The Cuntz semigroup of unital commutative AI-algebras

It is known that a C ∗ C^* -algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a C ∗ C^* -algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is ideal-filtered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered). We show that ideal-filteredness and property (V) are automatic for C ∗ C^* -algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of C ∗ C^* -algebras.

show abstract

Colored isomorphism of classifiable C-algebras

Elliott¹,

Jeffrey²

2022

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

It is shown that the coloured isomorphism class of a unital, simple, Z-stable, separable amenable C * -algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex. n k=1 u k ψϕ(a)u * k = a and n k=1 v k ϕψ(b)v * k = b for all a ∈ A and all b ∈ B.The present notion, coloured isomorphism, on the other hand, has order zero maps at the level of ultrapowers A ω and B ω (and so includes the case that the ultrapower maps are induced by a sequence of order zero maps at the level

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

The Cuntz semigroup of unital commutative AI-algebras

Cited by 3 publications

References 24 publications

Fraïssé theory for Cuntz semigroups

Fraïssé theory for Cuntz semigroups

The Global Glimm Property

Colored isomorphism of classifiable C-algebras

Contact Info

Product

Resources

About