We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connections to noncommutative CW-complexes and interval algebras. We prove that an inverse system of groupoids with Haar system preserving bonding maps has a limit, and that we get a corresponding direct system of groupoid C * -algebras. An explicit construction of an inverse system of groupoids is used to approximate a σ-compact groupoid G by second countable groupoids; if G is equipped with a Haar system and 2-cocycle then so are the approximation groupoids, and the maps in the inverse system are Haar system preserving. As an application of this construction, we show how to easily extend the Maximal Equivalence Theorem of Jean Renault to σ-compact groupoids.

No abstract

We consider a twist E over an étale groupoid G. When G is principal, we prove that the nuclear dimension of the reduced twisted groupoid C * -algebra is bounded by a number depending on the dynamic asymptotic dimension of G and the topological covering dimension of its unit space. This generalizes an analogous theorem by Guentner, Willett, and Yu for the C * -algebra of G. Our proof uses a reduction to the unital case where G has compact unit space, via a construction of "groupoid unitizations" G and E of G and E such that E is a twist over G. The construction of G is for r-discrete (hence étale) groupoids G which are not necessarily principal. When G is étale, the dynamic asymptotic dimension of G and G coincide. We show that the minimal unitizations of the full and reduced twisted groupoid C * -algebras of the twist over G are isomorphic to the twisted groupoid C * -algebras of the twist over G. We apply our result about the nuclear dimension of the twisted groupoid C * -algebra to obtain a similar bound on the nuclear dimension of the C * -algebra of an étale groupoid with closed orbits and abelian stability subgroups that vary continuously.

Abstract. Fix a von Neumann algebra N equipped with a suitable trace τ. For a path of self-adjoint Breuer-Fredholm operators, the spectral ow measures the net amount of spectrum which moves from negative to non-negative. We consider speci cally the case of paths of bounded perturbations of a xed unbounded self-adjoint BreuerFredholm operator a liated with N. If the unbounded operator is p-summable (that is, its resolvents are contained in the ideal L p ), then it is possible to obtain an integral formula which calculates spectral ow. is integral formula was rst proven by Carey and Phillips, building on earlier approaches of Phillips.eir proof was based on rst obtaining a formula for the larger class of θ-summable operators, and then using Laplace transforms to obtain a p-summable formula. In this paper, we present a direct proof of the p-summable formula, which is both shorter and simpler than theirs.

No abstract