Amalgamated free product 𝐶*-algebras and 𝐾𝐾-theory

Germain, Emmanuel

doi:10.1090/fic/012/04

Cited by 8 publications

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“…This homomorphism was considered in [8] and [9] and the following important result by E. Germain is a consequence of the Puppe exact sequences and Bott periodicity. We are heading against the main result in this section, Theorem 3.3, but first we need some more results and definitions.…”

Section: Kk-theory and Germain's Homomorphismmentioning

confidence: 99%

“…In that direction more general results in KK-theory are proved in [6] and [15]. About a decade later E. Germain obtained results in increasing generality in [7], [8] and [9]. Conjecture 1.1 in its final form appeared for the first time in [9].…”

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confidence: 98%

“…About a decade later E. Germain obtained results in increasing generality in [7], [8] and [9]. Conjecture 1.1 in its final form appeared for the first time in [9]. E. Germain defined a homomorphism G :…”

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confidence: 99%

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KK ‐theory of amalgamated free products of C *‐algebras

Eliasen¹

2007

Mathematische Nachrichten

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Key words C * -algebras, KK-theory, amalgamated free products, absorbing homomorphisms, countable direct sums of matrix algebras MSC (2000) 19K35 J. Cuntz has conjectured the existence of two cyclic six terms exact sequences relating the KK-groups of the amalgamated free product A1 * B A2 to the KK-groups of A1, A2 and B. First we establish automatic existence of strict and absorbing homomorphisms. Then we use this result to verify the conjecture when B is a countable direct sum of matrix algebras and the embeddings of B into A1 and A2 are quasiunital. Inspired by the proof we achieve the following nice classification result: A separable C * -algebra B is a countable direct sum of matrix algebras if and only if the unitary group of the multiplier algebra UM(B) is compact in the strict topology. Finally we prove the conjecture when the amalgamated free product has the property that any asymptotically split extension of A1 * B A2 is split.

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Section: Kk-theory and Germain's Homomorphismmentioning

confidence: 99%

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KK ‐theory of amalgamated free products of C *‐algebras

Eliasen¹

2007

Mathematische Nachrichten

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“…& Remark. Germain [10] defines a topological property called ''relative K-nuclearity'' for inclusions of unital C Ã -algebras TCS 1 ; TCS 2 : This property mostly serves as a sufficient condition for the full free product with amalgamation S ¼ S 1 Ã T S 2 to be ''dominated in K-theory'' by the reduced free product, which is equivalent to K-amenability in the case of C Ã -algebras of discrete quantum groups. However, when T is infinite-dimensional very little is known about relative K-nuclearity, even in our case of inclusions of C Ã -algebras of amenable discrete quantum groups.…”

Section: The Bass-serre Tree and K-amenabilitymentioning

confidence: 99%

K-amenability for amalgamated free products of amenable discrete quantum groups

Vergnioux¹

2004

Journal of Functional Analysis

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The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some useful properties of subgroups and amalgamated free products of discrete quantum groups. Using these properties and a quantum analogue of the Bass-Serre tree, we establish the K-amenability of amalgamated free products of amenable discrete quantum groups. rThe main goal of this paper is to prove the K-amenability of an amalgamated free product of amenable discrete quantum groups. The notion of K-amenability was first introduced by Cuntz [8] for discrete groups: the aim was to give a simpler proof to a result of Pimsner and Voiculescu [15] calculating the K-theory of the reduced C Ã -algebra of a free group. Cuntz proves that the K-theory of the reduced and full C Ã -algebras of the free groups are the same, and gives in [7] a very simple way to compute it in the full case.Julg and Valette [11] extend the notion of K-amenability to the locally compact case and establish the K-amenability of locally compact groups acting on trees with amenable stabilizers. By a construction of Serre [16], this includes the case of ARTICLE IN PRESS

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“…As applications, we obtain several results in KK-theory. Firstly, we show that our strong relative nuclearity implies Germain's relative K-nuclearity [Ge2], which is a relative counterpart of Skandalis's K-nuclearity [Sk]. In [Sk], Skandalis proved that nuclearity implies K-nuclearity by using Kasparov's generalized Voiculescu theorem.…”

Section: Introductionmentioning

confidence: 98%

Relative nuclearity for C⁎-algebras and KK-equivalences of amalgamated free products

Hasegawa

2015

Journal of Functional Analysis

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We prove a relative analogue of equivalence between nuclearity and CPAP. In its proof, the notion of weak containment for C * -correspondences plays an important role. As an application we prove KK-equivalence between full and reduced amalgamated free products of C * -algebras under a strengthened variant of 'relative nuclearity'. 1 4 KEI HASEGAWA introduced by Jian and Sepideh [JS], as well as relative amenability for von Neumann algebras ([Po][AD3] [OP]). In §5, we see that the examples listed above are actually (strongly) nuclear. §6 is devoted to proving Weyl-von Neumann-Voiculescu type results. Applications in KK-theory are given in the last three sections. In §7 we prove that strong relative nuclearity implies relative K-nuclearity. The proof of Theorem C and Theorem D are given in §8. In the final section, we establish six term exact sequences in KK-theory of reduced amalgamated free products and a KK-equivalence result for HNN-extensions.Notation. We basically follow the notation of Brown and Ozawa's book [BO]. For a C * -algebra A we denote by 1 A the unit of the multiplier algebra M(A) of A. The symbols B(H) and K(H) stand for the set of bounded operators and the set of compact ones on a Hilbert space H, respectively. We use the symbol ⊙ to denote the algebraic tensor product over C. For C *algebras A and B we denote by A ⊗ B and A ⊗ max B the minimal and the maximal tensor products, respectively. For a von Neumann algebra M , we denote by M * the (unique) predual of M . The von Neumann algebraic tensor product of M and another von Neumann algebra N is denoted by M⊗ N . We denote by CP(

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Amalgamated free product 𝐶*-algebras and 𝐾𝐾-theory

Cited by 8 publications

References 0 publications

KK ‐theory of amalgamated free products of C *‐algebras

KK ‐theory of amalgamated free products of C *‐algebras

K-amenability for amalgamated free products of amenable discrete quantum groups

Relative nuclearity for C⁎-algebras and KK-equivalences of amalgamated free products

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