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Tensor products and regularity properties of Cuntz semigroups

About this Title

Ramon Antoine, Francesc Perera and Hannes Thiel

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 251, Number 1199
ISBNs: 978-1-4704-2797-9 (print); 978-1-4704-4282-8 (online)
Published electronically: November 7, 2017
Keywords:Cuntz semigroup, tensor product, continuous poset, $C^{*}$-algebra

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Pre-completed Cuntz semigroups
  • Chapter 3. Completed Cuntz semigroups
  • Chapter 4. Additional axioms
  • Chapter 5. Structure of \texorpdfstring$\CatCu $Cu-semigroups
  • Chapter 6. Bimorphisms and tensor products
  • Chapter 7. \texorpdfstring$\CatCu $Cu-semirings and \texorpdfstring$\CatCu $Cu-semimodules
  • Chapter 8. Structure of \texorpdfstring$\CatCu $Cu-semirings
  • Chapter 9. Concluding remarks and open problems
  • Appendix A. Monoidal and enriched categories
  • Appendix B. Partially ordered monoids, groups and rings


The Cuntz semigroup of a -algebra is an important invariant in the structure and classification theory of -algebras. It captures more information than -theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups.Given a -algebra , its (concrete) Cuntz semigroup is an object in the category of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu (2008). To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter -semigroups.We establish the existence of tensor products in the category and study the basic properties of this construction. We show that is a symmetric, monoidal category and relate with for certain classes of -algebras.As a main tool for our approach we introduce the category of pre-completed Cuntz semigroups. We show that is a full, reflective subcategory of . One can then easily deduce properties of from respective properties of , for example the existence of tensor products and inductive limits. The advantage is that constructions in are much easier since the objects are purely algebraic.For every (local) -algebra , the classical Cuntz semigroup together with a natural auxiliary relation is an object of . This defines a functor from -algebras to which preserves inductive limits. We deduce that the assignment defines a functor from -algebras to which preserves inductive limits. This generalizes a result from Coward, Elliott, and Ivanescu (2008).We also develop a theory of -semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing -algebra has a natural product giving it the structure of a -semiring. For -algebras, it is an important regularity property to tensorially absorb a strongly self-absorbing -algebra. Accordingly, it is of particular interest to analyse the tensor products of -semigroups with the -semiring of a strongly self-absorbing -algebra. This leads us to define `solid' -semirings (adopting the terminology from solid rings), as those -semirings for which the product induces an isomorphism between and . This can be considered as an analog of being strongly self-absorbing for -semirings. As it turns out, if a strongly self-absorbing -algebra satisfies the UCT, then its -semiring is solid. We prove a classification theorem for solid -semirings. This raises the question of whether the Cuntz semiring of every strongly self-absorbing -algebra is solid.If is a solid -semiring, then a -semigroup is a semimodule over if and only if is isomorphic to . Thus, analogous to the case for -algebras, we can think of semimodules over as -semigroups that tensorially absorb . We give explicit characterizations when a -semigroup is such a semimodule for the cases that is the -semiring of a strongly self-absorbing [[ca]]satisfying the UCT. For instance, we show that a -semigroup tensorially absorbs the -semiring of the Jiang-Su algebra if and only if is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.

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