Fell Bundles Research Articles

A perspective on non-commutative quantum gravity

In this paper, we present some of the concepts underlying a program of non-commutative quantum gravity and recall some of the results. This program includes a novel approach to spectral triple categorification and also a precise connection between Fell bundles and Connes' non-commutative geometry. Motivated by topics in quantization of the non-commutative standard model and introduction of algebraic techniques and concepts into quantum gravity (following for example Crane, Baez and Barrett), we define spectral C*-categories, which are deformed spectral triples in a sense made precise. This definition gives to representations of a C*-category on a small category of Hilbert spaces and bounded linear maps, the interpretation of a topological quantum field theory. The construction passes two mandatory tests: (i) there is a classical limit theorem reproducing a Riemannian spin manifold manifesting Connes' and Schücker's non-commutative counterpart of Einstein's equivalence principle, and (ii) there is consistency with the experimental fermion mass matrix. We also present an algebra invariant taking the form of a partition function arising from a C*-bundle dynamical system in connection with C*-subalgebra theory.

Product systems over Ore monoids

We interpret the Cuntz–Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz–Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz–Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.

Irreducible induced representations of Fell bundle 𝐶*-algebras

We give precise conditions under which irreducible representations associated to stability groups induce to irreducible representations for Fell bundle C ∗ 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 ∗ C^* -algebras constructed from dynamical systems, our result percolates down to many different constructions including the many flavors of groupoid crossed products.

On Nuclearity of $C*$-algebras of Fell Bundles over Étale Groupoids

We show the nuclearity of the reduced C^* -algebra of a Fell bundle over an étale amenable groupoid, whose fibers over the unit space are all nuclear. We introduce the (minimal) tensor product of a Fell bundle and a C^* -algebra in order to show the uniqueness of tensor product norms.

FELL BUNDLES AND IMPRIMITIVITY THEOREMS: MANSFIELD’S AND FELL’S THEOREMS

AbstractIn 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.

Amenability for Fell bundles over groupoids

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.

Fell bundles and imprimitivity theorems: towards a universal generalized fixed point algebra

We apply the One-Sided Action Theorem from the first paper in this series to prove that Rieffel's Morita equivalence between the reduced crossed product by a proper saturated action and the generalized fixed-point algebra is a quotient of a Morita equivalence between the full crossed product and a universal fixed-point algebra. We give several applications, to Fell bundles over groups, reduced crossed products as fixed-point algebras, and C*-bundles.

A higher category approach to twisted actions on c* -algebras

AbstractC*-algebras form a 2-category with *-homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby–Smith twisted actions and the equivalence of such actions, covariant representations and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green, and Busby and Smith.The Packer–Raeburn Stabilization Trick implies that all Busby–Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalize this to actions of strict 2-groupoids.

Dynamical systems of type and their -algebras

AbstractGiven positive integers$n$and$m$, we consider dynamical systems in which (the disjoint union of)$n$copies of a topological space is homeomorphic to$m$copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra denoted by${\cal O}_{m,n}$, which in turn is obtained as a quotient of the well-known Leavitt C*-algebra$L_{m,n}$, a process meant to transform the generating set of partial isometries of$L_{m,n}$into a tame set. Describing${\cal O}_{m,n}$as the crossed product of the universal$(m,n)$-dynamical system by a partial action of the free group$\mathbb {F}_{m+n}$, we show that${\cal O}_{m,n}$is not exact when$n$and$m$are both greater than or equal to 2, but the corresponding reduced crossed product, denoted by${\cal O}_{m,n}^r$, is shown to be exact and non-nuclear. Still under the assumption that$m,n\geq 2$, we prove that the partial action of$\mathbb {F}_{m+n}$is topologically free and that${\cal O}_{m,n}^r$satisfies property (SP) (small projections). We also show that${\cal O}_{m,n}^r$admits no finite-dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.

Amenability of the Restricted Fourier Algebras

We discuss amenability of the restricted Fourier-Stieltjes algebras on inverse semigroups. We show that, for an E-unitary inverse semigroup, amenability of the restricted Fourier-Stieltjes algebra is related to the amenability of an associated Banach algebra on a Fell bundle.

Inverse semigroup expansions and their actions on $C^{\ast}$-algebras

In this work, we give a presentation of the prefix expansion ${\operatorname{\mathbf {Pr}} (G)}$ of an inverse semigroup $G$ as recently introduced by Lawson, Margolis and Steinberg which is similar to the universal inverse semigroup defined by the second named author in case $G$ is a group. The inverse semigroup ${\operatorname{\mathbf {Pr}} (G)}$ classifies the partial actions of $G$ on spaces. We extend this result and prove that Fell bundles over $G$ correspond bijectively to saturated Fell bundles over ${\operatorname{\mathbf {Pr}} (G)}$. In particular, this shows that twisted partial actions of $G$ (on $C^{*}$-algebras) correspond to twisted (global) actions of ${\operatorname{\mathbf {Pr}} (G)}$. Furthermore, we show that this correspondence preserves $C^{*}$-algebra crossed products.

