Semigroups, amenabiblity and fixed point theorems

FACULTAD DE MATEMÁTICAS DEPARTAMENTO DE ANÁLISIS MATEMÁTICO SEMIGROUPS, AMENABILITY AND FIXED POINT THEOREMS Trabajo de Fin de Máster realizado por: Javier Suárez Labat Tutora: María de los Ángeles Japón Pineda Catedrática de la Universidad de Sevilla Abstract This master thesis is focused on the study of amenability theory and its connection with fixed point theory for dynamical systems. A semigroup is said to be amenable if it admits a finitely additive probability measure defined in all of its subsets that is invariant by translations. We aim to give a general overview on amenability theory and its relationship with the existence on invariant means, which deeply makes use of techniques from functional analysis. We show techniques to produce examples of amenable and nonamenable groups. During this approach we prove and apply extensively Markov- Kakutani and Day’s fixed point theorems, which provide a characterization of amenability in terms of the existence of a common fixed point for a semigroup action. These results lie within the scope of the so-called “Kakutani-type” fixed point theorems in the context of topological dynamical systems. Last chapter will be devoted to the study of a wide class of this type of theorems for semigroup actions with special focus on the a Ryll-Narzdewski fixed point theorem. Some applications and further linear and non-linear extensions will be considered. i Contents Introduction 1 1 Preliminaries 4 1.1 Topological locally convex vector spaces . . . . . . . . . . . . . . . . 4 1.2 The space `∞(E)............................. 7 2 The dual of `∞(E): Bounded finitely additive measures and means 9 2.1 Bounded finitely additive measures . . . . . . . . . . . . . . . . . . . 9 2.2 Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Means and probability measures . . . . . . . . . . . . . . . . . . . . . 16 2.4 Properties of the set of means . . . . . . . . . . . . . . . . . . . . . . 20 3 Amenability 23 3.1 Algebraicprelude............................. 23 3.2 Left and right translations. Semigroup actions . . . . . . . . . . . . . 27 3.3 Amenability and means . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Examples ................................. 34 4 Amenability Characterization Through Fixed Point Theorems 39 4.1 Markov-Kakutani Theorem and Abelian Semigroups . . . . . . . . . . 40 4.2 Day’s Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . 45 5 Amenability by Examples. Paradoxical Decompositions 49 5.1 Amenability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Free groups and the von-Neumann conjecture . . . . . . . . . . . . . 56 5.3 Concrete examples of amenable and nonamenable groups . . . . . . . 57 5.4 Amenability, paradoxical decompositions and extension of measures . 60 6 Some “Kakutani-type" fixed point theorems for semigroups 65 6.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . 65 6.2 Extremal Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . 70 6.3 The Ryll-Narzdewski Fixed Point Theorem . . . . . . . . . . . . . . . 73 6.4 Someapplications............................. 75 6.5 Notes on further results . . . . . . . . . . . . . . . . . . . . . . . . . 79 Bibliography 81 Introduction This work is focused on the study of amenable semigroups and its properties with special emphasis on its connection with fixed point theory for dynamical systems. A semigroup is said to be amenable if it admits a finitely additive probability measure defined in all of its subsets that is invariant for the inner action of the semigroup. Amenability theory started with the measure problem. That is, the existence of a finitely additive extension of the Lebesgue measure to all subsets in Rnthat is invariant under rigid motions. This statement was proved to be true for n= 1,2in 1923 by Banach in [1]. Later, Banach and Tarski showed a paradoxical decomposition of the sphere in R3[2]. That is, a partition of the sphere that can be rearranged, by rigid motions, to create the original sphere twice. Amenability arises as the explanation of the difference between dimensions in the measure problem. The groups of isometries for one and two dimensions are amenable, that is, they each admit a finitely additive measure defined in all of their subsets that is invariant with respect to the group. On the other hand, the isometry group is not amenable for n≥3. The notion of amenability can also be considered for semigroups. Amenability theory was connected to functional analysis when in 1932 Banach related finitely additive measures and a set of linear operators called means. This is considered the birth of modern amenability. Taking advantage of this, in 1961 Mahlon M. Day gave a major result regarding amenable semigroups [7], which we cover in this manuscript. This result is known as Day’s fixed point Theorem and it characterizes amenable semigroups by a fixed point property, which shows an interplay among amenability theory, functional analysis and fixed point theory. Day’s fixed point Theorem is called a “Kakutani-type” fixed point result. That is, a statement of the form: Given a group or semigroup Sof continuous affine transformations on a compact convex subset Kof a locally convex vector space X 1 into itself. Then, under suitable conditions, Shas a common fixed point in K. The Kakutani-type name was coined by Namioka in [23] and still in recent day it has experienced new advances, including connections to amenability. We remark the works of various authors such us Wiśnicki [34] or Lau [19] and [20]. This work is structured as follows: In Chapter 1, we present some preliminary notions that will appear throughout the rest of this manuscript. We mainly include results on locally convex vector spaces and the Banach space `∞(E)of bounded functions from an arbitrary set Einto R. The second chapter covers the existence of an isometry between finitely additive measures of bounded variation on a set Eand the dual space of `∞(E). We make use of this isometry to introduce the concept of means and relate it to probability finitely additive measures. Then, in preparation of the next chapter, some properties of the set of means are given. During this chapter, various examples were added in order to illustrate each concept. Making use of the previous sections, Chapter 3is devoted to the notion of amenable semigroups, invariant means and their connection. The first Section is a summary on some algebraic notions regarding these structures. Another key concept on amenability are actions generated by semigroups. Specifically, we define a natural action of a semigroup Son the dual space of `∞(S)to state an equivalent definition of amenability in terms of the existence of invariant means. In this regard, Section 3.4 is dedicated to two remarkable examples: the additive semigroup N, which we relate to the study of Banach limits; and the main example of non-amenable groups, free groups. In Chapter 4we provide the Markov-Kakutani and Day’s fixed point theorems. The Markov-Kakutani theorem requires commutative operators and will be applied to prove that abelian semigroups are amenable. On the other hand, Day’s theorem is a proper characterization of amenability by fixed point properties, making it remarkably useful to prove that certain families of semigroups are amenable. This fact is mainly showcased in Section 5.1. In Chapter 5, we first showcase different techniques to generate amenable semigroups from any given amenable semigroup. Among others, we show that these examples of semigroups are amenable: subgroups of amenable groups, quotient groups of amenable groups and homomorphic images of amenable semigroups. As a conse- 2 quence every group containing a free subgroup is not amenable. This raised what is known as the von-Neumann conjecture which will be explained along Section 5.2. Besides within this Chapter, we will display some other characterizations of amenability through the study of paradoxical decompositions obtained by Følner and Tarski: a group is amenable if and only if it does not admit a paradoxical decomposition if and only if it satisfies the so called Følner condition. This will drive us to connect amenability with the measure problem. The most famous paradoxical decomposition is the one given by Banach and Tarski of the unit sphere in R3and the group of isometries. Chapter 6, inspired by Namioka [23], is devoted to various Kakutani-type results in the setting of topological dynamical systems. Markov-Kakutani and Day fixed point Theorems studied in Chapter 4 connected to amenability are just simple examples. Through this chapter, we pursue two main objectives. Firstly, we aim to prove the so called Ryll-Narzdewski fixed point Theorem and give some of its applications, as for instance, the existence of a Haar measure on every compact topological group. Lastly, several classical and recent common fixed point results under the actions of semigroups will be displayed, among which we find extensions and nonlinear counterparts of the Ryll-Narzdewski theorem. Although they do not lie within the scope of this master thesis, amenability has found in the recent years many applications in multiple areas such as Ergodic Theory, Banach Algebra, Cellular Automata and Group Theory. For more proper references regarding these topics the reader can consult the monographs [4], [16], [25] or [28]. 