On quantales that classify C*-algebras

Toposes, quantales and C* algebras in the atomic case

We start by reviewing the relation between toposes and Grothendieck quantales. We improve results of previous work on this relation by giving both a characterisation of the map from the tensor product of two internal sup-lattices to another sup-lattice and a description of the category of internal locales of a topos in terms of the associated Grothendieck quantale. We then construct a convolution product, corresponding to internal composition of matrices, on the set of positive lower semi-continuous functions on the underlying locale of the quantale attached to a topos. In good cases, this con…

On quantales and spectra of C*-algebras

We study properties of the quantale spectrum Max A of an arbitrary unital C*-algebra A. In particular we show that the spatialization of Max A with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-…

Quantales and Fell bundles

We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle $π:E\to G$ on an étale groupoid $G$ with $G_0$ locally compact Hausdorff, equipped with a suitable completion C*-algebra $A$ of its convolution algebra, we obtain a map of involutive quantales $p:\mathrm{Max}\ A\toΩ(G)$, where $\mathrm{Max}\ A$ consists of the closed linear subspaces of $A$ and $Ω(G)$ is the topology of $G$. We study various properties of $p$ which mimick, to various degrees, those of open maps of topological spaces. These are closely related to proper…

对合量子格与量子格赋值的对合拓扑空间

Involutive quantales and quantale-enriched involutive topological spaces

本文扩展量子格赋值拓扑空间以包含对合，通过量子框架平衡拓扑化单位$C^*$-代数谱，并识别纯态子空间为强Hausdorff分离的对合空间。

将映射分类到一致迹序列代数中

Classifying maps into uniform tracial sequence algebras

本文利用迹数据对从核C*-代数到核Z-稳定C*-代数的一致迹序列代数的*-同态进行分类。

*同态的分类 I：单位单核C*-代数

Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras

本文对满足万有系数定理的单位可分核C*-代数到张量吸收Jiang-Su代数的单位单可分核C*-代数的单位嵌入分类，为单位分类定理的稳定有限情形提供新自包含证明。