与离散群相关的非交换L_p-空间的逼近性质

Discrete Conduche Fibrations and C*-algebras

The higher rank graphs of Kumjian and Pask are discrete Conduche fibrations over the monoid of k-tuples of natural numbers for some k in which every morphism in the base has a finite preimage under the the fibration. We examine the generalization of this construction to discrete Conduche fibrations with the same finiteness condition and a lifting property for completions of cospans to commutative squares, over any category satisfying a strong version of the right Ore condition, including all categories with pullbacks and right Ore categories in which all morphisms are monic.

Bimodules over ${\rm VN}(G)$, harmonic operators and the non-commutative Poisson boundary

Starting with a left ideal $J$ of $L^1(G)$ we consider its annihilator $J^{\perp}$ in $L^{\infty}(G)$ and the generated ${\rm VN}(G)$-bimodule in $\mathcal{B}(L^2(G))$, ${\rm Bim}(J^{\perp})$. We prove that ${\rm Bim}(J^{\perp})=({\rm Ran} J)^{\perp}$ when $G$ is weakly amenable discrete, compact or abelian, where ${\rm Ran} J$ is a suitable saturation of $J$ in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the ${\rm VN}(G)$-bimodule generated by the space of jointly ha…

Characterizing graph C*-correspondences

Every separable nondegenerate C*-correspondence over a commutative C*-algebra with discrete spectrum is isomorphic to a graph correspondence.

Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces

Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 <= p <= infinity. One can associate to phi a Schur multiplier on the Schatten classes S_p(L^2 G), as well as a Fourier multiplier on Lp(LG), the non-commutative Lp-space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the Lp-Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups. For a discrete group G and in the speci…

On the rough-paths approach to non-commutative stochastic calculus

We study different possibilities to apply the principles of rough paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of rough paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelli's controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy.

A universal, non-commutative C*-algebra associated to the Hecke algebra of double cosets

Let G be a discrete group and $Γ$ an almost normal subgroup. The operation of cosets concatanation extended by linearity gives rise to an operator system that is embeddable in a natural C* algebra. The Hecke algebra naturally embeds as a diagonal of the tensor product of this algebra with its opposite. When represented on the $l^2$ space of the group, by left and right convolution operators, this representation gives rise to abstract Hecke operators that in the modular group case, are unitarily equivalent to the classical operators on Maass wave forms

非对易Bloch理论概述

Non-commutative Bloch theory. An Overview

对于在阿贝尔群作用下不变的微分算子，布洛赫理论是分析谱性质的首选工具。通过对这一问题的非交换性研究，我们引入了Hilbert C*-模上椭圆算子的非交换Bloch理论。它将C*-代数的性质与模算子的谱性质联系起来，如能带结构、康托谱的弱泛性和离散谱的不存在。例如，它适用于在投射群作用下不变的微分算子，例如具有周期磁场的薛定谔算子。

Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture

We investigate various notions of peaking behaviour for states on a $\mathrm{C}^*$-algebra, where the peaking occurs within an operator system. We pay particularly close attention to the existence of sequences of elements forming an approximation of the characteristic function of a point in the state space. We exploit such characteristic sequences to localize the $\mathrm{C}^*$-algebra at a given state, and use this localization procedure to verify a variation of Arveson's hyperrigidity conjecture for arbitrary operator systems.

非交换巴拿赫代数的Gelfand理论

Gelfand theory for non-commutative Banach algebras

设$A$是Banach代数.我们称一对$(G, B)$为$A$的Gelfand理论，如果满足下列公理：（G1）$B$是$C^\ast$-代数，$G : A \to B$是同态，（G2）赋值$L \mapsto G^{-1}(L)$分别是$B$和$A$的极大模左理想集之间的双射;（G3）对于$B$的每个极大模左理想$L$，由$B$诱导的线性映射$G_L : A / G^{-1}(L) \to B /L $有稠密值域.交换Banach代数的Gelfand理论很容易被这些公理刻画。任意巴拿赫代数的Gelfand理论具有交换Gelfand理论的许多性质。我们证明了有单位元的齐次巴拿赫代数总是有Gelfand理论。对于具有离散谱的阈$C^\ast$-代数，我们证明了恒等式是唯一的Gelfand理论（直到一个适当的等价概念）.

受限群代数的空间离散化

Spatial discretization of restricted group algebras

本文通过有限截面方法研究有限生成离散群的受限群代数的空间离散化，重点分析了有限截面序列生成的代数的拟交换子理想及其序列稳定性，并证明了交换群和自由非交换群的有限截面序列代数是分形的。

Tilings的非交换几何与Gap标号

Non Commutative Geometry of Tilings and Gap Labelling

对于给定的镶嵌，构造了一个非交换空间和相应的C*-代数.这包括由平铺的平移引起的广群上的拓扑的定义。该代数也是一个或多个粒子系统的离散模型的观测量的代数的平铺或其周期性识别。它的标度有序K_0群给出了Schroedinger算子的间隙标号。该群是针对一维平铺及其笛卡尔积计算的。它的图像下的状态下，调查的平铺下不变的替代。该图像的一部分由确定的平铺的外壳上的不变测度给出。从一维平铺的笛卡尔积的结果指出，通过积分态密度的值标记的间隙已经完全由该措施确定。

非交换几何计算程序用于无序拓扑绝缘体

A Computational Non-Commutative Geometry Program for Disordered Topological Insulators

本文提出基于非交换几何原理的计算程序，用于研究无序拓扑绝缘体的拓扑相变及稳定性，展示了非交换布里渊 торus 在描述电子动力学中的应用。