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Tensor products and regularity properties of Cuntz semigroups

About this Title

Ramon Antoine, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain, Francesc Perera, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain and Hannes Thiel, Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany

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
MSC: Primary 06B35, 06F05, 15A69, 46L05.; Secondary 06B30, 06F25, 13J25, 16W80, 16Y60, 18B35, 18D20, 19K14, 46L06, 46M15, 54F05.

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Table of Contents


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


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. We 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 (2008). To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $\mathrm {Cu}$-semigroups.

We establish the existence of tensor products in the category $\mathrm {Cu}$ and study the basic properties of this construction. We 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 our approach we introduce the category $\mathrm {W}$ of pre-completed Cuntz semigroups. We 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.

For every (local) $C^*$-algebra $A$, the classical Cuntz semigroup $W(A)$ together with a natural auxiliary relation is an object of $\mathrm {W}$. This defines a functor from $C^*$-algebras to $\mathrm {W}$ which preserves inductive limits. We deduce that the assignment $A\mapsto \mathrm {Cu}(A)$ defines a functor from $C^*$-algebras to $\mathrm {Cu}$ which preserves inductive limits. This generalizes a result from Coward, Elliott, and Ivanescu (2008).

We also develop a theory of $\mathrm {Cu}$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing $C^*$-algebra has a natural product giving it the structure of a $\mathrm {Cu}$-semiring. For $C^*$-algebras, it is an important regularity property to tensorially absorb a strongly self-absorbing $C^*$-algebra. Accordingly, it is of particular interest to analyse the tensor products of $\mathrm {Cu}$-semigroups with the $\mathrm {Cu}$-semiring of a strongly self-absorbing $C^*$-algebra. This leads us to define ‘solid’ $\mathrm {Cu}$-semirings (adopting the terminology from solid rings), as those $\mathrm {Cu}$-semirings $S$ for which the product induces an isomorphism between $S\otimes _{\mathrm {Cu}} S$ and $S$. This can be considered as an analog of being strongly self-absorbing for $\mathrm {Cu}$-semirings. As it turns out, if a strongly self-absorbing $C^*$-algebra satisfies the UCT, then its $\mathrm {Cu}$-semiring is solid. We prove a classification theorem for solid $\mathrm {Cu}$-semirings. This raises the question of whether the Cuntz semiring of every strongly self-absorbing $C^*$-algebra is solid.

If $R$ is a solid $\mathrm {Cu}$-semiring, then a $\mathrm {Cu}$-semigroup $S$ is a semimodule over $R$ if and only if $R\otimes _{\mathrm {Cu}}S$ is isomorphic to $S$. Thus, analogous to the case for $C^*$-algebras, we can think of semimodules over $R$ as $\mathrm {Cu}$-semigroups that tensorially absorb $R$. We give explicit characterizations when a $\mathrm {Cu}$-semigroup is such a semimodule for the cases that $R$ is the $\mathrm {Cu}$-semiring of a strongly self-absorbing $C^*$-algebra satisfying the UCT. For instance, we show that a $\mathrm {Cu}$-semigroup $S$ tensorially absorbs the $\mathrm {Cu}$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.

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