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Fairly amenable semigroups

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Deprez, JT (2014) Fairly amenable semigroups. PhD thesis, University of Tasmania.

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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 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 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” 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 ℓ1(S) and the dual action in ℓ1(S) upside-down: attempt convolution
in ℓ1(S) and the dual action in ℓ1(S). In this scenario, the curious will consider
such ill-defined expressions as 0 S. Fortunately, wherever the convolution partial
action of s on ϕ 2 ℓ1(S), i.e. s ϕ, 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(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)
Keywords: groups, semigroups, algebra, analysis, amenability, measure theory
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Date Deposited: 26 Oct 2014 23:48
Last Modified: 15 Sep 2017 01:06
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