Density and invariant means in left amenable semigroups
Følner conditions and densities are usual tools associated to the amenability of semigroups. In the present paper the authors consider the Følner density, a particularly well-behaved notion of density denoted as \(d\) that was introduced in a previous paper of theirs [Semigroup Forum 73, No. 2, 273--300 (2006; Zbl 1111.22004)]. The Følner density is defined through so-called Følner nets, that is, nets \(( F_\alpha)_{\alpha \in D} \) of finite subsets of \(S\) such that \((\frac{|sF_\alpha \triangle F_\alpha|}{|F_\alpha|})_{\alpha\in D}\) converges to 0. If \(\text{LIM}_0(S)\) denotes the subset of \(\ell_\infty(S)^\ast\) consisting of all accumulation points of Følner nets in a semigroup \(S\), then \(\text{LIM}_0(S)\) forms a weak\(^\ast\) closed subset of the set \(\text{LIM}(S)\) of all left invariant means of \(S\). The authors prove that \(\text{LIM}(S)\) is actually the weak\(^\ast\) closed convex hull of \(\text{LIM}_0(S)\), when \(S\) is left cancellative and left amenable. This is a crucial step for the product formula for the Følner density that constitutes one of the main objectives of the paper. This product formula states that \(d(A\times B)=d(A)d(B)\) whenever \(A\) and \(B\) are subsets of left amenable, left cancellative semigroups \(S\) and \(T\), respectively. It is also shown that \(\text{LIM}_0(\mathbb N)\) is a convex set. As a consequence, \(\text{LIM}_0(S)= \text{LIM}(S)\) when \(S=\mathbb N\), but it is left open which other semigroups may satisfy this equality. The final part of the paper concerns properties of invariant means of semigroups viewed as measures on their Stone-Čech compactification. It is proved, for instance, that any two distinct extreme points of \(\text{LIM}(S)\) are mutually singular or that \(\mu(\{p\})>0\) for some \(\mu\in\text{LIM}(S)\) and \(p\in \beta S\) if and only if there is a Følner net \(\langle F_\alpha\rangle_{\alpha\in D}\) for which \(\langle| F_\alpha|\rangle_{\alpha\in D}\) is bounded (and that in this case minimal left ideals of \(\beta S\) are finite groups).
scientific article; zbMATH DE number 5625460
scientific article