C ÓMO SE RESUELVEN LOS PROBLEMAS DE ATRASOS HORARIOS ?

Now recall De nition 2.7. is formulation was given in Day (1957) directly in terms of semigroups, and is the most popular. Means and le-invariance are exactly as previously de ned.

De nition 4.3 A semigroupSis(classically) le-amenableif there exists a le-invariant mean onS, similarly for right- and bi-amenable.

In spite of the lack of inverses and cancellativity in a general semigroupS, the two actions—the convolution action ∗ inℓ1(S) and the dual action · inℓ∞(S)—work almost as well as they do for groups, with some minor tweaking.

Supposeϕ∈ℓ∞(S)andϕˆ ∈ℓ1_(S)∗_{are equivalent under the usual identi cation} ofℓ1_(S)∗_andℓ∞_{(S), i.e. such that}

ˆ

ϕ(f) =⟨f, ϕ⟩, and ϕ(s) =ϕˆ(χ_{s}

)

for allf ∈ ℓ1(S), s ∈ S. Onℓ1(S)we have the convolution off1, f2 ∈ ℓ1(G)given again as usual (note how this varies from convolution for a group):

(f1∗f2) (x) :=

∑

s,t∈S st=x

f1(s)f2(t).

Henceforth, a summation (or similar) over “st=x” shall mean “over all pairss, t∈ Ssuch thatst=x.” In particular, the convolution action carries over toℓ1_(S)from the group case as well. For allx ∈S,

{s∗f}(x) = ∑

t∈s−1_{x}

f(t).

While the∗action was well-de ned for a groupGonℓ∞(G), here, the expression

s∗ϕfors ∈ Sandϕ ∈ ℓ∞(S)is only well-de ned in some cases, such as for the subspaceℓ1_(S).

Lemma 4.4 e le convolution action and the dual le action are duals of one an- other, i.e.

⟨ϕ, s∗f⟩=⟨s·ϕ, f⟩

Proof ⟨ϕ, s∗f⟩=∑ t∈S {s∗f}(t)ϕ(t) =∑ t∈S   ∑ u∈s−1_{t} f(u)  _ϕ(t) =∑ t∈S ∑ u∈s−1_{t} f(u)ϕ(su) =∑ u∈S f(u)ϕ(su) =∑ u∈S f(u){s·ϕ}(u) =⟨s·ϕ, f⟩ as required. □

So far so good—amenability ofSis now (loosely) connected to the amenability ofℓ1_(S).

Lemma 4.5 SupposeSis a semigroup with a le zeroz.Sis right amenable because the meanδzgiven byδz(f) = f(z)for allf∈ℓ∞(S)is a right invariant mean forS. Furthermore, ifSis le amenable thenδzis the only le invariant mean.

Proof e meanδzis a right-invariant mean. 1. δzis a mean.

P: For allf,δz(f)must clearly be within the range of values taken byf, i.e. inf x∈Sf(x)≤δz(f) =f(z)≤sup_x∈Sf(x). 2. δzis right-invariant. P: For alls∈S, δz(f·s) ={f·s}(z) =f(zs) =f(z) ∵zis a le zero.

Now supposeSis le amenable with a le invariant meanm. m(f) =m(z·f)

by le invariance, but{z·f}(t) = f(zt) = f(z)for allt ∈S, and thereforem(f) =

f(z) = δz(f). □

Corollary 4.6 Any semigroupSwith zero0is amenable, and the invariant mean is unique.

Proof By Lemma 4.5, the meanm given bym(f) = f(0)for allfis the unique

le- and right-invariant mean forS. □

Together with eorem 4.2, this demonstrates that De nition 2.1 and 4.3 are not equivalent. Unlike the case with groups, it seems that under De nition 4.3, a semi- group can be amenable and be scarcely any bigger than non-amenable subsemi- groups and subgroups—merely adjoin a zero. Semigroup amenability is even more

ckle than that, however.

Corollary 4.7 Suppose a semigroupShas two distinct le zeroesz1andz2. enS isnotle amenable.

Proof AssumeSis le amenable. Applying Lemma 4.5 to bothz1andz2, the only le invariant mean mis given by both m(f) = f(z1) andm(f) = f(z2)for all

f∈ℓ∞(S). Any functionfsuch thatf(z1)̸=f(z2)suffices to show a contradiction.

□

4.2.1 Breakdown between definitions

It is reasonably easy to show that every le measurable semigroup is le amenable (Klawe, 1977, p102), but this may be the best we get between the two. De nition 4.3 sets a low bar.

No group is “trivially” amenable in the way that a semigroup with zero is. Con- sider the le action of a (semi)group element on an indicator function. If we are operating within agroup, we may do the following:

(gχA) (x) =    1 ifgx∈A 0 ifgx /∈A =    1 ifx∈g−1A 0 ifx /∈g−1_A =χg −1_A(x).

is is clearly not available to all semigroups. At minimum we require some substi- tute forg−1_{A, and in fact if we attempt to deduce a nitely-additive measureµfrom} some le-invariant meanmwe obtain for alls∈SandA⊆Sthe condition

wheres−1_{Adenotes the preimage, i.e. s}−1_{A =} {_t ∈_S:st∈_A}_{. (Paterson, 1988,} Exercise 0.32). On groups, this is the same as the usual measure invariance, since preimages coincide with inverses. On semigroups in general, it isnotthe same. Example 4.8 Consider again a non-trivial semigroup with zero. Here the invariant mean is given bym(f) = f(0). Suppose that we were to try to obtain an invariant

nitely-additive measureµfrom this de nition by settingµ(A) =m(χA). en

µ(A) =    1 if0∈A 0 if0 /∈A =|A∩{0}|.

isµis nitely additive (for disjointA, B,0is in at most one of them) and has total measure 1 (0 ∈ S). However, µ(0(S\ {0})) = µ({0}) = 1andµ((S\ {0})) = 0, soµis not invariant (in that multiplication by0alters the value). Instead, it satis es

preimage invariancegiven byµ(s−1A) = µ(A) = µ(As−1). In particular, note that 0−1A=    S if0∈A ∅ if0 /∈A . * * *

It seems that neither De nition 2.1 or De nition 2.7, when extended to semigroups, capture reasonable intuition about semigroups. Both de nitions are ckle with re- spect to the presence or absence of zeroes. Why shouldF0

2be considered amenable, when the sole reason is the presence of a zero?