Alex Kumjian
The author provides some definitions and structural results about Fell bundles, defined as C^*-algebra bundles over topological groupoids. Such bundles are a mutual generalization of semi-direct products of groups with C^*-algebras and C^*-algebra bundles over topological spaces. In particular a Morita equivalence theorem with semi-direct products is established.
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Given an action of a groupoid by isomorphisms on a Fell bundle (over another groupoid), we form a semidirect-product Fell bundle, and prove that its $C^{*}$-algebra is isomorphic to a crossed product.
Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.
We establish conditions under which the universal and reduced norms coincide for a Fell bundle over a groupoid. Specifically, we prove that the full and reduced C*-algebras of any Fell bundle over a measurewise amenable groupoid coincide, and also that for a groupoid G whose orbit space is T_0, the full and reduced algebras of a Fell bundle over G coincide if the full and reduced algebras of the restriction of the bundle to each isotropy group coincide.
We define the Zappa-Szép product of a Fell bundle by a groupoid, which turns out to be a Fell bundle over the Zappa-Szép product of the underlying groupoids. Under certain assumptions, every Fell bundle over the Zappa-Szép product of groupoids arises in this manner. We then study the representation associated with the Zappa-Szép product Fell bundle and show its relation to covariant representations. Finally, we study the associated universal C*-algebra, which turns out to be a C*-blend, generalizing an earlier result about the Zappa-Szép product of groupoid C*-algebras. In the case of discrete…
In this paper, we show that if E is a Fell bundle over an amenable étale locally compact Hausdorff groupoid such that every fiber on the unit space is nuclear, then $C?^*r(E)$ is also nuclear. In order to show this result, we introduce (minimal) tensor products of Fell bundles with fixed $C^*$-algebras.
We investigate some consequences of a recent stabilization result of Ionescu, Kumjian, Sims, and Williams, which says that every Fell bundle $C^*$-algebra is Morita equivalent to a canonical groupoid crossed product. First we use the theorem to give conditions that guarantee the $C^*$-algebras associated to a Fell bundle are either nuclear or exact. We then show that a groupoid is exact if and only if it is "Fell exact", in the sense that any invariant ideal gives rise to a short exact sequence of reduced Fell bundle $C^*$-algebras. As an application, we show that extensions of exact groupoids…
We consider fiberwise singly generated Fell-bundles over etale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo-Martin-Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of traces on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a des…
Given a continuous open surjective morphism $π:G\to H$ of étale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its C*-algebra $C^*_r(E)$ is isomorphic to $C^*_r(G)$. This is related to results of Fell concerning C*-algebraic bundles over groups. The case $H=X$, a locally compact space, was treated earlier by Ramazan. We conclude that $C^*_r(G)$ is strongly Morita equivalent to a crossed product, the C*-algebra of a Fell bundle arising from an action of the groupoid $H$ on a C*-bundle over $H^0$. We apply the theory to groupoid morphisms obtained from ext…
Given a saturated Fell bundle A over an inverse semigroup S which is semi-abelian in the sense that the fibers over the idempotents of S are commutative, we construct a twisted etale groupoid L such that A can be recovered from L in a canonical way. As an application we recover most of Renault's recent result on the classification of Cartan subalgebras of C*-algebras through twisted etale groupoids.
We generalize Renault's notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable $C^*$-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the $C^*$-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel's approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid a…
C$^*$-algebras of Fell bundles over étale groupoids
本文在不使用 Renault 分解定理的前提下，构造了非饱和 Fell 丛在非豪斯多夫局部紧 étale 群胚上的全 C$^*$-代数，推广了 Muhly-Williams 标准构造。
A universal property for groupoid C*-algebras. II. Fell bundles
本文研究Hausdorff局部紧群状上的可能不饱和上半连续Fell包，建立其完整截面C*-代数在任意C*-代数上的表示的普遍性质，并证明其函子性和精确性，定义准轨道空间和映射，扩展Renault的积分与分解定理。
We study the groupoid C*-algebras associated to the equivalence relation induced by a quotient map on a locally compact Hausdorff space. This C*-algebra is always a Fell algebra, and if the quotient space is Hausdorff, it is a continuous-trace algebra. We show that the C*-algebra of a locally compact, Hausdorff and principal groupoid is a Fell algebra if and only if the groupoid is one of these relations, extending a theorem of Archbold and Somerset about etale groupoids. The C*-algebras of these relations are, up to Morita equivalence, precisely the Fell algebras with trivial Dixmier-Douady i…
We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff étale groupoids. We interpret these as actions on C*-algebras by Hilbert bimodules and describe the section algebras of these Fell bundles. Our constructions give saturated Fell bundles over non-Hausdorff étale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms. That is, the Packer-Raeburn Stabilisation Trick doe…
We study the $C^*$-algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer--Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced $C^*$-algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the $C^*$-algebra …
We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.