A classic Morita equivalence result for Fell bundle $C^*$-algebras

We show how to extend a classic Morita Equivalence Result of Green's to the $C^*$-algebras of Fell bundles over transitive groupoids. Specifically, we show that if $p:{\mathcal B}\to G$ is a saturated Fell bundle over a transitive groupoid $G$ with stability group $H=G(u)$ at $u\in G^{(0)}$, then $C^* (G,{\mathcal B})$ is Morita equivalent to $C^*(H,{\mathcal C})$, where ${\mathcal C}={\mathcal B}_{| H}$. As an application, we show that if $p:{\mathcal B}\to G$ is a Fell bundle over a group $G$ and if there is a continuous $G$-equivariant map $\sigma:$ Prim $A\to G/H$, where $A=B(e)$ is the $C^*$-algebra of $\mathcal B$ and $H$ is a closed subgroup, then $C^*(G,{\mathcal B})$ is Morita equivalent to $C^* (H,{\mathcal C}^{I})$ where ${\mathcal C}^{I}$ is a Fell bundle over $H$ whose fibres are $A/I$-$A/I$-imprimitivity bimodules and $I=\bigcap\{ P:\sigma(P)=eH\}$. Green's result is a special case of our application to bundles over groups.

Twisted actions and regular Fell bundles over inverse semigroups

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 Sieben's twisted actions essentially correspond to twisted etale groupoids with topologically trivial twists.

Integrability of dual coactions on Fell bundle C*-algebras

We study integrability for coactions of locally compact groups. For abelian groups, this corresponds to integrability of the associated action of the Pontrjagin dual group. The theory of integrable group actions has been previously studied by Ruy Exel, Ralf Meyer and Marc Rieffel. Our goal is to study the close relationship between integrable group coactions and Fell bundles. As a main result, we prove that dual coactions on C*-algebras of Fell bundles are integrable, generalizing results by Ruy Exel for abelian groups.

A horizontal categorification of Gel'fand duality

In the setting of C⁎-categories, we provide a definition of spectrum of a commutative full C⁎-category as a one-dimensional unital Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gel'fand duality theorem generalizing the usual Gel'fand duality between the categories of commutative unital C⁎-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well known, the somehow unconventional way we “glue” them together seems to shed some new light on the subject.

Formula omitted]-algebras and Fell bundles associated to a textile system

The notion of textile system was introduced by M. Nasu in order to analyze endomorphisms and automorphisms of topological Markov shifts. A textile system is given by two finite directed graphs G and H and two morphisms p , q : G → H , with some extra properties. It turns out that a textile system determines a first quadrant two-dimensional shift of finite type, via a collection of Wang tiles, and conversely, any such shift is conjugate to a textile shift. In the case the morphisms p and q have the path lifting property, we prove that they induce groupoid morphisms π , ρ : Γ ( G ) → Γ ( H ) between the corresponding étale groupoids of G and H. We define two families A ( m , n ) and A ¯ ( m , n ) of C ∗ -algebras associated to a textile shift, and compute them in specific cases. These are graph algebras, associated to some one-dimensional shifts of finite type constructed from the textile shift. Under extra hypotheses, we also define two families of Fell bundles which encode the complexity of these two-dimensional shifts. We consider several classes of examples of textile shifts, including the full shift, the Golden Mean shift and shifts associated to rank two graphs.

Graphs of $C^*$-correspondences and Fell bundles

We define the notion of a �-system of C � -correspondences associated to a higher-rank graph �. Roughly speaking, such a system assigns to each vertex ofa C � - algebra, and to each path ina C � -correspondence in a way which carries compositions of paths to balanced tensor products of C � -correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of C � -correspondences to associate a C � -algebra to each �-system. We then construct a Fell bundle over the path groupoid Gand show that the C � -algebra of the �-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.

CATEGORIFIED NONCOMMUTATIVE MANIFOLDS

We construct a noncommutative geometry with generalised 'tangent bundle' from Fell bundle C*-categories (E) beginning by replacing pair groupoid objects (points) with objects in E. This provides a categorification of a certain class of real spectral triples where the Dirac operator D is constructed from morphisms in a category. Applications for physics include quantization via the tangent groupoid and new constraints on D finite (the fermion mass matrix).

Fell bundles associated to groupoid morphisms

Given a continuous open surjective morphism $\pi :G\rightarrow H$ 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 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.

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

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.