3 Chapter 1 Preliminaries In this first section, we aim to provide the reader with some general context in regard to topological locally convex vector spaces. Also, we will give some results on which subsequent sections are based. Since this section is designed as a simple refresh of a basic functional analysis course, some proofs will be excluded. Nevertheless, these results will each have a referenced proof. The main references for this Section are [9] and [27]. 1.1 Topological locally convex vector spaces We recall the following definition of topological vector spaces. Definition 1.1.1. A topological vector space (X, τ)is a vector space Xover a field K=R,Ctogether with a topology τsuch that (X, τ)is Hausdorff and the maps (x, y)→x+yand (λ, x)→λx are continuous. It follows from the definition that the maps y→v+yfor each fixed v∈X and y→ay for each fixed scalar a6= 0 are homeomorphisms from Xonto itself. Hence, the neighborhood system of the origin determines the whole topology. The dual space X∗of a topological vector space Xis the space of all linear and continuous maps x∗: (X, τ)→R[C]. The evaluation x∗(x)can be noted as x∗xor as hx∗, xi. In this work, we are interested in those topological vector spaces that have a base of convex neighborhoods of the origin. These spaces are said to be locally convex. 4 For the following proposition, we need to give a meaning to two operations in the space ba(E). Let m1,m2∈ba(E)and λ∈R. Then, we have the following operations •The addition m1+m2is defined as (m1+m2)(A) = m1(A) + m2(A)for each A⊂E. •The scalar multiplication λm1is defined as (λm1)(A) = λm1(A)for each A⊂E. It is easy to check that, with these operations, ba(E)is a vector space. Proposition 2.1.5. The mapping that assigns each measure m∈ba(E)its variation evaluated at E,|m|(E) = kmk, is a norm. Proof. It is easy to check that |m|(E)=0if and only if m(A)=0for all A⊆E, since |m(A)|+|m(E\A)|>0for any subset Awith non-zero measure. Let m1,m2∈ba(E)and λ∈R. Let {Ai}n i=1 be a disjoint partition of E. Then, n X i=1 |(m1+m2)(Ai)|= n X i=1 |m1(Ai) + m2(Ai)| ≤ n X i=1 |m1(Ai)|+ n X i=1 |m2(Ai)|, which implies |m1+m2|(E)≤ |m1|(E) + |m2|(E). Similarly, it holds that n X i=1 |λm1(Ai)|=|λ| n X i=1 |m1(Ai)| and thus |λm|(E) = |λ||m|(E). Proposition 2.1.6. Let m∈ba(E)be a bounded finitely additive measure on E. Then, for all A, B ∈ P(E), one has: 1) m(∅) = 0; 2) m(A∪B) = m(A) + m(B)−m(A∩B); 3) If A⊂B, then m(B\A) = m(B)−m(A). If m∈ba(E)is positive, the following also hold: 4) m(A∪B)≤m(A) + m(B); 5) If A⊂B, then m(A)≤m(B); 11 Proof. First property derives from m(E) = m(E∪∅) = m(E) + m(∅). For all A, B ∈ P(E), we have m(A∪B) = m(A) + m(B\A)and m(B) = m(A∩B) + m(B\A), which gives 2) and, if m is non-negative, implies 4). Finally, since B=A∪(B\A) and A∩(B\A) = ∅; we have m(B) = m(A) + m(B\A), yielding 5). Now applying the same reasoning as for 4),5) holds. We now define the concept of finitely additive probability measure. We will give this definition its full meaning applying the bijection in Theorem 2.2.1. Definition 2.1.7. A map m:P(E)→[0,1] is called a finitely additive probability measure on Eif it satisfies the following properties: 1. m(E) = 1, 2. m(A∪B) = m(A) + m(B)for all A, B ∈ P(E)such that A∩B=∅. The set of all probability measures on a set Eis noted as PM(E). Trivially, the set of probability measures on Eis a subset of ba(E). Example. Let x∈Ebe any element. The map defined as mx(A) =    1if x∈A, 0if x /∈A; is a finitely additive probability measure. Clearly mx(E)=1, since x∈E. Also, since any two disjoint subsets A, B ⊂Everify that xis in one and only one of them, the finitely additive property holds. Note that, as probability measures are positive, they verify all properties in Proposition 2.1.6. 2.2 Representation Theorem The upcoming results might be recognized by anyone already familiar with the Riesz representation theorem. We aim to prove that the Banach spaces `∞(E)∗and ba(E) are isometrically identified. For each subset A⊂E, we denote by χAthe characteristic map of A, that is, the map χA:E→Rdefined by χA(x) = 1 if x∈Aand χA(x) = 0 if x /∈A. Trivially 12 χA∈`∞(E)for all A⊆E. The bijection relies on the following statement. For each µ∈`∞(E)∗, one can define a measure m as m:P(E)−→ R A−→ m(A) = µ(χA). Observe that by linearity of µ, m(A∪B) = µ(χA∪B) = µ(χA+χB) = µ(χA) + µ(χB) for all A, B ⊂Esuch that A∩B=∅. Moreover, for any disjoint partition {Ai}n i=1 of E, n X i=1 |m(Ai)|= n X i=1 |µ(χAi)| = n X i=1 µ(χAi)sgn µ(χAi) =µ n X i=1 χAisgn µ(χAi)!