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OverviewThe 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. Full Product DetailsAuthor: Ramon Antoine , Francesc Perera , Hannes ThielPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.298kg ISBN: 9781470427979ISBN 10: 1470427974 Pages: 191 Publication Date: 30 March 2018 Audience: Professional and scholarly , College/higher education , Professional & Vocational , Postgraduate, Research & Scholarly Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction Pre-completed Cuntz semigroups Completed Cuntz semigroups Additional axioms Structure of Cu-semigroups Bimorphisms and tensor products Cu-semirings and Cu-semimodules Structure of Cu-semirings Concluding remarks and Open Problems Appendix A. Monoidal and enriched categories Appendix B. Partially ordered monoids, groups and rings Bibliography Index of Terms Index of SymbolsReviewsAuthor InformationRamon Antoine, Universitat Autonoma de Barcelona, Spain. Francesc Perera, Universitat Autonoma de Barcelona, Spain. Hannes Thiel, Universitat Munster, Germany. Tab Content 6Author Website:Countries AvailableAll regions |
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