arXiv subjects

Jean N. Renault

Publications and source records attributed to Jean N. Renault.

5 recordsLinked to original sources

arXiv2021

Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions $Σ$ of groupoids $G$ by bundles $A$ of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid $G$ by a given bundle $A$. There is a natural action of $Σ$ on the dual of $A$, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of $A$ with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of $G$ on the dual of $A$. We prove that the full $C^*$-algebra of this twist is isomorphic to the full $C^*$-algebra of $Σ$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

math.OA↗

arXiv2020

C*-Algebras of extensions of groupoids by group bundles

Given a normal subgroup bundle $\mathcal A$ of the isotropy bundle of a groupoid $Σ$, we obtain a twisted action of the quotient groupoid $Σ/\mathcal A$ on the bundle of group $C^*$-algebras determined by $\mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(Σ)$. Restricting to the case where $\mathcal A$ is abelian, we describe $C^*(Σ)$ as the $C^*$-algebra associated to a $\mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $Σ/\mathcal A$ on the Pontryagin dual space of $\mathcal A$. We give some illustrative examples of this result.

math.OA↗

arXiv2015

Amenability of Groupoids Arising from Partial Semigroup Actions and Topological Higher Rank Graphs

We consider the amenability of groupoids $G$ equipped with a group valued cocycle $c:G\to Q$ with amenable kernel $c^{-1}(e)$. We prove a general result which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and if there is countable set $D\subset G$ such that $c(G^{u})D=Q$ for all $u\in G^{(0)}$. We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our methods yield a nice alternative groupoid approach to these important constructions.

math.OA↗

arXiv2012

Uniqueness theorems for topological higher-rank graph C*-algebras

We consider the boundary-path groupoids of topological higher-rank graphs. We show that the all such groupoids are topologically amenable. We deduce that the C*-algebras of topological higher-rank graphs are nuclear and prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem. We then provide a necessary and sufficient condition for simplicity of a topological higher-rank graph C*-algebra, and a condition under which it is also purely infinite.

math.OA↗

arXiv1997

The Brauer Group of a Locally Compact Groupoid

We define the Brauer group $\Br(G)$ of a locally compact groupoid $G$ to be the set of Morita equivalence classes of pairs $(\A,α)$ consisting of an elementary C*-bundle $\A$ over $G^{(0)}$ satisfying Fell's condition and an action $α$ of $G$ on $\A$ by $*$-isomorphisms. When $G$ is the transformation groupoid $X\times H$, then $\Br(G)$ is the equivariant Brauer group $\Br_H(X)$. In addition to proving that $\Br(G)$ is a group, we prove three isomorphism results. First we show that if $G$ and $H$ are equivalent groupoids, then $\Br(G)$ and $\Br(H)$ are isomorphic. This generalizes the result that if $G$ and $H$ are groups acting freely and properly on a space $X$, say $G$ on the left and $H$ on the right then $\Br_G(X/H)$ and $\Br_H(G/ X)$ are isomorphic. Secondly we show that the subgroup $\Br_0(G)$ of $\Br(G)$ consisting of classes $[\A,α]$ with $\A$ having trivial Dixmier-Douady invariant is isomorphic to a quotient $\E(G)$ of the collection $\Tw(G)$ of twists over $G$. Finally we prove that $\Br(G)$ is isomorphic to the inductive limit $\Ext(G,T)$ of the groups $\E(G^X)$ where $X$ varies over all principal $G$ spaces $X$ and $G^X$ is the imprimitivity groupoid associated to $X$.

funct-an↗