Fairly amenable semigroups

Amenability developed alongside modern analysis, as it is a central property lacking in a group used to show, for example, the Banach-Tarski paradox (Wagon, 1993). The first working definition was given by von Neumann (1929), in terms of finitely-additive measures. A number of useful theorems are capable of being shown using this basic definition. The firrst modern definition of amenability was given by M. M. Day (1957), whose concept involved invariant means. For groups this coincides exactly with the von Neumann condition: each invariant mean corresponds to an invariant finitely-additive measure, corresponding via Lebesgue integration. This advance was significant as it opened the door to the application of abstract harmonic analysis, fixed-point theorems, and an industry of consequences. Amenable groups support almost-invariant finite means, and via decomposition this is culminated as the F‚àö‚àèlner condition, a statement about finite sets. Abelian groups are amenable as a simple consequence of the Markov-Kakutani fixed-point theorem. A theorem of B. E. Johnson (1972) led to the development of amenable Banach algebras and C*-algebras, neatly encoding amenability in the mechanics of cohomology theory. While amenability is directly generalisable from groups to semigroups, the two key definitions do not correspond in the same way as they do for groups: extracting a finitely-additive measure from a left-invariant mean yields what might be called a left preimage-invariant measure, and for groups these merely correspond to the inverse elements. A simple but surprising consequence of Day's definition of amenability is that semigroups with a zero element are both left and right amenable (Day, 1957). Yet they cannot support a (totally) invariant finitely-additive measure (van Douwen, 1992, p231). On the other hand, all semigroups with more than one distinct left zero are not left amenable (Paterson, 1988), and in particular there are many non-amenable finite semigroups, which is another contrast to the group case: all finite groups are amenable. This standard definition of amenability for semigroups is therefore unintuitive and, perhaps, unsatisfactory. Restricting to better-behaved classes of semigroups, such as the inverse semigroups, does little to improve this. The first new result of the present work is that there is a weakening of invariance that can be used in the context of finitely-additive measures to generalise group amenability to semigroups in a different way. For a semigroup S, a finitely-additive measure 2 [0; 1]P(S) will be called left fairly invariant if, for all s 2 S and A S such that sjA is an injection, (sA) = (A). When a semigroup supports such a finitely-additive measure, then it is left fairly amenable. Fair amenability is a generalisation of group amenability, and retains some of the useful theorems. Some of the results shown using this formulation include: a semigroup is left fairly amenable when it satisfies a weakened Strong F‚àö‚àèlner Condition, finite semigroups are all fairly amenable, semigroups with involution are either fairly amenable on both the left and the right or not at all, adjoining a zero does not cause a non-fairly amenable semigroup to become fairly amenable, directed unions of fairly amenable semigroups are fairly amenable, and a variety of examples which are fairly amenable or not fairly amenable. The name amenable‚ÄövÑvp is, as the story goes, supposed to be a pun, since amenable groups support invariant means. Thus an important question for fair amenability is, what condition for a mean is equivalent to the fair invariance of the corresponding finitely-additive measure? One approach is to flip the duality between the convolution action in ‚Äövëv¨1(S) and the dual action in ‚Äövëv¨1(S) upside-down: attempt convolution in ‚Äövëv¨1(S) and the dual action in ‚Äövëv¨1(S). In this scenario, the curious will consider such ill-defined expressions as 0 S. Fortunately, wherever the convolution partial action of s on ˜ìvØ 2 ‚Äövëv¨1(S), i.e. s ˜ìvØ, is well-defined and bounded, then the integral with respect to a left fairly-invariant measure can be readily computed. It is shown that a semigroup S left fairly amenable if, and only if, there exists a mean m such that m(˜ìvØ) = m(s ˜ìvØ) for all s 2 S and ˜ìvØ 2 ‚Äövëv¨1(S) such that s ˜ìvØ 2 ‚Äövëv¨1(S). Hence the nomenclature fairly amenable‚ÄövÑvp is justified as a pun also. Some variations on fair amenability and related results are also explored. As a variation on the * partial action, an operator ‚Äöv§vµ is introduced on ‚Äövëv¨1(S), which induces a full action of S. One drawback of‚Äöv§vµcompared to *is that, in order to express fair amenability, an additional condition is required to limit the scope of invariance appropriately. Finally, inner ‚Äöv§vµ invariance and its fair‚ÄövÑvp variant are briefly explored.