The KK-theory of amalgamated free products

Fima, Pierre; Germain, Emmanuel

doi:10.1016/j.aim.2020.107174

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…For information regarding K-theory and KK-theory we refer to [1]. A detailed introduction to reduced and universal amalgamated free products can be found in [25,Section 2].…”

Section: K-theory and The Uct For Amalgamated Free Productsmentioning

confidence: 99%

“…The identification of the K-theory groups of amalgamated free products has been achieved by Fima and Germain in [25]. Concretely speaking, [25, Corollary 4.12] implies that if A 1 , B, A 2 are unital separable C * -algebras, with unital inclusions α 1 ∶ B → A 1 and α 2 ∶ B → A 2 , then for every separable C * -algebra C there are the following six-term short exact sequences in K-theory and KK-theory:…”

Section: K-theory and The Uct For Amalgamated Free Productsmentioning

confidence: 99%

“…Here the map γ is the dual of the KK-pairing K * (A) ⊗ KK * (A, C) → K * (C), and as such is natural in A and C. We shall now record a consequence of the results of [25].…”

Section: K-theory and The Uct For Amalgamated Free Productsmentioning

confidence: 99%

“…Germain was first to establish unconditional results on the K-theory of free product C * -algebras in [27,28]. An extension of this work to amalgamated free products was only recently obtained by Fima and Germain in [25].…”

Section: Introductionmentioning

confidence: 99%

“…Our key ingredients to establish this KK-equivalence are Fima's and Germain's work on amalgamated free products [25], combined with the dévissage procedure for graph product C * -algebras obtained by Caspers and Fima [9]. The K-theory computations for full group C * -algebras of right-angled Coxeter groups then make use of six-term exact sequences established in [25].…”

Section: Introductionmentioning

confidence: 99%

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Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Raum¹,

Skalski²

2021

Preprint

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Exploiting the graph product structure and results concerning amalgamated free products of C * -algebras we provide an explicit computation of the K-theoretic invariants of right-angled Hecke C * -algebras, including concrete algebraic representants of a basis in K-theory. On the way, we show that these Hecke algebras are KK-equivalent with their undeformed counterparts and satisfy the UCT. Our results are applied to study the isomorphism problem for Hecke C * -algebras, highlighting the limits of K-theoretic classification, both for varying Coxeter type as well as for fixed Coxeter type.

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“…For information regarding K-theory and KK-theory we refer to [1]. A detailed introduction to reduced and universal amalgamated free products can be found in [25,Section 2].…”

Section: K-theory and The Uct For Amalgamated Free Productsmentioning

confidence: 99%

Section: K-theory and The Uct For Amalgamated Free Productsmentioning

confidence: 99%

“…Here the map γ is the dual of the KK-pairing K * (A) ⊗ KK * (A, C) → K * (C), and as such is natural in A and C. We shall now record a consequence of the results of [25].…”

Section: K-theory and The Uct For Amalgamated Free Productsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Raum¹,

Skalski²

2021

Preprint

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Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Raum

Skalski

2022

Advances in Mathematics

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Untitled

2022

Trans. Amer. Math. Soc. Ser. B

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Motivated by the theory of Cuntz-Krieger algebras we define and study C * -algebras associated to directed quantum graphs. For classical graphs the C * -algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear.We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to KK-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C * -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of KK-theory.We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation.We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.

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The KK-theory of amalgamated free products

Cited by 7 publications

References 12 publications

Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Classifying right-angled Hecke C*-algebras via K-theoretic invariants

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