≤ kµk, since kχAisgn µ(χAi)k= 1 for every Aiand Ai∩Aj=∅for all i6=j. Thus, taking supremum on the disjoint partitions {Ai}n i=1 of E, we get kmk ≤ kµk(2.1) Hence, m is a measure of bounded variation i.e. m ∈ba(E). Theorem 2.2.1. The map Φ : `∞(E)∗→ba(E)that assigns each mean µ∈`∞(E)∗ the measure m∈ba(E)defined as: m(A) = Φµ(A) = µ(χA), for all A⊂E, is linear and bijective. Moreover, kµk=kmk. Thus, ba(E)with its natural norm is a Banach space that is isometrically identified as the dual of `∞(E). The proof of this theorem is divided in several steps in the form of the following lemmas. The proof is strongly based on the density of the subspace of simple functions 13 in `∞(E). Although this fact can be known, we introduce all the details on the sake of completeness. The objective is, given a measure, to define a bounded linear operator in a dense subspace of `∞(E)and extend it by continuity to provide an operator on all `∞(E). Specifically, we call S(E)the set of all simple functions defined on E, that is, functions that take finite many values of R. In fact, we will use the following representation of these functions: Let x∈S(E)and {λ1, . . . , λn} ⊂ Rthe values xtakes, then x= n X i=1 λiχAi, where Ai=x−1(λi). When there is no worry of confusion, we also note the function with constant value 1as χE= 1 for simplicity of notation. Lemma 2.2.2. The vector subspace S(E)is dense in `∞(E). Proof. Let x∈`∞(E). Now set α= infExand βE= supEx. Since xis bounded |α|,|β|<+∞, we can define, for every n∈N, the finite set {λi}n i=1, where λi= α+i(β−α)/n for i= 1, . . . , n. Now consider the map yn:E−→ Rdefined by yn(a) = min{λi:x(a)≤λi}, for all a∈E. The map ynis a simple function since it takes values in the set {λ1, . . . , λn}. This function also satisfies by construction that kx−ynk∞≤(β−α)/n. Consequently, S(E)is dense in `∞(E)in regards to k·k∞, since we have defined a sequence of simple functions {yn}convergent to xfor an arbitrary x∈`∞(E). Let m be a finitely additive measure on E. We define the map µ:S(E)→Rsuch as, for all x∈S(E), µ(x) = µ n X i=1 λiχAi!= n X i=1 λim(Ai),(2.2) where {λ1, . . . , λn}are the finite values of xand Ai=x−1(λi). 14 Lemma 2.2.3. For each measure m∈ba(E), the map µ:S(E)−→ Rdefined above is linear and continuous with |µ(x)| ≤ kxkkmk, for all x∈S(E). Proof. Let x, y ∈S(E)and ψ, η ∈R. Denote V(respectively W) the set of values taken by x(resp. y). Then the family of subsets {x−1(α)∩y−1(β)}(α,β)∈V×Wis a partition of E. By definition of µwe get: µ(ψx +ηy) = X (α,β)∈V×W m(x−1(α)∩y−1(β))(ψα +ηβ) = =ψX (α,β)∈V×W m(x−1(α)∩y−1(β))α+ +ηX (α,β)∈V×W m(x−1(α)∩y−1(β))β= =ψµ(x) + ηµ(y). Consequently, µis linear. Let x∈S(E),V={λ1, . . . , λn}be the set of values taken by xand Ai= x−1({λi}). Let λ= max{|λi|: 1 ≤i≤n}. Then, since all Aiare disjoint, |µ(x)|= n X i=1 λim(Ai)≤λ n X i=1 |m(Ai)|!≤λkmk=kxkkmk. Hence, we have that kµk ≤ |m(E)|<+∞i.e. µis continuous. Lemma 2.2.4. Let Xbe a normed vector space and let Ybe a dense vector subspace of X. Suppose that ϕ:Y−→ Ris a continuous linear map. Then the exists a continuous linear map ˜ϕ:X−→ Rsuch that ˜ϕ|Y=ϕwith k˜ϕk=kϕk. Proof. Let x∈X. Since Yis dense, there exists a sequence {yn}n≥0in Ysuch that lim yn=x. For all, p, q ≥0, we have that |ϕ(yp)−ϕ(yq)|=|ϕ(yp−yq)| ≤ kϕkkyp−yqk. Thus, {ϕ(yn)}is a Cauchy sequence and it converges as Ris complete. Now set ˜ϕ(x) = lim ϕ(yn). Notice this definition does not depend on the sequence. Let {y0 n} be another sequence on ywhich tends to xthen it holds that |ϕ(yn)−ϕ(y0 n)| ≤ kϕkkyn−y0 nk. 15 Linearity is inherited by the linearity of ϕ. Let x, ˆx∈Xand {yn},{ˆyn} ⊂ Y sequences convergent to xand ˆxrespectively. Let α, β ∈R. Then lim ϕ(αyn+βˆyn) = lim(αϕ(yn)+βϕ(ˆyn)) = αlim ϕ(yn)+βlim ϕ(ˆyn) = α˜ϕ(x)+β˜ϕ(ˆx). It holds that kϕ(yn)k ≤ kϕkkynk, for all n≥0. Thus k˜ϕ(x)k ≤ kϕkkxk,(2.3) for all x∈X. This shows that ˜ϕis continuous. Also, if x∈Ywe can take yn=x for all n≥0, so ˜ϕ|Y=ϕ. It remains to prove that k˜ϕk=kϕk. From (2.3) we have k˜ϕk≤kϕkand, since ˜ϕ(x) = ϕ(x)for all x∈S(E), we have that: k˜ϕk= sup{|˜ϕ(x)|:x∈X, kxk= 1} ≥ {|˜ϕ(x)|:x∈Y, kxk= 1}=kϕk. Thus k˜ϕk=kϕk. Proof of Theorem 2.2.1. Let m ∈ba(E). Apply the extension in Lemma 2.2.4 with Y=S(E),X=`∞(E)and ϕ=µfrom (2.2). Call µthe extension of µ. By construction of µ, it holds that, µ(χA) = µ(χA) = m(A), for all A⊂E. Thus φ(µ) = m and φis surjective. It remains to show that φis injective. Consider µ1, µ2∈`∞(E)∗such that φ(µ1) = φ(µ2), which implies µ1(χA) = µ2(χA), for every A⊂E. By linearity, µ1and µ2agree on every element in S(E). Since S(E)is dense in `∞(E)and µ1, µ2are continuous they agree in all of `∞(E). Lastly, combine the norm inequalities in (2.1) and 2.2.3 to get kµk=kµk ≤ kmk ≤ kµk, i.e. kmk=kµk. 2.3 Means and probability measures Now that we have our representation theorem identifying `∞(E)∗with ba(E)assembled, we introduce the notion of means as operators in `∞(E)∗that satisfy certain 16 properties. The next objective is to prove the relationship between means and probability finitely additive measures. Definition 2.3.1. Let µin `∞(E)∗. Then µis called a mean on `∞(E)if it satisfies: inf{f(x) : x∈E} ≤ µ(f)≤sup{f(x) : x∈E},(2.4) for all f∈`∞(E). The set of all means on the set Eis noted by M(E). The following basic operators are examples of means. Example. Let Ebe any set. For each e∈E, the functor: δe:`∞(E)−→ R f−→ f(e), is trivially a mean, since infEf≤δe(f) = f(e)≤supEf. Thus, the set of means on E,M(E), is non-empty. We now prove equivalent conditions for linear and continuous maps in `∞(E)∗to be means. Recall that the notation 1∈`∞(E)corresponds to the constant function of value 1,χE. Proposition 2.3.2. Let µ∈ M(E)be a mean. Then µsatisfies the following properties: µ(f)≥0if f≥0; (I) µ(1) = 1; (II) kµk= 1.