Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Tensor Products and Regularity Properties of Cuntz Semigroups
 
Ramon Antoine Universitat Autónoma de Barcelona, Barcelona, Spain
Francesc Perera Universitat Autónoma de Barcelona, Barcelona, Spain
Hannes Thiel Universität Münster, Münster, Germany
Tensor Products and Regularity Properties of Cuntz Semigroups
eBook ISBN:  978-1-4704-4282-8
Product Code:  MEMO/251/1199.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $47.00
Tensor Products and Regularity Properties of Cuntz Semigroups
Click above image for expanded view
Tensor Products and Regularity Properties of Cuntz Semigroups
Ramon Antoine Universitat Autónoma de Barcelona, Barcelona, Spain
Francesc Perera Universitat Autónoma de Barcelona, Barcelona, Spain
Hannes Thiel Universität Münster, Münster, Germany
eBook ISBN:  978-1-4704-4282-8
Product Code:  MEMO/251/1199.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $47.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2512018; 191 pp
    MSC: 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 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.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Pre-completed Cuntz semigroups
    • 3. Completed Cuntz semigroups
    • 4. Additional axioms
    • 5. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semigroups
    • 6. Bimorphisms and tensor products
    • 7. \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semirings and \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semimodules
    • 8. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semirings
    • 9. Concluding remarks and open problems
    • A. Monoidal and enriched categories
    • B. Partially ordered monoids, groups and rings
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2512018; 191 pp
MSC: 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 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.

  • Chapters
  • 1. Introduction
  • 2. Pre-completed Cuntz semigroups
  • 3. Completed Cuntz semigroups
  • 4. Additional axioms
  • 5. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semigroups
  • 6. Bimorphisms and tensor products
  • 7. \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semirings and \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semimodules
  • 8. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semirings
  • 9. Concluding remarks and open problems
  • A. Monoidal and enriched categories
  • B. Partially ordered monoids, groups and rings
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.