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.

arXiv comment34 pages; to appear in Journal of the London Mathematical Society

References in corpus (6)

• A simple C*-algebra with a finite and an infinite projection
• Tensor products and regularity properties of Cuntz semigroups
• Nuclear dimension and Z-stability of non-simple C*-algebras
• Group C*-algebras as decreasing intersection of nuclear C*-algebras
• The Cuntz semigroup and stability of close C*-algebras
• On C*-algebras which cannot be decomposed into tensor products with both factors infinite-dimensional

Cited by in corpus (2)

• A unitary Cuntz semigroup for C*-algebras of stable rank one
• Nowhere scattered multiplier algebras