Quasi-local Algebras and Asymptotic Expanders

Abstract

In this paper, we study the relation between the uniform Roe algebra and the uniform quasi-local algebra associated to a discrete metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic expanders. We show that being asymptotic expanders is a coarse property, and it implies non-uniformly local amenability. Moreover, we also analyse some C * -algebraic properties of uniform quasi-local algebras. In particular, we show that a uniform quasi-local algebra is nuclear if and only if the underlying metric space has Property A.

On the structure of asymptotic expanders

“…Original C * -algebraic motivation. Asymptotic expanders were introduced in [19]. The driving motivation behind this definition was the study of the relation between the uniform Roe algebra C * u (X) and the uniform quasi-local algebra C * uq (X) for a discrete metric space X (see Subsection 2.3 for precise definitions of these algebras).…”

“…The quasi-local operators form an algebra-denoted C * uq (X)-that is somewhat easier to handle, and it follows from the definition that C * u (X) ⊆ C * uq (X). In some special cases, it has been proved that these two algebras coincide [10,32,33], but it is not known whether this is always the case (we refer to [19,33] and references therein for further discussion and motivation).…”

“…The idea in [19] was to focus on the study of the averaging projection P X , where X is a coarse disjoint union of a sequence of finite metric spaces (see Subsections 2.1 and 2.4). One of the main results in [19] is a complete characterisation of the metric spaces for which the averaging projection P X is quasi-local.…”

“…The idea in [19] was to focus on the study of the averaging projection P X , where X is a coarse disjoint union of a sequence of finite metric spaces (see Subsections 2.1 and 2.4). One of the main results in [19] is a complete characterisation of the metric spaces for which the averaging projection P X is quasi-local. Namely, if X = n∈N X n is the coarse disjoint union of a sequence of finite metric spaces {X n } with |X n | → ∞, then P X is quasi-local if and only if the following condition is satisfied:…”

“…(AsEx) for every 0 < α ≤ 1 2 , there exist 0 < c < 1 and R > 0, such that |∂ R A| > c|A| for every A ⊂ X n satisfying α|X n | ≤ |A| ≤ 1 2 |X n |; where ∂ R A is the R-boundary of A, i.e., the set of points in the complement of A that are at most at distance R from A (see [19,Theorem B]). This characterisation led the authors of [19] to introduce the concept of asymptotic expanders: Definition ([19, Definition 3.9]). A sequence {X n } n∈N of finite metric spaces with |X n | → ∞ is a sequence of asymptotic expanders if it satisfies the condition (AsEx).…”