K-theory for the tame C⁎-algebra of a separated graph

Ara, Pere; Exel, Ruy

doi:10.1016/j.jfa.2015.05.010

Cited by 8 publications

(12 citation statements)

References 17 publications

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“…By injectivity of ∂ 1 and exactness of the sequence K 1 (C * L) Ker(Id − α * ) = Z.p F , where p F denotes the constant map taking the value p F on Z, it corresponds to the class [1] of 1 in K 0 (C * B) (see Corollary 1). Moreover, we know from [28, Lemma 2] that ∂ 1 [u] = − [1].…”

Section: Computations On the Right-hand Sidementioning

confidence: 99%

“…On the analytical side, for the case of classical lamplighter groups

L = (Z / n double-struckZ) ≀ Z

computations for the K ‐theory of

C^{*} L

have implicitly appeared in different contexts (see, for instance [, Example 6.10; , Theorem 15; , Theorem 4.12; , Section 6.5], or [, Example 3]). Our approach is different from the previous ones in two ways: first, we treat lamplighter groups of finite groups in full generality, and second, we specify generators for the K ‐groups and show their relevance for the Baum–Connes assembly map.…”

Section: Introductionmentioning

confidence: 99%

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K ‐homology and K ‐theory for the lamplighter groups of finite groups

Flores

Pooya

Valette

2017

Proc. London Math. Soc.

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Let F be a finite group. We consider the lamplighter group L=F≀Z over F. We prove that L has a classifying space for proper actions normalE̲L which is a complex of dimension 2. We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that the assembly map μiL:KiLfalse(E̲0.16emLfalse)→Kifalse(C∗Lfalse)false(i=0,1false) is an isomorphism. Actually, K0false(C∗Lfalse) is free abelian of countable rank, with an explicit basis consisting of projections in C∗L, while K1false(C∗Lfalse) is infinite cyclic, generated by the unitary of C∗L implementing the shift. Finally we show that, for F abelian, the C∗‐algebra C∗L is completely characterized by |F| up to isomorphism.

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Section: Computations On the Right-hand Sidementioning

confidence: 99%

“…On the analytical side, for the case of classical lamplighter groups

L = (Z / n double-struckZ) ≀ Z

computations for the K ‐theory of

C^{*} L

Section: Introductionmentioning

confidence: 99%

K ‐homology and K ‐theory for the lamplighter groups of finite groups

Flores

Pooya

Valette

2017

Proc. London Math. Soc.

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“…More information around partial actions may be found in the surveys [45,115,116,170,250,251,260]. Notice also that partial actions and/or related structures have been mentioned in [12,13,24,46,62,67,90,92,142,154,161,166,167].…”

Section: (G U(a))→pic(a G )→Pic(a) G →H 2 (G U(a))→b(a/a α )→ → H 1mentioning

confidence: 99%

Recent developments around partial actions

Dokuchaev

2018

São Paulo J. Math. Sci.

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We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids, the globalization problem for partial actions, Morita theory of partial actions, twisted partial actions, partial projective representations and the Schur multiplier, cohomology theories related to partial actions, Galois theoretic results, ring theoretic properties and ideals of partial crossed products. Among the applications we consider in more detail the case of the Carlsen-Matsumoto C *-algebra related to an arbitrary subshift, but also mention many others. The total number of publications directly related to partial actions and partial representations is more than 130, so that it is impossible even to describe briefly the content of all of them within the constraints of the present survey. Thus, the majority of them are only cited with respects to specific topics, trying to give an idea about the involved matter. In order to complete the picture, we refer the reader to a recent book by Ruy Exel, to our previous surveys, as well as to those by other authors.

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“…Indeed, as shown in [10], there are examples of crossed-product functors that are not correspondence functors for which the above result fails even for crossed products by ordinary actions. In the recent paper by Ara and Exel [4] (see in particular Corollary 6.9) the authors have applied the result by McClanahan to some interesting partial actions of free groups associated to separated graphs in order to deduce that certain full and reduced crossed products have the same K-theory (and the K-theory is effectively computed in [4]). By the above result these computations extend to the respective exotic crossed product related to any given correspondence crossed-product functor.…”

Section: More Generally Every Other Morita Enveloping Actionmentioning

confidence: 99%

Maximality of dual coactions on sectional $C^{*}$-algebras of Fell bundles and applications

Buss

Echterhoff

2016

Studia Math.

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Abstract. In this paper we give a simple proof of the maximality of dual coactions on full cross-sectional C * -algebras of Fell bundles over locally compact groups. This result was only known for discrete groups or for saturated (separable) Fell bundles over locally compact groups. Our proof, which is derived as an application of the theory of universal generalised fixed-point algebras for weakly proper actions, is different from these previously known cases and works for general Fell bundles over locally compact groups. As applications we extend certain exotic crossed-product functors in the sense of Baum, Guentner and Willett to the category of Fell bundles and the category of partial actions and we obtain results about the K-theory of (exotic) cross-sectional algebras of Fell-bundles over K-amenable groups. As a bonus, we give a characterisation of maximal coactions of discrete groups in terms of maximal tensor products.

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K-theory for the tame C⁎-algebra of a separated graph

Abstract: Abstract. A separated graph is a pair (E, C) consisting of a directed graph E and a set C = v∈E 0 C v

Cited by 8 publications

References 17 publications

K ‐homology and K ‐theory for the lamplighter groups of finite groups

K ‐homology and K ‐theory for the lamplighter groups of finite groups

Recent developments around partial actions

Maximality of dual coactions on sectional $C^{*}$-algebras of Fell bundles and applications

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