(III) Conversely, if µ∈`∞(E)∗satisfies any two of the conditions (I),(II) and (III), then µis a mean. Proof. We start assuming µ∈`∞(E)∗is a mean and proving the three conditions above. First and second properties are immediately derived from the inequality in the definition 2.3.1: µ(f)≥inf{f(x) : x∈E} ≥ 0,∀f≥0 17 and 1 = inf{1(x) : x∈E} ≤ µ(1) ≤sup{1(x) : x∈E}= 1. Consequently, (II) and (2.3.1) implies (III) since, for any f∈`∞(E), kµ(f)k=|µ(f)| ≤ sup{|f(x)|:x∈E}=kfk thus, kµk ≤ 1and, 1 = |µ(1)| ≤ kµkk1k=kµk. We will now prove that any two conditions in the statement imply (2.4): 1. First assume (I) and (II). Then let f∈`∞(E)and define a= inf{f(x) : x∈E}, b= sup{f(x) : x∈E}. thus f−a, b −f≥0. Now, simply apply the assumptions: 0≤µ(f−a) = µ(f)−µ(a) = µ(f)−aµ(1) = µ(f)−a=⇒µ(f)≥a, 0≤µ(b−f) = µ(b) + µ(−f) = bµ(1) −µ(f) = b−µ(f) =⇒µ(f)≤b. 2. Assuming (I) and (III) we will proof µis a mean by proving (II) holds by our last reasoning. We will prove µ(1) = 1 proving both inequalities. First, (I) and linearity yield µ(1) ≥1, since µ(0) ≥0and µ(1) −1 = µ(0) ≥0. Now apply (III) and get µ(1) = |µ(1)| ≤ kµkk1k= 1. 3. Lastly, assuming (II) and (III) we will prove (I). Let f≥0in `∞(E)and define aand blike in 1). Then we have b≥a≥0and b−f≥0. We also have, by (II), that µ(c) = cµ(1) = cfor all constants c. Finally, also applying linearity and (III) we have b−µ(f) =µ(b−f)≤ kµkkb−fk=kb−fk= = sup{b−f(x) : x∈E}=b+ sup{−f(x) : x∈E}=b−a. Thus, µ(f)≥a≥0for all f≥0. 18 Taking advantage of the results in Section 2.2, we can now easily prove the existence of a bijection between the set of probability measures PM(E)and the set of means M(E). For each subset A⊂E, we denote by χAthe characteristic map of A, that is, the map χA:E→Rdefined by χA(x)=1if x∈Aand χA(x)=0otherwise. The bijection relies on the following statement. Let µ∈ M(E)be a mean, then one can define the measure m ∈ PM(E)such as m:P(E)−→ [0,1] A−→ m(A) = µ(χA). Observe that m(E) = µ(χE) = µ(1) = 1 and, by linearity of µ, m(A∪B) = µ(χA∪B) = µ(χA+χB) = µ(χA) + µ(χB) for all A, B ⊂Esuch that A∩B=∅. Thus, m is indeed a measure in PM(E). Theorem 2.3.3. The map Φ : `∞(E)∗→ba(E)from Theorem 2.2.1 defines a bijection between M(E)and PM(E). More formally, the map Φ : M(E)→ PM(E)that assigns each mean µ∈ M(E) the measure m∈ba(E)defined as: m(A) = Φµ(A) = µ(χA), for all A⊂E, is bijective. Proof. Since Φ : `∞(E)∗→ba(E)is a bijection, it suffices to prove that Φ(M(E)) = PM(E). 1. From Theorem 2.3.3, we have that Φis an isometry. Hence, Φ|M(E)is injective and kΦ(µ)k=kµk= 1, for all µ∈ M(E). Moreover, since µ(f)≥0for any f≥0, we have that Φ(µ)(A) = µ(χA)≥0. Thus, Φ(µ)is a finitely additive positive measure verifying kΦ(µ)k= Φ(µ)(E) = 1, i.e. Φ(µ)∈ PM(E). Hence, Φ(M(E)) ⊆ PM(E). 19 2. On the other hand, from the definition of Φwe have that, given a measure m∈ PM(E), then the operator Φ−1(m)verifies kΦ−1(m)k=kmk= 1 and µ(1) = µ(χE) = m(E) = 1. Thus, by Proposition 2.3.2, Φ−1(m)is a mean. Hence, Φ−1(PM(E)) ⊆ M(E). Combining both inclusions, we conclude that Φ : M(E)→ PM(E)is bijective. 2.4 Properties of the set of means Our next interest is to provide various properties of the set of means that will be used in plenty of the upcoming results. In this segment, various techniques from functional analysis, such us the Hahn-Banach separation Theorem 1.1.4, will be applied. Proposition 2.4.1. The subset M(E)of means on Eis a convex and compact subset of `∞(E)∗with respect to the ω∗-topology. Proof. Let µ1, µ2∈ M(E)and α∈[0,1] arbitrary. Then, by Proposition 2.3.2, to show (αµ1+ (1 −α)µ2)is a mean it suffices to prove that (αµ1+ (1 −α)µ2)(1) = 1 and (αµ1+ (1 −α)µ2)(f)≥0, for all f≥0. The first holds, since µ1(1) = 1 = µ2(1). Now let f≥0. Then µ1(f), µ2(f)≥0and αµ1(f) + (1 −α)µ2(f)≥0 for every α∈[0,1]. It remains to prove that M(E)is compact. Let {µi}i∈Ibe a net in M(E)converging to u∈`∞(E)∗. Then for all f∈`∞(E)we have: inf Ef≤µi(f)≤sup E f, and taking limits we get inf Ef≤u(f)≤sup E f. Thus, every convergent net in M(E)converges to a mean i.e. M(E)is closed in `∞(E)∗with respect to the ω∗-topology. Lastly, since M(E)is contained in {u∈`∞(E)∗:kuk ≤ 1}, which is compact by the Banach-Alaoglu Theorem 1.2.2, M(E)is compact. 