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Fairly amenable semigroups
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Abstract
Amenability developed alongside modern analysis, as it is a central property lacking
in a group used to show, for example, the BanachTarski paradox (Wagon, 1993).
The first working definition was given by von Neumann (1929), in terms of finitelyadditive
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 finitelyadditive
measure, corresponding via Lebesgue integration. This advance was significant as it
opened the door to the application of abstract harmonic analysis, fixedpoint theorems,
and an industry of consequences. Amenable groups support almostinvariant
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 MarkovKakutani fixedpoint 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 finitelyadditive measure from a leftinvariant mean yields what might be called a
left preimageinvariant 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 finitelyadditive 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
nonamenable 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 betterbehaved
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 finitelyadditive measures to generalise group
amenability to semigroups in a different way. For a semigroup S, a finitelyadditive
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 finitelyadditive 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 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 nonfairly 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” 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
finitelyadditive measure? One approach is to flip the duality between the convolution
action in ℓ1(S) and the dual action in ℓ1(S) upsidedown: attempt convolution
in ℓ1(S) and the dual action in ℓ1(S). In this scenario, the curious will consider
such illdefined expressions as 0 S. Fortunately, wherever the convolution partial
action of s on ϕ 2 ℓ1(S), i.e. s ϕ, is welldefined and bounded, then the integral
with respect to a left fairlyinvariant 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(ϕ) = m(s ϕ) for all s 2 S and ϕ 2 ℓ1(S) such that s ϕ 2 ℓ1(S). Hence
the nomenclature “fairly amenable” 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 ⊛ is introduced on ℓ1(S), which induces
a full action of S. One drawback of⊛compared to *is that, in order to express
fair amenability, an additional condition is required to limit the scope of invariance
appropriately. Finally, inner ⊛ invariance and its “fair” variant are briefly explored.
Item Type:  Thesis  PhD 

Authors/Creators:  Deprez, JT 
Keywords:  groups, semigroups, algebra, analysis, amenability, measure theory 
Additional Information:  Copyright the Author 
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