[论文解读] Tensor products and regularity properties of Cuntz semigroups

Ramon Antoine, Francesc Perera+1 more|Oct 2, 2014

Advanced Operator Algebra Research参考文献 87被引用 33

一句话总结

本文在Cuntz半群范畴$Χ$中建立了张量积的存在性，引入了预完备化Cuntz半群范畴$Χ$作为技术工具。定义了'稳固'$Χ$-半环——强自吸收$C^*$-代数在半群设定下的类比——并证明了满足UCT的强自吸收$C^*$-代数的Cuntz半群是稳固的，通过此类半环上的半模将这一结果与Toms-Winter猜想联系起来。

ABSTRACT

The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(A\otimes B)$ with $Cu(A)\otimes_{Cu}Cu(B)$ for certain classes of C*-algebras. As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic. We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.

研究动机与目标

• 通过引入预完备化Cuntz半群范畴$Χ$，为Cuntz半群建立范畴框架。
• 在范畴$Χ$中建立张量积的存在性，并将$Χ(A \bigotimes B)$与$Χ(A) \bigotimes_{Χ} Χ(B)$对某些$C^*$-代数的关系进行关联。
• 在半群设定下，定义并研究作为强自吸收$C^*$-代数类比的'稳固'$Χ$-半环。
• 刻画作为稳固$Χ$-半环上半模的$Χ$-半群，特别是与Jiang-Su代数的$Χ$-半群的关系。
• 通过Jiang-Su代数的$Χ$-半环上半模的刻画，建立Toms-Winter猜想的半群层级类比。

提出的方法

• 将范畴$Χ$的预完备化Cuntz半群引入为纯粹代数设定，使得张量积和归纳极限等构造更容易处理。
• 证明$Χ$是$Χ$的全且可反射的子范畴，从而实现从$Χ$到$Χ$的性质传递。
• 通过利用双态射范畴中表示对象的普遍性质，在$Χ$中构造张量积。
• 为$Χ$-半环$R$与$Χ$-半群$S$定义$R \bigotimes S$上的乘积，使$R \bigotimes_{Χ} S$成为一个具有自然半环结构的$Χ$-半群。
• 引入'稳固'$Χ$-半环的概念，即满足自然映射$R \bigotimes_{Χ} R \to R$为同构的半环。
• 利用Grothendieck完备化与正锥函子，关联部分有序环与可消去、锥形半环，确保与张量积的相容性。

实验结果

研究问题

• RQ1 范畴$Χ$的Cuntz半群是否具有良好的张量积结构？
• RQ2 $C^*$-代数张量积的Cuntz半群如何与它们各自Cuntz半群的张量积相关？
• RQ3 在范畴与代数上，哪些$Χ$-半环的性质使其类似于强自吸收$C^*$-代数？
• RQ4 在何种条件下，$C^*$-代数的$Χ$-半群是稳固$Χ$-半环上的半模？
• RQ5 能否通过Jiang-Su代数的$Χ$-半环上半模的结构，形式化并证明半群层级的Toms-Winter猜想？

主要发现

• 预完备化Cuntz半群范畴$Χ$具有所有小余极限，包括归纳极限与张量积。
• $A \to Χ(A)$的映射是从$C^*$-代数到$Χ$的函子，且保持归纳极限。
• 满足UCT的强自吸收$C^*$-代数的$Χ$-半群是一个稳固的$Χ$-半环。
• 一个$Χ$-半群$S$是Jiang-Su代数的$Χ$-半环上的半模，当且仅当$S$是几乎无穿孔且几乎可除的。
• $X$-半环$R$与$X$-半群$S$的张量积$R \bigotimes_{Χ} S$同构于$S$，当且仅当$S$是$R$上的半模。
• $X$-半环与半模的理论提供了Toms-Winter猜想的半群层级类比，其中Jiang-Su代数的$X$-半环起着核心作用。