Tensor products and regularity properties of Cuntz semigroups / Ramon Antoine, Francesc Perera, Hannes Thiel.

Format:

Book

Author/Creator:

Antoine, Ramon, 1973- author.

Perera, Francesc, 1970- author.

Thiel, Hannes, 1982- author.

Series:

Memoirs of the American Mathematical Society ; Volume 251, Number 1199.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 251, Number 1199

Language:

English

Subjects (All):

C*-algebras.

Tensor products.

Tensor algebra.

Semigroups.

Physical Description:

1 online resource (206 pages).

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2017]

Summary:

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.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Background

1.2. The categories and

1.3. Tensor products

1.4. Multiplicative structure of -semigroups. Solid -semirings

Acknowledgments

Chapter 2. Pre-completed Cuntz semigroups

2.1. The categories \CatPreW and \CatW

2.2. The pre-completed Cuntz semigroup of a *-algebra

Chapter 3. Completed Cuntz semigroups

3.1. The category \CatCu

3.2. The completed Cuntz semigroup of a *-algebra

Chapter 4. Additional axioms

Chapter 5. Structure of Cu-semigroups

5.1. Ideals and quotients

5.2. Functionals

5.3. Soft and purely noncompact elements

5.4. Predecessors, after Engbers

5.5. Algebraic semigroups

5.6. Nearly unperforated semigroups

Chapter 6. Bimorphisms and tensor products

6.1. Tensor product as representing object

6.2. The tensor product in \CatPreW

6.3. The tensor product in \CatCu

6.4. Examples and Applications

Chapter 7. Cu-semirings and Cu-semimodules

7.1. Strongly self-absorbing *-algebras and solid \CatCu-semirings

7.2. Cuntz semigroups of purely infinite C*-algebras

7.3. Almost unperforated and almost divisible -semigroups

7.4. The rationalization of a semigroup

7.5. The realification of a semigroup

7.6. Examples and Applications

Chapter 8. Structure of Cu-semirings

8.1. Simple -semirings

8.2. Algebraic -semirings

8.3. Classification of solid -semirings

Chapter 9. Concluding remarks and open problems

Appendix A. Monoidal and enriched categories

Appendix B. Partially ordered monoids, groups and rings

B.1. The category of monoids

B.2. The categories PrePOM and POM of positively (pre)ordered monoids

B.3. The category of partially ordered groups

B.4. The category of partially ordered rings

Bibliography

Index of Terms

Index of Symbols.

Back Cover.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-4282-5