Rapid decay and localizability for Fell bundles over etale Groupoids
本文引入Fell bundles over局部紧Hausdorff étale groupoids的快速衰减性质，通过卷积范数分析控制，证明多项式增长的Fell bundles自然满足RDP，并探讨了RDP与动力系统的关系，揭示了群与groupoid理论间的边界。
Functoriality for groupoid and Fell bundle $C^*$-algebras
本文定义 étale 群胚及 Fell 丛的新态射，建立到 $C^*$-代数的函子，并刻画 Cartan 对、Weyl 群胚与扭曲间的范畴等价及余极限保持性。
A compactification of Fell is applied to locally compact non-Hausdorff groupoids and yields locally compact Hausdorff groupoids. In the etale case, this construction provides a geometric picture for the left-regular representations introduced by Khoshkam and Skandalis.
Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica-Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz-Nica-Pimsner algebra of the product system. We apply these results to improve on existing suf…
KMS states on the $\mathrm{C}^*$-algebras of Fell bundles over {é}tale groupoids
本文证明了étale群胚上饱和Fell丛的全C*-代数上KMS态与迷向群上Fell丛C*-代数的可测态场之间的一一对应，建立了积分-分解定理，并通过群胚交叉积、矩阵代数等例子加以验证。
Approximation properties of Fell bundles over inverse semigroups and non-Hausdorff groupoids
本文在逆半群上 Fell 丛的框架下证明 Fell 吸收技巧的类比，并利用 Exel 逼近性质证明全截面与约化截面 C*-代数同构，进而得到具有逼近性质且单位纤维为核的 Fell 丛的截面 C*-代数是核的，且结果同样适用于 étale 非豪斯多夫群胚。
We show that if E is an equivalence of upper semicontinuous Fell bundles B and C over groupoids, then there is a linking bundle L(E) over the linking groupoid L such that the full cross-sectional algebra of L(E) contains those of B and C as complementary full corners, and likewise for reduced cross-sectional algebras. We show how our results generalise to groupoid crossed-products the fact, proved by Quigg and Spielberg, that Raeburn's symmetric imprimitivity theorem passes through the quotient map to reduced crossed products.
If $p \colon \mathcal B\to G$ is a Fell bundle over an étale groupoid, then we show that there is an norm reducing injective linear map $j \colon C^*_r(G;\mathcal B)\to Γ_{0}(G;\mathcal B)$ generalizing the well know map $j \colon C^*_{r}(G)\to C_{0}(G)$ in the case of an étale groupoid.
Given a Fell bundle $\mathscr C\overset{q}{\to}Ξ$ over the discrete groupoid $Ξ$, we study the symmetry of the associated Hahn algebra $\ell^{\infty,1}(Ξ\!\mid\!\mathscr C)$ in terms of the isotropy subgroups of $Ξ$. We prove that $Ξ$ is symmetric (resp. hypersymmetric) if and only if all of the isotropy subgroups are symmetric (resp. hypersymmetric). We also characterize hypersymmetry using Fell bundles with constant fibers, showing that for discrete groupoids, 'hypersymmetry' equals 'rigid symmetry'.
Sections of Fell Bundles over Étale Groupoids
本文在无第二可数、局部紧或Hausdorff假设下，直接由截面卷积构造étale群胚上Fell丛的约化与本质C*-代数，避免j-映射，并推广Varela态射使构造函子化。
In this paper we study amenability, nuclearity and tensor products of $C^{\ast}$-Fell bundles by the method of induced representation theory.
In this paper, we prove that the induced representation theories of two equivalent Fell bundles are essentially identical; and we apply our results to carry the induced representation theory and imprimitivity theorems of saturated Fell bundles to arbitrary Fell bundles.
We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain regular Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the stable Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby-Smith twisted actions in the group case. As an application we describe twisted etale groupoid C*-algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sie…
In the third and latest paper in this series, we recover the imprimitivity theorems of Mansfield and Fell using our technique of Fell bundles over groupoids. Also, we apply the Rieffel Surjection of the first paper in the series to relate our version of Mansfield's theorem to that of an Huef and Raeburn, and to give an automatic amenability result for certain transformation Fell bundles.
