Amenability of coarse spaces and $$\mathbb {K}$$ K -algebras

Abstract

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite Communicated by Efim Zelmanov. extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.

Amenability and uniform Roe algebras

“…First we recall the notion of amenability on metric spaces and C-algebras. We refer to [5] (and references cited therein) for motivation, analysis or proofs in a more general context. In the last subsection we also include the definition and some properties of semi-pre-C * -algebras as introduced by Ozawa in [39].…”

“…Finally, there is a natural strengthening of the notion of a Følner net which we call a proper Følner net. In the context of metric spaces and algebras, it corresponds to the Følner net being exhaustive, where its subtle difference from general Følner nets plays an important rôle in the study of amenability in these contexts (see [5,). We introduce the new class of properly Følner C * -algebras in Definition 3.13, and prove analogs of Theorems 1 and 2 in this more restrictive context (see Theorems 3.17 (ii) and 4.12).…”

Amenability and paradoxicality in semigroups and C⁎-algebras

“…For the second part we follow a similar route to that of [4,Proposition 3.4]. Recall from [29] that a semigroup satisfying the Klawe condition is amenable if and only if for every ε > 0 and finite F ⊂ S there is a (ε, F)-Følner set F ⊂ S such that |F | = |sF | for every s ∈ F (see Theorem 2.6 in [29] in relation with the notion of strong Følner condition).…”

“…Later, Følner provided in [28] a very useful combinatorial characterization of amenability in terms of nets of finite subsets of the group that are almost invariant under left multiplication. This alternative approach was then used to study amenability in the context of algebras over arbitrary fields by Gromov [31, §1.11] and Elek [22] (see also [8,14,4] as well as Definition 2.1 (5)), in operator algebras by Connes [16,17] (see also Definition 2.2 (2)) and in metric spaces by Ceccherini-Silberstein, Grigorchuk and de la Harpe in [13] (see also [4]).…”

Notions of Infinity in Quantum Physics

“…It is therefore clear that all finite structures are normally amenable, while infinite structures might be or not. We refer to [29,15,16,18,23,26,1,2] for additional motivation and results on this body of work.…”