eBook ISBN:  9781470442828 
Product Code:  MEMO/251/1199.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $47.00 
eBook ISBN:  9781470442828 
Product Code:  MEMO/251/1199.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $47.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 251; 2018; 191 ppMSC: Primary 06; 15; 46; Secondary 13; 16; 18
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 precompleted 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.

Table of Contents

Chapters

1. Introduction

2. Precompleted Cuntz semigroups

3. Completed Cuntz semigroups

4. Additional axioms

5. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemigroups

6. Bimorphisms and tensor products

7. \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemirings and \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemimodules

8. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemirings

9. Concluding remarks and open problems

A. Monoidal and enriched categories

B. Partially ordered monoids, groups and rings


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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 precompleted 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.

Chapters

1. Introduction

2. Precompleted Cuntz semigroups

3. Completed Cuntz semigroups

4. Additional axioms

5. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemigroups

6. Bimorphisms and tensor products

7. \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemirings and \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemimodules

8. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cusemirings

9. Concluding remarks and open problems

A. Monoidal and enriched categories

B. Partially ordered monoids, groups and rings