We give a notion of equivalence for Fell bundles over groups, not necessarily saturated nor separable, and show that equivalent Fell bundles have Morita-Rieffel equivalent cross-sectional $C^*$-algebras. Our notion is originated in the context of partial actions and their enveloping actions. The equivalence between two Fell bundles is implemented by a bundle of Hilbert bimodules with some extra structure. Suitable cross-sectional spaces of such a bundle turn out to be imprimitivity bimodules for the cross-sectional $C^*$-algebras of the involved Fell bundles. We show that amenability is preser…
Ground states on the \(\mathrm{C}^*\)-algebras of Fell bundles over étale groupoids
该研究针对平展群胚上的 Fell 丛的 C*代数，建立其基态空间与对应边界群胚 C*代数态空间的仿射同胚，探讨基态与 KMS_∞ 态的关系，并将结果应用于两类扭曲 C*代数以得到基态的显式刻画。
Fell bundle ladder
本文利用阶梯技术在相关Fell丛的理想之间建立双射，揭示了不同Fell丛理想结构的对应关系。
Fell algebras, groupoids, and projections
本文构造紧谱且Dixmier-Douady不变量平凡的Fell代数，以区别于连续迹$C^*$-代数，并探讨局部Hausdorff谱上满投影的存在性及其与Wieler螺线管的联系。
We introduce notions of weak and strong equivalence for non-saturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional $C^*-$algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.
We consider Fell bundles over discrete groups, and the C*-algebra which is universal for representations of the bundle. We define deformations of Fell bundles, which are new Fell bundles with the same underlying Banach bundle but with the multiplication deformed by a two-cocycle on the group. Every graph algebra can be viewed as the C*-algebra of a Fell bundle, and there are are many cocycles of interest with which to deform them. We thus obtain many of the twisted graph algebras of Kumjian, Pask and Sims. We demonstate the utility of our approach to these twisted graph algebras by proving tha…
Imprimitivity theorems and self-similar actions on Fell bundles
该研究提出群胚在其他群胚及Fell丛上的自相似作用概念，推导得到这类动力学下的新非本原性定理，推广了此前涉及群与群胚作用的相关非本原性定理。
We give a formula for the Dixmier-Douady class of a continuous-trace groupoid crossed product that arises from an action of a locally trivial, proper, principal groupoid on a bundle of elementary $C^*$-algebras that satisfies Fell's condition.
We prove a version of Wordingham's theorem for left regular representations in the setting of Fell bundles of inverse semigroups and use this result to discuss the various associated cross sectional C*-algebras.
The Rieffel Correspondence for Equivalent Fell Bundles
本文研究等价 Fell 丛中理想的对应关系，建立了广义 Rieffel 对应。
Symmetry for algebras associated to Fell bundles over groups and groupoids
本文证明局部紧群上 Fell 丛对应的 Banach *-代数在群离散刚性对称时具有对称性，并将其推广到加权版本、积分核及离散群胚情形，同时研究了变换群胚的对称性与持久性。
The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding
本文提出了一种基于II型C*-代数的规范群oid构造，通过规范对角嵌入将代数嵌入到群oidC*-代数中，展示了其在K-理论和拓扑学中的应用。
Our goal in this paper and two sequels is to apply the Yamagami-Muhly-Williams equivalence theorem for Fell bundles over groupoids to recover and extend all known imprimitivity theorems involving groups. Here we extend Raeburn's symmetric imprimitivity theorem, and also, in an appendix, we develop a number of tools for the theory of Fell bundles that have not previously appeared in the literature.
We define weakly proper Fell bundles and construct exotic fixed point algebras for such bundles. Three alternative constructions of such algebras are given. Under a kind of freeness condition, one of our constructions implies that every exotic cross sectional C*-algebra of a weakly proper Fell bundle is Morita equivalent to an exotic fixed point algebras. The other constructions are used to show that ours generalizes that of Buss and Echterhoff on weakly proper actions on C*-algebras. We also generalize to Fell bundles the fact that every C*-action which is proper in Kasparov's sense is amenab…
Deformation of Fell bundles
本文通过直接变形Fell束并取横截代数，研究了局部紧群上Fell束的变形，并证明其与对偶共作用的变形等价。
We study the C*-algebras of Fell bundles. In particular, we prove the analogue of Renault's disintegration theorem for groupoids. As in the groupoid case, this result is the key step in proving a deep equivalence theorem for the C*-algebras of Fell bundles.
