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