The Cuntz semigroup of unital commutative AI-algebras

Eduard, Vilalta,

doi:10.48550/arxiv.2104.08165

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…Then Lsc(X, M) is a Cuntz category semigroup whenever X is a second-countable compact Hausdorff space with finite covering dimension and M is countably based (see [4,Theorem 5.17]). It was also shown in [66,Corollary 4.22] that Lsc(X, N) is a Cuntz category object when X is a compact metric space. It was proved in [14,Theorem 10.1] and [38,Theorem 6.11] that Cu(C 0 (X)) ≅ Lsc(X, N), via the rank map, if X is [0, 1] or (0, 1].…”

Section: The Cuntz Categorymentioning

confidence: 98%

Colored isomorphism of classifiable C-algebras

Elliott¹,

Jeffrey²

2022

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

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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

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Section: The Cuntz Categorymentioning

confidence: 98%

Colored isomorphism of classifiable C-algebras

Elliott¹,

Jeffrey²

2022

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

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“…Remark 5.2. A Cu-semigroup S is said to be sup-semilattice ordered if suprema exist and we have x + (y ∨ z) = (x + y) ∨ (x + z) for every x, y, z ∈ S; see, for example, [Vil21].…”

Section: Property (V)mentioning

confidence: 99%

The Global Glimm Property

Thiel¹,

Eduard²

2022

Preprint

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It is known that a 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 * -algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is idealfiltered (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 *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 * -algebras.

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The Cuntz semigroup of unital commutative AI-algebras

Cited by 2 publications

References 12 publications

Colored isomorphism of classifiable C-algebras

Colored isomorphism of classifiable C-algebras

The Global Glimm Property

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