20 We conclude this section with the definition of free groups. Definition 3.1.9. Let Mbe a set. The free group Fwith generating set Mis the group of all finite concatenations, or words, of letters from {σ, σ−1:σ∈M}, where two words are equivalent if they differ in finite pairs of adjacent letters of the form σσ−1or σ−1σ, called trivial syllables. In a free group F, a word with no trivial syllables is called a reduced word and Fmay be taken to consist of all reduced words. A free group with ngenerators is noted as Fn. This definition can be extended to semigroups, with a free semigroup generated by Mbeing the set of finite concatenations {σ:σ∈M}. Notice that, in free semigroups, trivial syllables might not exist. 3.2 Left and right translations. Semigroup actions In this section we work with the definition of a semigroup acting on a set. Then, we will define the specific actions of semigroups in the spaces ba(S)and `∞(S)∗. Although the definitions are given separately, the isometry seen in last section indicates a relation between such actions. Definition 3.2.1. Let Sbe a semigroup. An action of Son a set Eis defined as a mapping of the form S×E−→ E (s, x)7−→ s(x). that satisfies, (st)x=s(tx), for all s, t ∈Sand for all x∈E. When no confusion might arise, we will note actions as s(x) = sx. Notice that a semigroup operation defines a natural action of the semigroup on itself. More formally, when E=S,(s, t)7→ st is called the left action of the semigroup Son itself. Another relation we can consider is sometimes called the right anti-action of Son itself as it is the converse of the left action, that is (s, t)7→ ts. 27 Definition 3.2.2. Let Sbe a semigroup. We define the left [right] mappings of Son the space ba(S), S×ba(S)−→ ba(S) (s, m)7−→ sm[ms], as sm(A) = m(s−1A) [ms(A) = m(As−1)], for each A⊂Sand s∈S, where s−1A={t∈S:st ∈A}[As−1={t∈S:ts ∈A}]. The definition of s−1A[As−1] is not required for groups, since s−1exists for every element of the group and thus s−1Aagrees with the usual concept s−1A={s−1t:t∈ A}. It is worth remark, that the commutative property yields that left and right mappings are equivalent. More formally, if Sis a commutative semigroup, then st ∈Aif and only if ts ∈Ayielding s−1A={t∈S:st ∈A}={t∈S:ts ∈A}=As−1. Example. Let (Z,+) be the additive group of integers. Let m ∈ PM(Z)be a measure and let k∈Z. Then the definition above translates to: for each A⊂Z, km(A) = m(−k+A) = m(A−k) = m({n−k:n∈A}). Notice that −k+A=A−kbecause the addition is commutative. Now we translate this notion to the semigroup (N,+). In this case, we must use the formal definition of s−1A={t∈S:st ∈A}. Let k∈Nand let A⊂N. Then, the definition remains as follows: km(A) = m(k−1A) = m({n∈N:k+n∈A}) = m({n−k:n∈A}∩N). Notice how with this simple semigroup example, we can show that s−1Acan be empty even if Ais nonempty. For example, if A={1, . . . , m}then (m+ 1)−1A={n∈N: m+1+n∈A}=∅. Example. Another illustrative example is the unit circumference ∂Din Cwhich, along with the usual product, yields a group. Each z∈∂Drepresents a rotation. Thus, let m∈ PM(∂D)and let w∈∂D, then w−1=wand wm(A) = m(w−1A) = m(wA). 28 Proposition 3.2.3. For any given measure m∈ba(S)and for every s∈S, the mapping sm[ms]is a measure in ba(S). Proof. It suffices to prove that, for any given s∈S, every disjoint covering of S, {Ai}n i=1, verifies that {s−1Ai}n i=1 is also a disjoint covering of S. First, let t∈s−1Ai∩s−1Aj. Then st ∈Ai∩Ajwhich only makes sense if i=j. On the other hand, let t∈S. Then, st ∈Sand hence st lies inside one and only one Aiand, by definition, t∈s−1Ai. Thus {s−1Ai}n i=1 is a disjoint covering of S. Recall that |m|=m for any positive measure m. Thus, as a trivial case of last property, we have that any probability measure m in PM(S)verifies that sm,ms∈ PM(S). We now introduce the definition of left and right translations of functions in `∞(S). For each s∈S, the semigroup structure induces two linear operators `sand rsfrom `∞(S)to `∞(S)for each s∈S, defined respectively as, `sf(t) = f(st)and rsf(t) = f(ts), for each f∈`∞(S). Thus, we have the following left [right] translation. S×`∞(S)−→ E (s, f)7−→ `sf[rsf]. Since |`sf(t)|,|rsf(t)|≤kfk, for all t∈S;`sfand rsfare in `∞(S). Their adjoints, `∗ sand r∗ s, are defined as `∗ sµ(f) = µ(`sf)and r∗ sµ(f) = µ(rsf), for each µ∈`∞(S)∗and f∈`∞(S). Analogously, we have the following maps, S×`∞(S)∗−→ E (s, µ)7−→ `∗ sµ[r∗ sµ]. The definition of the left operator on `∞(S)∗does hold the condition of action as `∗ s(`∗ tµ)(f) = `∗ tµ(`sf) = µ(`t`sf) = µ(`stf) = `∗ stµ(f). On the other hand, the right translation verifies r∗ sr∗ tµ= r∗ tsµ. These operators will be used to define amenability in terms of means. Next result will yield the continuity of left and right translations and their adjoints. The proof is only detailed