Cuntz semigroups of ultraproduct C∗‐algebras

Abstract

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C *-algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C *-algebras. As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C *-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.

Abstract Bivariant Cuntz Semigroups

“…We refer to this functor as the τ -construction. In [APT18c] we show that the τ -construction can also be used to compute the Cuntz semigroup of ultraproduct C * -algebras. In our setting, the functor τ applied to the semigroup of generalized Cu-morphisms Cu[S, T ] yields the internal-hom of S and T ; see Definition 5.3.…”

Abstract bivariant Cuntz semigroups II

“…Further, we show in [APT18c] that Cu is complete and cocomplete, and that the functor that assigns to each C * -algebra its Cuntz semigroup is compatible with products and ultraproducts.…”

The Global Glimm Property

“…A Cu-morphism is an order-preserving monoid morphism that also preserves suprema of increasing sequences and the way-below relation. The resulting category Cu of Cu-semigroups and Cu-morphisms is bicomplete (it has arbitrary limits and colimits); see [APT20c]. Further, every *homomorphism ϕ : A → B induces a Cu-morphism Cu(ϕ) : Cu(A) → Cu(B), and the functor Cu : C * → Cu is continuous; see [APT18, Corollary 3.2.9].…”