With each Fell bundle over a discrete group G we associate a partial action of G on the spectrum of the unit fiber. We discuss the ideal structure of the corresponding full and reduced cross-sectional C*-algebras in terms of the dynamics of this partial action.
This paper is aimed at investigating links between Fell bundles over Morita equivalent groupoids and their corresponding reduced C*-algebras. Mainly, we review the notion of Fell pairs over a Morita equivalence of groupoids, and give the analogue of the Renault's Equivalence Theorem for the reduced C*-algebras of equivalent Fell systems. Eventually, we will use this theorem to connect the reduced C*-algebra of an S^1-central groupoid extension to that of its associated Dixmier-Douady bundle.
Fell bundles and Haagerup properties
该研究为离散群上的Fell丛引入新的Haagerup型性质，探讨其与相关C*-代数Haagerup性质的关系，扩展了非扭曲情形下的相关结论。
Exactness and Fell bundles with the approximation property over inverse semigroups
本文研究了具有逼近性质的Fell包 over逆半群的减少横截代数的精确性，证明其等价于Fell包的单位纤维的精确性，并扩展了作者关于局部紧豪斯多夫第二可数群oid作用于可分C*-代数的结果。
W*-Amenability for Fell bundles over discrete groups
本文研究了离散群上的$W^*$-Fell bundles的amenability特性，通过近似性质和条件期望进行表征，并证明了amenability在子群限制下的传递性，扩展了经典群amenability的结果。
We give precise conditions under which irreducible representations associated to stability groups induce to irreducible representations for Fell bundle C*-algebras. This result generalizes an earlier result of Echterhoff and the second author. Because the Fell bundle construction subsumes most other examples of C*-algebras constructed from dynamical systems, our result percolates down to many different constructions including the many flavors of groupoid crossed products.
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. 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.
In this paper we develop the notion of Herz-Schur multipliers to the context of Fell bundle, and we give a necessary and sufficient condition if the reduced cross-sectional $C^{\ast}$-algebra $C_r^{\ast}(\mathfrak{B})$ of Fell bundle $\mathfrak{B}$ over discrete group $G$ is nuclear in terms of this generalized notion. As an application, we prove that if $C_r^{\ast}(\mathfrak{B})$ is nuclear, then for any subgroup $H \subset G$, the $C^{\ast}$-algebra $C_r^{\ast}(\mathfrak{B}_H)$ is nuclear.
We show that every Fell bundle B over a locally compact group G is "proper" in a sense recently introduced by Ng. Combining our results with those of Ng we show that if B satisfies the "approximation property" then it is amenable in the sense that the full and reduced cross-sectional C*-algebras coincide.
Conditional expectations on Fell bundles
本文证明 Fell 丛到子丛的连续条件期望蕴含子丛的全截面 C*-代数包含于丛的相应代数，且存在条件期望；约化情形类似，且可顺从性遗传给子丛。
Building on previous papers by Anantharaman-Delaroche (AD) we introduce and study the notion of AD-amenability for partial actions and Fell bundles over discrete groups. We prove that the cross-sectional C*-algebra of a Fell bundle is nuclear if and only if the underlying unit fibre is nuclear and the Fell bundle is AD-amenable. If a partial action is globalisable, then it is AD-amenable if and only if its globalisation is AD-amenable. Moreover, we prove that AD-amenability is preserved by (weak) equivalence of Fell bundles and, using a very recent idea of Ozawa and Suzuki, we show that AD-ame…
We define and study fibrations of topological groupoids. We interpret a groupoid fibration L->H with fibre G as an action of H on G by groupoid equivalences. Our main result shows that a crossed product for an action of L is isomorphic to an iterated crossed product first by G and then by H. Here groupoid action means a Fell bundle over the groupoid, and crossed product means the section C*-algebra.
We introduce the notion of groupoid pre-equivalences and prove that they give rise to groupoid equivalences by taking certain quotients. Then, given an equivalence of Fell bundles $\mathscr{B}$ and $\mathscr{C}$ and another equivalence between $\mathscr{C}$ and $\mathscr{D}$, we construct an equivalence between $\mathscr{B}$ and $\mathscr{D}$ out of the tensor product bundle. As a consequence, we obtain that Fell bundle equivalence is indeed an equivalence relation.
We show that every groupoid C*-algebra is isomorphic to its opposite, and deduce that there exist C*-algebras that are not stably isomorphic to groupoid C*-algebras, though many of them are stably isomorphic to twisted groupoid C*-algebras. We also prove that the opposite algebra of a section algebra of a Fell-bundle over a groupoid is isomorphic to the section algebra of a natural opposite bundle.
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