Tensor products and regularity properties of cuntz semigroups

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 Cu(A) is an object in the category 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 Cu-semigroups. We establish the existence of tensor products in the category Cu and study the basic properties of this construction. We show that Cu is a symmetric, monoidal category and relate Cu(A ⊗ B) with Cu(A) ⊗_Cu Cu(B) for certain classes of C^∗-algebras. As a main tool for our approach we introduce the category W of pre-completed Cuntz semigroups. We show that Cu is a full, reflective subcategory of W. One can then easily deduce properties of Cu from respective properties of W, for example the existence of tensor products and inductive limits. The advantage is that constructions in 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 W. This defines a functor from C^∗-algebras to W which preserves inductive limits. We deduce that the assignment A → Cu(A) defines a functor from C^∗-algebras to Cu which preserves inductive limits. This generalizes a result from Coward, Elliott, and Ivanescu (2008). We also develop a theory of 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 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 Cu-semigroups with the Cu-semiring of a strongly self-absorbing C^∗-algebra. This leads us to define 'solid' Cu-semirings (adopting the terminology from solid rings), as those Cu-semirings S for which the product induces an isomorphism between S ⊗_Cu S and S. This can be considered as an analog of being strongly self-absorbing for Cu-semirings. As it turns out, if a strongly self-absorbing C^∗-algebra satisfies the UCT, then its Cu-semiring is solid. We prove a classification theorem for solid 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 Cu-semiring, then a Cu-semigroup S is a semimodule over R if and only if R⊗_Cu S is isomorphic to S. Thus, analogous to the case for C^∗-algebras, we can think of semimodules over R as Cu-semigroups that tensorially absorb R. We give explicit characterizations when a Cu-semigroup is such a semimodule for the cases that R is the Cu-semiring of a strongly self-absorbing C^∗-algebra satisfying the UCT. For instance, we show that a Cu-semigroup S tensorially absorbs the 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.