算子代数：非正式综述

An Introduction to Abstract Classification Theory in the Operator Algebraic Setting

In the setting of modern mathematical logic and model theory, classification theory has been one of the landmark achievements of the field. Likewise, the classification of UHF-algebras and AF-algebras were substantial contributions to the field of operator algebra theory. These seemingly disparate topics of study in mathematics, model theory and operator algebras, have in recent years become closely related in many respects. I here attempt to bridge the gap between these two topics by discussing how operator algebraic classifications may be understood in terms of model-theoretic classification…

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。

Fraïssé theory in operator algebras

We overview the development of Fraïssé theory in the setting of continuous model theory, and some of the its recent applications to $\mathrm{C}^*$-algebra theory and functional analysis.

Structure of nuclear C*-algebras: From quasidiagonality to classification, and back again

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the regularity conjecture and its interplay with internal and external approximation properties.

Some remarks on free products of rigid $C^*$-2-categories

In this informal expository note, we present a universal, formulaic construction of the free product of rigid $C^*$-2-categories. This construction allows for a straightforward, purely categorical formulation of the free composition of subfactors and planar algebras considered by Bisch and Jones. As an application, we explain the results of Tarrago and Wahl on free wreath products of compact quantum groups in this categorical language.

Positive combinations of projections in von Neumann algebras and purely infinite simple C*-algebras

We give an overview of the question: which positive elements in an operator algebra can be written as a linear combination of projections with positive coefficients. A special case of independent interest is the question of which positive elements can be written as a sum of finitely many projections. We focus on von Neumann algebras, on purely infinite simple C*-algebras, and on their associated multiplier algebras.