On Nuclearity of $C*$-algebras of Fell Bundles over Étale Groupoids

Takeishi, Takuya

doi:10.4171/prims/132

Cited by 13 publications

(14 citation statements)

References 8 publications

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“…If C * r (G, Σ) is nuclear, then by [25,Theorem 5.4], G must be amenable. Since G is amenable, it acts properly on a continuous field H of affine Euclidean spaces by [26,Lemme 3.5].…”

Section: Preliminariesmentioning

confidence: 99%

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Cartan subalgebras and the UCT problem

Barlak

2017

Advances in Mathematics

101

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Abstract. We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us to deduce, among other things, that a crossed product O 2 ⋊ α Z p satisfies the UCT if there is some automorphism γ of O 2 with the property that γ(D 2 ) ⊆ O 2 ⋊ α Z p is regular, where D 2 denotes the canonical masa of O 2 . We prove that this condition is automatic if γ(D 2 ) ⊆ O 2 ⋊ α Z p is not a masa or α(γ(D 2 )) is inner conjugate to γ(D 2 ). Finally, we relate the UCT problem for separable, nuclear, M 2 ∞ -absorbing C*-algebras to Cartan subalgebras and order two automorphisms of O 2 .

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“…If C * r (G, Σ) is nuclear, then by [25,Theorem 5.4], G must be amenable. Since G is amenable, it acts properly on a continuous field H of affine Euclidean spaces by [26,Lemme 3.5].…”

Section: Preliminariesmentioning

confidence: 99%

“…The proof of this theorem relies heavily on Tu's results and techniques. To make these applicable to our setup, we make use of recent results by Takeishi [25] and by van Erp and Williams [5].…”

Section: Introductionmentioning

confidence: 99%

Cartan subalgebras and the UCT problem

Barlak

2017

Advances in Mathematics

101

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“…We note the following theorem of Takeishi [34] which classifies when C * r (Σ; G) is nuclear. The case when the twist is trivial can be found in [7, Theorem 5.6.18], and for more general groupoids in [1, Corollary 6.2.14].…”

Section: Theorem 22 (Cf [3 Corollary 27]) Let D Be a Cartan Subalgebra Of A Unital C * -Algebra A With Faithful Conditional Expectationmentioning

confidence: 99%

“…Takeishi [34] showed nuclearity of A is equivalent to the amenability of G. Let G (0) ⊆ H ⊆ G be an open subgroupoid, and let Σ H → H be the corresponding subtwist of Σ → G. In Theorem 4.2, amenability of G is used to show that, viewing…”

Section: Introductionmentioning

confidence: 99%

Intermediate C*-algebras of Cartan embeddings

Brown

Exel

Fuller

et al. 2021

Proc. Amer. Math. Soc. Ser. B

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Let A A be a C ∗ ^* -algebra and let D D be a Cartan subalgebra of A A . We study the following question: if B B is a C ∗ ^* -algebra such that D ⊆ B ⊆ A D \subseteq B \subseteq A , is D D a Cartan subalgebra of B B ? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A A onto B B , and the case when A A is nuclear and D D is a C ∗ ^* -diagonal of A A . In both cases there is a one-to-one correspondence between the intermediate C ∗ ^* -algebras B B , and a class of open subgroupoids of the groupoid G G , where Σ → G \Sigma \rightarrow G is the twist associated with the embedding D ⊆ A D \subseteq A .

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“…Proof. If G is amenable, then C * (G, E ) is amenable by [27,Theorem 4.1] and Lemma 6.2. Since N T X ∼ = C * (G, E ) by Theorem 5.1, we have N T X nuclear, and then N O X (as defined in [26]) is nuclear because it is a quotient of N T X .…”

Section: Applicationsmentioning

confidence: 99%

Groupoid Fell bundles for product systems over quasi-lattice ordered groups

Rennie¹,

Robertson²,

Sims³

2017

Math. Proc. Camb. Phil. Soc.

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Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica-Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz-Nica-Pimsner algebra of the product system. We apply these results to improve on existing sufficient conditions for nuclearity of the Nica-Toeplitz algebra and the Cuntz-Nica-Pimsner algebra, and for the Cuntz-Nica-Pimsner algebra to coincide with its co-universal quotient.2010 Mathematics Subject Classification. 46L05.

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On Nuclearity of $C*$-algebras of Fell Bundles over Étale Groupoids

Cited by 13 publications

References 8 publications

Cartan subalgebras and the UCT problem

Cartan subalgebras and the UCT problem

Intermediate C*-algebras of Cartan embeddings

Groupoid Fell bundles for product systems over quasi-lattice ordered groups

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