Abstract bivariant Cuntz semigroups II

Abstract

<jats:title>Abstract</jats:title><jats:p>We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative <jats:inline-formula id="j_forum-2018-0285_ineq_9999_w2aab3b7e2689b1b6b1aab1c15b1b1Aa"><jats:alternatives><jats:tex-math>{\mathrm{Cu}}</jats:tex-math></jats:alternatives></jats:inline-formula>-semirings, and we obtain, under mild conditions, a new characterization of solid <jats:inline-formula id="j_forum-2018-0285_ineq_9998_w2aab3b7e2689b1b6b1aab1c15b1b3Aa"><jats:alternatives><jats:tex-math>{\mathrm{Cu}}</jats:tex-math></jats:alternatives></jats:inline-formula>-semirings <jats:italic>R</jats:italic> by the condition that <jats:inline-formula id="j_forum-2018-0285_ineq_9997_w2aab3b7e2689b1b6b1aab1c15b1b7Aa"><jats:alternatives><jats:tex-math>{R\cong\llbracket R,R\rrbracket}</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>

Abstract Bivariant Cuntz Semigroups

“…In particular, this product equips S, S with the structure of a (not necessarily commutative) Cu-semiring. These structures will be further analysed in a subsequent paper; see [APT18b].…”

Cuntz semigroups of ultraproduct C∗‐algebras

“…They defined the category of abstract Cuntz semigroups (also called ‐semigroups) and showed that the process of assigning to each ‐algebra the (concrete) Cuntz semigroup of its stabilization is a sequentially continuous functor. The properties of the category and its relation with the category of ‐algebras was further developed in [3–5]. It is therefore interesting to seek efficient methods to compute the Cuntz semigroup of products and ultraproducts.…”

Cuntz semigroups of ultraproduct C*-algebras

“…They defined the category Cu of abstract Cuntz semigroups (also called Cu-semigroups) and showed that the process of assigning to each C * -algebra A the (concrete) Cuntz semigroup of its stabilisation is a sequentially continuous functor. The properties of the category Cu and its relation with the category of C * -algebras was further developed in [APT18], [APT17], and [APT20]. It is therefore interesting to seek efficient methods to compute the Cuntz semigroup of products and ultraproducts.…”