Free products with amalgamation over central 𝐶*-subalgebras

Courtney, Kristin; Shulman, Tatiana

doi:10.1090/proc/14746

Cited by 5 publications

(4 citation statements)

References 31 publications

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“…Specifically, the main result of [19] is that pushouts of separable commutative C * -algebras are RFD. More generally, the main theorem of [13] proves this for central pushouts A * C B with A and B separable and strongly RFD, i.e. such that all of their quotients are RFD.…”

Section: Introductionmentioning

confidence: 80%

“…The literature on RFD C * -algebras is rather substantial, as is that on RF groups; so much so, in fact, that it would be impossible to do it justice. For samplings (the best a short note such as this one can do) we direct the reader to, say, [14,2,10,3,17,19,20,13,24] (for RFD C * -algebras) and [5,4,6] or [21, §6.5], [23,Chapters 6,14,15], [12,Chapter 2] (for RF groups), and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…We are concerned here with the types of "permanence" properties for residual finiteness as studied, say, in [10,3,19,20,13,24,5], to the effect that various types of pushouts (also known as amalgamated free products) of RFD or RF objects are again such. Specifically, the main result of [19] is that pushouts of separable commutative C * -algebras are RFD.…”

Section: Introductionmentioning

confidence: 99%

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Residual finiteness for central pushouts

Chirvăsitu¹

2020

Preprint

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We prove that pushouts A * C B of residually finite-dimensional (RFD) C * -algebras over central subalgebras are always residually finite-dimensional provided the fibers A p and B p , p ∈ spec C are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman. This then allows us to prove that certain central pushouts of amenable groups have RFD group C * -algebras. Along the way, we discuss the problem of when, given a central group embedding H ≤ G, the resulting C * -algebra morphism is a continuous field: this is always the case for amenable G but not in general.

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Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Residual finiteness for central pushouts

Chirvăsitu¹

2020

Preprint

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“…Definition 4.9. (Following [21] in the case of C*-algebras.) An algebra A is strongly residually finite-dimensional (strongly RFD) if A/I is residually finitedimensional for every ideal I of A.…”

Section: 2mentioning

confidence: 99%

The finite dual coalgebra as a quantization of the maximal spectrum

Reyes¹

2021

Preprint

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In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A • act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. We investigate cases where the finite dual coalgebra of a twisted tensor product is a crossed product coalgebra of the respective finite duals. This is achieved by interpreting the finite dual as a topological dual. Sufficient conditions for this result to be applied to Ore extensions, smash product algebras, and crossed product bialgebras are described. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. Finally, we implement these techniques for quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions.

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The finite dual coalgebra as a quantization of the maximal spectrum

Reyes

2024

Journal of Algebra

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Free products with amalgamation over central 𝐶*-subalgebras

Cited by 5 publications

References 31 publications

Residual finiteness for central pushouts

Residual finiteness for central pushouts

The finite dual coalgebra as a quantization of the maximal spectrum

The finite dual coalgebra as a quantization of the maximal spectrum

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