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 |
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 |
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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 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.
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Table of Contents
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Chapters
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1. Introduction
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2. Pre-completed Cuntz semigroups
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3. Completed Cuntz semigroups
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4. Additional axioms
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5. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semigroups
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6. Bimorphisms and tensor products
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7. \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semirings and \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semimodules
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8. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semirings
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9. Concluding remarks and open problems
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A. Monoidal and enriched categories
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B. Partially ordered monoids, groups and rings
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Additional Material
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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 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.
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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
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8. Structure of \texorpdfstring{$\ensuremath {\mathrm {Cu}}$}Cu-semirings
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9. Concluding remarks and open problems
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A. Monoidal and enriched categories
-
B. Partially ordered monoids, groups and rings