for left maps as it is analogous for the right case, rsand r∗ s. 29 Proposition 3.2.4. The left [right] translations of Son `∞(S)are linear and normcontinuous with k`sk= 1 [krsk= 1]. Consequently, its adjoint `∗ s[r∗ s]is linear on `∞(S)∗and k`∗ sk= 1 [kr∗ sk= 1]. Proof. Linearity of both operators is easily checked by definition. Let f1, f2∈`∞(S), µ1, µ2∈`∞(S)∗and α, β ∈R, then `s(αf1(t) + βf2(t)) = αf1(st) + βf2(st) = α`sf1(t) + β`sf2(t), `∗ s(αµ1(f) + βµ2(f)) = αµ1(`sf) + βµ2(`sf) = α`∗ sµ1(f) + β`∗ sµ2(f). As it was shown in the definition of `s,|`s(f)|≤kfk. Thus `sis continuous and k`s(f)k ≤ 1. In fact, k`sk= 1 since `s(1) = 1. Hence, k`∗ sk=k`sk= 1, since taking adjoints preserves the norm. Corollary 3.2.5. The operators `sand `∗ s[rsand r∗ s]are weak and weak∗continuous respectively. Proof. Both statements are immediate, since strongly continuous and affine implies weakly continuous (Proposition 1.1.6) and Proposition 1.1.3 implies `∗ sis weak∗continuous. 3.3 Amenability and means We now introduce the definition of amenable semigroups. Definition 3.3.1. Let Sbe a semigroup. A measure m∈ PM(S)is called left [right] invariant if sm(A) = m(A) [ms(A) = m(A)], for each s∈Sand for each A⊂S. If mis both left and right invariant, then it is called a two-sided invariant measure or simply an invariant measure. Definition 3.3.2. A semigroup Sis called left amenable if it admits a measure m∈ PM(S)that is left invariant. Analogously, Sis called right amenable if it admits a right invariant measure. If the semigroup Sadmits a measure m∈ PM(S)that is left and right invariant then Sis said to be two-sided amenable. 30 Recall that left and right maps agree on commutative semigroups. Thus, sm=ms for each m ∈ PM(S)and for every s∈S. That is, any measure on a commutative semigroup that is left or right invariant is in fact two-sided invariant i.e. left, right and two-sided amenability are equivalent for commutative semigroups. Example. To visualize the idea of an amenable semigroup we will show some initial examples. •If Gis a finite group, the it is amenable. We define the cardinal measure as the number of elements in a set C,|C|. Then, the following measure is left invariant, m(A) = |A| |G|. Since every element of Ghas an inverse, g−1Ahas the same number of elements as Aand thus m is indeed a left invariant measure in G(in fact, it is a two-sided invariant measure). •Let Sbe any semigroup with a null element 0, that is an element that verifies s0 = 0s= 0 for all s∈S. Then Sis amenable with measure: m(A) =    1if 0∈A, 0if 0/∈A. It is indeed left and right invariant since 0∈Aif and only if 0∈s−1A[As−1]. We only detail left invariance since right invariance is analogous. First if 0∈A, then s0[0s]∈Afor all s∈Sand 0∈s−1A={t∈S:st ∈A}. Also, if 0∈s−1Afor any s∈Sthen s0 = 0 ∈A. In fact, This is the only left invariant measure on a semigroup with a 0element. Notice that 0−1A={t∈S: 0t∈A}=   Sif 0∈A, ∅if 0/∈A. Now, by definition of left invariant measure, m must verify that m(A) = m(0−1A) =    1if 0∈A, 0if 0/∈A. 31 Notice we have proved that Rwith the product is amenable. Another example of semigroup with a null element is the semigroup of linear operators on a vector space X with the composition. Since T(0) = 0 for every linear operator T, then the constant operator defined N(x) = 0 for all x∈Xis a null element for the composition NT =N=TN for every linear operator Ton X. Thus such semigroup is amenable. Definition 3.3.3. Let Sbe a semigroup. •A mean µon M(S)is called left [right] invariant if µ(`sf) = µ(f) [µ(rsf) = µ(f)], for each s∈Sand for all f∈`∞(S). •A mean µon M(S)is called a two-sided invariant mean if it is both left and right invariant. Notice that if a mean µis left [right] invariant then it must verify `∗ sµ=µ[r∗ sµ=µ] as elements of `∞(S)∗, for all s∈S. Unsurprisingly, as it is shown in the following proposition, from the construction of the isometry defined in Theorem 2.3.3 naturally follows that invariant means are in correspondence with invariant probability measures. Hence, next result yields an equivalent definition of amenability. Proposition 3.3.4. Let µ∈ M(S)and Φthe isometry in Theorem 2.3.3. Then µis left [right] invariant if and only if the measure m= Φ(µ)is left [right] invariant. Proof. As always the proof will only be detailed for left invariance. If µis a left invariant mean, then m = Φ(µ)is clearly invariant since, for all A⊂Sand s∈S, m(A) = µ(χA) = `∗ sµ(χA) = µ(`sχA) = µ(χs−1A) = m(s−1A) = sm(A). Now let m ∈ PM(S)be a left invariant probability measure. Then µ= Φ−1(m) is left invariant for simple functions. Indeed it holds that, `∗ sµ(χA) = µ(χs−1A) = sm(A) = m(A) = µ(χA), for all A⊂Sand s∈S. Thus, by linearity of µand `∗ sfor all s∈S, and continuity of `∗ s(Proposition 3.2.4); µis left invariant. This discussion is summarized in the following corollary. 