Fell bundles associated to groupoid morphisms

Alex Kumjian; Birant Ramazan; Valentin Deaconu

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Fell bundles associated to groupoid morphisms

Valentin Deaconu, Alex Kumjian, Birant Ramazan

Operator Algebras · Submitted 23 Dec 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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 $H^0$.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,991 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.93 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Given a continuous open surjective morphism $\pi :G\to H$ of \'etale 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 extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. We also prove a structure theorem for abelian Fell bundles.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Given a continuous open surjective morphism $\pi :G\to H$ of \'etale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its C-algebra $C^_r(E)$ is isomorphic to $C^*_r(G)$.	contribution 0.71	0.97	0.77	asserted 1.00
s1	This is related to results of Fell concerning C*-algebraic bundles over groups.	background 0.99	0.05	0.02	asserted 0.92
s2	The case $H=X$, a locally compact space, was treated earlier by Ramazan.	background 1.00	0.04	0.02	asserted 1.00
s3	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 $H^0$.	result 0.55	0.93	0.66	asserted 1.00
s4	We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property.	contribution 0.93	0.91	0.03	asserted 0.99
s5	We also prove a structure theorem for abelian Fell bundles.	result 0.63	0.94	0.05	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

Given a continuous open surjective morphism $\pi :G\to H$ of \'etale 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 extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. We also prove a structure theorem for abelian Fell bundles.

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2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Ramazan [ref] , where a notion of Abelian Fell bundle (which contains our commutative -categories as a special case) is introduced and a structure theorem for them (in terms of “twisted coverings of groupoids”) is proved.

A Horizontal Categorification of Gelfand Duality · Introduction · 0.84

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