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Tensor Products and Regularity Properties of Cuntz Semigroups.
・ISBN 978-1-4704-2797-9 paper US$ 78.00
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| 著者・編者 | Antoine, Ramon / Perera, Francesc / Thiel, Hannes, |
|---|---|
| シリーズ | Memoirs of the American Mathematical Society |
| 出版社 | (American Mathematical Society, US) |
| 出版年月 | 2018 |
| ページ数 | 191 pp. |
| 言語 | ENG |
| ニュース番号 | ＜M25-13260＞ |
解説
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. The authors systematically study the lattice and category theoretic aspects of Cuntz semigroups.
Given a $C^*$-algebra $A$, its (concrete) Cuntz semigroup $\mathrm{Cu}(A)$ is an object in the category $\mathrm{Cu}$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, the authors call the latter $\mathrm{Cu}$-semigroups.
The authors establish the existence of tensor products in the category $\mathrm{Cu}$ and study the basic properties of this construction. They show that $\mathrm{Cu}$ is a symmetric, monoidal category and relate $\mathrm{Cu}(A\otimes B)$ with $\mathrm{Cu}(A)\otimes_{\mathrm{Cu}}\mathrm{Cu}(B)$ for certain classes of $C^*$-algebras.
As a main tool for their approach the authors introduce the category $\mathrm{W}$ of pre-completed Cuntz semigroups. They show that $\mathrm{Cu}$ is a full, reflective subcategory of $\mathrm{W}$. One can then easily deduce properties of $\mathrm{Cu}$ from respective properties of $\mathrm{W}$, for example the existence of tensor products and inductive limits. The advantage is that constructions in $\mathrm{W}$ are much easier since the objects are purely algebraic.
Given a $C^*$-algebra $A$, its (concrete) Cuntz semigroup $\mathrm{Cu}(A)$ is an object in the category $\mathrm{Cu}$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, the authors call the latter $\mathrm{Cu}$-semigroups.
The authors establish the existence of tensor products in the category $\mathrm{Cu}$ and study the basic properties of this construction. They show that $\mathrm{Cu}$ is a symmetric, monoidal category and relate $\mathrm{Cu}(A\otimes B)$ with $\mathrm{Cu}(A)\otimes_{\mathrm{Cu}}\mathrm{Cu}(B)$ for certain classes of $C^*$-algebras.
As a main tool for their approach the authors introduce the category $\mathrm{W}$ of pre-completed Cuntz semigroups. They show that $\mathrm{Cu}$ is a full, reflective subcategory of $\mathrm{W}$. One can then easily deduce properties of $\mathrm{Cu}$ from respective properties of $\mathrm{W}$, for example the existence of tensor products and inductive limits. The advantage is that constructions in $\mathrm{W}$ are much easier since the objects are purely algebraic.