# Amenability for Fell bundles over groupoids

@article{Sims2012AmenabilityFF, title={Amenability for Fell bundles over groupoids}, author={Aidan Sims and Dana P. Williams}, journal={Illinois Journal of Mathematics}, year={2012}, volume={57}, pages={429-444} }

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 T0, 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.

## 34 Citations

### Some consequences of stabilization theorem for Fell bundles over exact groupoids

- MathematicsJournal of Operator Theory
- 2019

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…

### Reduced C*-algebras of Fell bundles over inverse semigroups

- Mathematics
- 2015

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…

### Reduced C*-algebras of Fell bundles over inverse semigroups

- MathematicsIsrael Journal of Mathematics
- 2017

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…

### A stabilization theorem for Fell bundles over groupoids

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2017

We study the C *-algebras associated with upper semi-continuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer–Raeburn ‘stabilization trick’, we…

### Groupoid Semidirect Product Fell Bundles I- Actions by Isomorphisms

- Mathematics
- 2021

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.

### A non-amenable groupoid whose maximal and reduced $C^*$-algebras are the same

- Mathematics
- 2015

We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to…

### Amenability for actions of \'etale groupoids on $C^*$-algebras and Fell bundles

- Mathematics
- 2022

. 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…

### Groupoid Fell bundles for product systems over quasi-lattice ordered groups

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2017

Abstract 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…

### Irreducible induced representations of Fell bundle C * -algebras

- Mathematics
- 2013

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…

### IRREDUCIBLE INDUCED REPRESENTATIONS OF FELL

- Mathematics
- 2013

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…

## References

SHOWING 1-10 OF 22 REFERENCES

### An equivalence theorem for reduced Fell bundle C*-algebras

- Mathematics
- 2013

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…

### Equivalence and disintegration theorems for Fell bundles and their C*-algebras

- Mathematics
- 2008

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…

### RENAULT'S EQUIVALENCE THEOREM FOR GROUPOID CROSSED PRODUCTS

- Mathematics
- 2007

We provide an exposition and proof of Renault's equivalence theo- rem for crossed products by locally Hausdorff, locally compact groupoids. Our approach stresses the bundle approach, concrete…

### Morita Equivalence and Continuous-Trace $C^*$-Algebras

- Mathematics
- 1998

The algebra of compact operators Hilbert $C^*$-modules Morita equivalence Sheaves, cohomology, and bundles Continuous-trace $C^*$-algebras Applications Epilogue: The Brauer group and group actions…

### Higher Rank Graph C-Algebras

- Mathematics
- 2000

Building on recent work of Robertson and Steger, we associate a C{algebra to a combinatorial object which may be thought of as a higher rank graph. This C{algebra is shown to be isomorphic to that of…

### A Classic Morita Equivalence Result for Fell Bundle C*-algebras

- Mathematics
- 2009

We show how to extend a classic Morita Equivalence Result of Green's to the \cs-algebras of Fell bundles over transitive groupoids. Specifically, we show that if $p:\B\to G$ is a saturated Fell…

### Haar measure for measure groupoids

- Mathematics
- 1978

It is proved that Mackey's measure groupoids possess an analogue of Haar measure for locally compact groups; and many properties of the group Haar measure generalize. Existence of Haar measure for…

### Remarks on the Ideal Structure of Fell Bundle C*-Algebras

- Mathematics
- 2009

We show that if $p:\B\to G$ is a Fell bundle over a locally compact groupoid $G$ and that $A=\Gamma_{0}(G^{(0)};\B)$ is the \cs-algebra sitting over $G^{(0)}$, then there is a continuous $G$-action…

### On higher rank graph C ∗ -algebras

- Mathematics
- 2000

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C…