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.

Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f : X → Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f : X → Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y ) ≤ asdim(X) + n − 1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension.

We extend the symbol calculus and study the limit operator theory for σ-compact,étale and amenable groupoids, in the Hilbert space case. This approach not only unifies various existing results which include the cases of exact groups and discrete metric spaces with Property A, but also establish new limit operator theories for group/groupoid actions and uniform Roe algebras of groupoids. In the process, we extend a monumental result by Exel, Nistor and Prudhon, showing that the invertibility of an element in the groupoid C * -algebra of a σ-compact amenable groupoid with a Haar system is equivalent to the invertibility of its images under regular representations.

The purpose of this paper is to investigate the duality between large scale and small scale. It is done by creating a connection between C*-algebras and scale structures. In the commutative case we consider C*-subalgebras of C b (X), the C*-algebra of bounded complexvalued functions on X. Namely, each C*-subalgebra C of C b (X) induces both a small scale structure on X and a large scale structure on X. The small scale structure induced on X corresponds (or is analogous) to the restriction of C b (h(X)) to X, where h(X) is the Higson compactification. The large scale structure induced on X is a generalization of the C 0 -coarse structure of N.Wright. Conversely, each small scale structure on X induces a C*-subalgebra of C b (X) and each large scale structure on X induces a C*-subalgebra of C b (X). To accomplish the full correspondence between scale structures on X and C*-subalgebras of C b (X) we need to enhance the scale structures to what we call hybrid structures. In the noncommutative case we consider C*-subalgebras of bounded operators B(l 2 (X)).

The aim of this paper is to prove all well-known metrization theorems using partitions of unity. To accomplish this, we first discuss sufficient and necessary conditions for existence of U -small partitions of unity (partitions of unity subordinated to an open cover U of a topological space X).