Tensor products and regularity properties of Cuntz semigroups

Abstract

The Cuntz semigroup of a \ca{} is an important invariant in the structure and classification theory of \ca{s}. 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 \ca{} $ A$ , its (concrete) Cuntz semigroup $ \Cu(A)$ is an object in the category $ \CatCu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu in \cite{CowEllIva08CuInv}. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $ \CatCu$ -semigroups. We establish the existence of tensor products in the category $ \CatCu$ and study the basic properties of this construction. We show that $ \CatCu$ is a symmetric, monoidal category and relate $ \Cu(A\otimes B)$ with $ \Cu(A)\otimes_\CatCu\Cu(B)$ for certain classes of \ca{s}. As a main tool for our approach we introduce the category $ \CatW$ of pre-completed Cuntz semigroups. We show that $ \CatCu$ is a full, reflective subcategory of $ \CatW$ . One can then easily deduce properties of $ \CatCu$ from respective properties of $ \CatW$ , e.g.\ the existence of tensor products and inductive limits. The advantage is that constructions in $ \CatW$ are much easier since the objects are purely algebraic. For every (local) \ca{} $ A$ , the classical Cuntz semigroup $ W(A)$ together with a natural auxiliary relation is an object of $ \CatW$ . This defines a functor from \ca{s} to $ \CatW$ which preserves inductive limits. We deduce that the assignment $ A\mapsto\Cu(A)$ defines a functor from \ca{s} to $ \CatCu$ which preserves inductive limits. This generalizes a result from \cite{CowEllIva08CuInv}. We also develop a theory of $ \CatCu$ -semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing \ca{} has a natural product giving it the structure of a $ \CatCu$ -semiring. For \ca{s}, it is an important regularity property to tensorially absorb a strongly self-absorbing \ca{}. Accordingly, it is of particular interest to analyse the tensor products of $ \CatCu$ -semigroups with the $ \CatCu$ -semiring of a strongly self-absorbing \ca{}. This leads us to define solid'\'''\'' $ \CatCu$ -semirings (adopting the terminology from solid rings), as those $ \CatCu$ -semirings $ S$ for which the product induces an isomorphism between $ S\otimes_\CatCu S$ and $ S$ . This can be considered as an analog of being strongly self-absorbing for $ \CatCu$ -semirings. As it turns out, if a strongly self-absorbing \ca{} satisfies the UCT, then its $ \CatCu$ -semiring is solid. We prove a classification theorem for solid $ \CatCu$ -semirings. This raises the question of whether the Cuntz semiring of every strongly self-absorbing \ca{} is solid. If $ R$ is a solid $ \CatCu$ -semiring, then a $ \CatCu$ -semigroup $ S$ is a semimodule over $ R$ if and only if $ R\otimes_{\CatCu}S$ is isomorphic to $ S$ . Thus, analogous to the case for \ca{s}, we can think of semimodules over $ R$ as $ \CatCu$ -semigroups that tensorially absorb $ R$ . We give explicit characterizations when a $ \CatCu$ -semigroup is such a semimodule for the cases that $ R$ is the $ \CatCu$ -semiring of a strongly self-absorbing \ca{} satisfying the UCT. For instance, we show that a $ \CatCu$ -semigroup $ S$ tensorially absorbs the $ \CatCu$ -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.