Amalgamated free products of <i>C</i>*-bundles

Blanchard, Etienne

doi:10.1017/s0013091506001556

Cited by 6 publications

(15 citation statements)

References 29 publications

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“…We conclude that RR(A) ≤ n, and hence RR(A) ≤ RR(B). We prove (7). Observe that ι ⊗ id Mn : M n (A) → M n (A) is also positive existential, and so is its unitization.…”

Section: Proofmentioning

confidence: 84%

“…In the general case, permanence properties of this kind were systematically studied in [27] and, in a different setting, in Section 2 of [18], in the context of continuous model theory. For the classes in (6), (7) and (8), the result is new even in the separable case. (1) Let D be either a strongly self-absorbing C * -algebra or the compact operators.…”

Section: Proofmentioning

confidence: 90%

“…Let A be as in the statement, and let B ⊆ A be a separable subalgebra. By Subsection 2.4.7 in [8] (see also Lemma 4.3 in [7]), there exists a separable C(Y )-subalgebra C ⊆ A containing B. An inspection of the proof of the said proposition shows that the fibers of C are also potentially D-absorbing.…”

Section: Proofmentioning

confidence: 99%

“…Most of these are well known: Theorem 3.15 just collects the relevant references, and provides a proof where one is needed. The only original results are (2), (7), (8), and (9). Theorem 3.15.…”

Section: Structure Of the Crossed Productmentioning

confidence: 99%

“…For (7), one uses an identical argument as for (9), replacing O 2 by M n ∞ and noting that unique n-divisibility of an abelian group is equivalent to absorbing Z[1/n] tensorially. Also, (8) follows by applying (7) for all n ∈ N.…”

Section: Proofmentioning

confidence: 99%

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Rokhlin dimension: duality, tracial properties, and crossed products

Gardella¹,

Hirshberg²,

Santiago³

2019

Ergod. Th. Dynam. Sys.

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We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C * -algebras, including: D-absorbing C * -algebras, where D is a strongly self-absorbing C * -algebra, stable C * -algebras, C * -algebras with finite nuclear dimension (or decomposition rank), C * -algebras with finite stable rank (or real rank), and C * -algebras whose K-theory is either trivial, rational, or n-divisible for n ∈ N. The combination of nuclearity and the UCT is also shown to be preserved by these actions. Some of these results are new even in the wellstudied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property.At the core of our arguments is a certain local approximation of the crossed product by a continuous C(X)-algebra with fibers that are stably isomorphic to the underlying algebra. The space X is computed in some cases of interest, and we use its description to construct a Z 2 -action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

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“…We conclude that RR(A) ≤ n, and hence RR(A) ≤ RR(B). We prove (7). Observe that ι ⊗ id Mn : M n (A) → M n (A) is also positive existential, and so is its unitization.…”

Section: Proofmentioning

confidence: 84%

Section: Proofmentioning

confidence: 90%

Section: Proofmentioning

confidence: 99%

Section: Structure Of the Crossed Productmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

See 3 more Smart Citations

Rokhlin dimension: duality, tracial properties, and crossed products

Gardella¹,

Hirshberg²,

Santiago³

2019

Ergod. Th. Dynam. Sys.

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Residual finiteness for central pushouts

Chirvăsitu

2021

Proc. Amer. Math. Soc.

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Fields of locally compact quantum groups: Continuity and pushouts

Chirvăsitu

2021

Int. J. Math.

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We prove that discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of [Formula: see text]-algebras are again free via a Fell-topology characterization for [Formula: see text]-field continuity, recovering a result of Blanchard’s in a somewhat more general setting.

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Amalgamated free products of C*-bundles

Abstract: Given two unital continuous C * -bundles, A and B, over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).

Cited by 6 publications

References 29 publications

Rokhlin dimension: duality, tracial properties, and crossed products

Rokhlin dimension: duality, tracial properties, and crossed products

Residual finiteness for central pushouts

Fields of locally compact quantum groups: Continuity and pushouts

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