptions it is enough to determine a kind of differential structure on such a 'noncommutative space'. We construct an analogue of a Riemannian metric in this setting, formulate a Poincaré-type inequality, provide existence and uniqueness results for quasi-linear elliptic and parabolic PDEs defined in terms of the constructed noncommutative calculus.

Nonlinear partial differential equations on noncommutative Euclidean spaces

Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the anal…

The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces

The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C^*$-modules and their maps. Here we give a systematic exposition of this theory; a reference tool for `noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.

Square functions for Ritt operators on noncommutative $L^p$-spaces

For any Ritt operator $T$ acting on a noncommutative $L^p$-space, we define the notion of \textit{completely} bounded functional calculus $H^\infty(B_γ)$ where $B_γ$ is a Stolz domain. Moreover, we introduce the `column square functions' $\norm{x}_{T,c,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1}|T^{k-1}(I-T)^α(x)|^2\Big)^{1/2}}_{L^p(M)}$ and the `row square functions' $\norm{x}_{T,r,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1} |\Big(T^{k-1}(I-T)^α(x)\Big)^*|^2\Big)^{1/2}}_{L^p(M)}$ for any $α>0$ and any $x\in L^p(M)$. Then, we provide an example of Ritt operator which admits a completely bounde…

Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I

In this work, we solve the problem explicitly stated at the end of a paper of Junge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groups including all amenable groups and free groups. More precisely, we prove that the Hodge-Dirac operator of the canonical "hidden" noncommutative geometry associated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on a bisector which implies a positive answer to the quoted problem. Our result can be seen as a strengthening of the dimension free estimates …

Noncommutative Gelfand Duality and Applications I: The Existence of invariant subspaces

Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a characterization of unitary groups of C*-algebras, and, for arbitrary bounded Hilbert space operators, (i) A spectral theorem cum continuous functional calculus, and (ii) A proof of the general Invariant Subspace Theorem. Also described is a nonabelian generalization of Pontryagin duality of abelian locally compact groups.

C*-代数上的微分算子及其在光滑泛函演算与切群胚上施瓦茨函数中的应用

Differential operators on C*-algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid

本文在C*-代数上引入微分算子的非交换概念，证明其公共闭域在光滑泛函演算下封闭，并由此推出Connes切群胚上的施瓦茨函数具有相同封闭性。

Dixmier traces and extrapolation description of noncommutative Lorentz spaces

We study the relationships between Dixmier traces, zeta-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this investigation is that of operator Lorentz spaces possessing an extrapolation description. We demonstrate the applicability of our results to Hörmander-Weyl pseudo-differential calculus. In that context, we prove that the Dixmier trace of a pseudo-differential operator coincide with the `Dixmier integral' of its symbol.

Moduli space and structure of noncommutative 3-spheres

We analyse the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the $C^\ast$-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pai…

Moyal-Groenewold 平面上的谐振子：一种通过李群与扭曲 Weyl 元组的方法

The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples

本文通过李群与扭曲 Weyl 元组方法，证明了 Moyal-Groenewold 平面上谐振子具有有界函数演算，并建立了扭曲转移原理，推进了非交换空间上函数演算与演化方程最大正则性的研究。