Invariant means and the Stone-Čech compactification

On a Geometric Property of the Set of Invariant Means on a Group

“…Therefore suppt n$Es~KEK. On the other hand, since suppt p^, is invariant [15,Theorem 4.3] and K is minimal invariant we conclude that K = s~K = suppt jli¿. Since G is generated by 5 we see that x~K = K for each xEG, i.e., K is an invariant subset of ßG.…”

The Myhill property for cellular automata on amenable semigroups

“…Remark 7.2. Recall that a semigroup S is said to be left-reversible if any two left-principal ideals in S intersect, i.e., aS ∩ bS = ∅ for all a, b ∈ S. As every left-amenable semigroup is clearly left-reversible, one deduces from Ore's theorem that if S is a cancellative leftamenable semigroup, then S embeds in an amenable group, its group of left-quotients G := {st −1 : s, t ∈ S} (see [16,Corollary 3.6]). When S is a cancellative commutative semigroup, e.g., S = N for which G = Z, given any finite subset F ⊂ G, we can always find t ∈ S such that t + F ⊂ S (if F = {s i − t i : s i , t i ∈ S, 1 ≤ i ≤ n}, we can take t = 1≤i≤n t i ).…”

Amenable transformation semigroups

“…A finite abstract semigroup has a left invariant mean if and only if each pair of right ideals has a nonempty intersection (Rosen [10]); in S, of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700010399 In the case of the /-and r-semigroups, the set of all means becomes a semigroup under the Arens multiplication, and this semigroup has a number of interesting properties (see [11] for details). For example, if X ( = S) has a left invariant mean, then the smallest closed two-sided ideal in the semigroup of means consists of all the left invariant means.…”