Fairly amenable semigroups

by JT Deprez

Institution: University of Tasmania
Year: 2014
Keywords: groups; semigroups; algebra; analysis; amenability; measure theory
Record ID: 1054151
Full text PDF: http://eprints.utas.edu.au/18589/1/Whole-Deprez-thesis.pdf


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 Følner 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 Følner Condition, finite semigroups…