32 Corollary 3.3.5. Let Sbe a semigroup. Then the following assertions are equivalent. a) S is left amenable. b) There exists a probability measure m∈ PM(S)such that m(s−1A) = m(A), for all s∈Sand for all A⊂S. c) There exists a mean µ∈`∞(S)∗such that µ(`sf) = µ(f)for all s∈Sand for all f∈`∞(S). That is, `∗ sµ=µfor all s∈S. Analogous for right and two-sided amenability. Notice that condition c)asserts that a left invariant mean is a fixed point of the adjoint operators `∗ sfor all s∈S. To conclude this segment, we expose the main utility of the set of finite means M1(S)and Theorem 2.4.3. Using the notation in Day’s article [6], we have the following definition. Definition 3.3.6. Let Sbe a semigroup. A net {mα}⊂M1(S)of finite means is said to converge to left invariance if ω∗-lim `∗ smα−mα= 0 i.e. ω∗-lim `smα(f)−mα(f) = 0, for all s∈Sand for all f∈`∞(S). Proposition 3.3.7. Let Sbe a semigroup. The following are equivalent: a) Sis left amenable. b) There exists a net (µα)of finite means that is convergent to left invariance. Proof. By Corollary 3.3.5, Sis left amenable if and only if there exists a left invariant mean in `∞(S)∗. If µ∈ M(S)is a left invariant mean, the existence of a net (µα)of finite means converging to m is immediate by density of M1(S)(Theorem 2.4.3). Thus, the net must verify 0 = `∗ sµ−µ=ω∗-lim `∗ sµα−µα, for all s∈S, since the action `∗ sis ω∗-to-ω∗continuous (Corollary 3.2.5). 33 On the other hand, let (µα)be a net of finite means on M1(S)converging to left invariance. Then, taking a subnet if needed, by compactness of M(S)there exists a mean µsuch that ω∗-lim µα=µ. Now, since `∗ sis continuous for every s∈S, we have that 0 = ω∗-lim `∗ sµα−µα=`∗ sµ−µ. Thus, m is a left invariant mean on M(S). As a consequence of the previous argument, we can remark that every cluster point of a net of finite means convergent to left invariance corresponds to a left invariant mean on M(S). 3.4 Examples In this section we will study some examples of amenable and non-amenable semigroups. We will start with the semigroup of positive integers with the additive operation and the relation of left invariant means with the concept of Banach limits. Proposition 3.4.1. The additive semigroup of positive integers (N,+) is an amenable semigroup. Proof. Let δkbe the mean defined in the example after Proposition 2.3.2 for each k∈N. That is, δk(f) = f(k)for each f∈`∞(N). Consider the sequence {µn}n∈Ndefined as µn=1 n n X k=1 δk. Each µnis a finite mean as it is an affine combination of δk. We now prove that this net converges to left and right invariance. Since addition is commutative, left and right operators agree i.e. `s=γs, for all s∈N; so it suffices to show that lim nµn(`sf)−µn(f) = 0, 34 for all f∈`∞(N). Note that, by definition and linearity of µn,µn(`sf)is defined as µn(`sf) = 1 n n X k=1 δk(`sf) = 1 n n X k=1 f(s+k). Thus, for each fixed s∈Nand f∈`∞(N), we can assume that n>s, yielding the following |µn(f)−µn(`sf)| ≤ 1 n n X k=1 |f(k)−f(s+k)|= =|f(1) + ···+f(s)−f(n+ 1) −···−f(s+n)| n≤ ≤2sC n n→∞ −−−→ 0, where C= sup{|f(k)|:k∈N}. Thus, {µn(f)}is convergent to left and right invariance for all f∈`∞(N)i.e. µnis convergent to both-side invariance. Thus, by Proposition 3.3.7, there exists a both-side invariant mean µ∈`∞(N)∗. In the next note, we relate this proposition to Banach limits. Definition 3.4.2. A Banach limit is a continuous linear functional φ:`∞(N)→R that verifies the following properties for each f∈`∞(N): 1. If f≥0, then φ(f)≥0. 2. φ(f) = φ(Sf), where S:`∞(N)→`∞(N)is the shift operator i.e. Sf(n) = f(n+ 1) for all n≥1. 3. If (f(n))n∈Nis a convergent sequence, then φ(f) = limnf(n). The next Proposition is an interesting relation between means, amenability and Banach limits. Notice that we specify the group Nas the additive semigroup of positive integers, since as the multiplicative semigroup the statement would not hold. Proposition 3.4.3. Let Nbe the additive semigroup of positive integers. Then the set of all Banach limits on `∞(N)is exactly the set of all invariant means in `∞(N)∗. 35 Proof. Notice that a continuous linear functional φ:`∞(N)→Rverifies 1.and 3. on the definition of a Banach limit then it is a mean. This holds from Proposition 2.3.2. Property 1.is in fact the property I in Proposition 2.3.2 and 3.implies φ(1) = limn1 = 1, where 1is the constant function f(n) = 1, for all n≥1. Now let µbe an invariant mean in `∞(N)∗. Then, µis a continuous linear functional on `∞(N)by definition and the shift operator Sis the map `1. Thus, by invariance of µ,µ(Sf) = µ(`1f) = µ(f), verifying 2.. Lastly, 3.holds from 2.and the definition of mean since Skf(n) = f(n+k) = `kf(n)i.e. Sk=`k. Hence, if f∈`∞(N)defines a convergent sequence f(n)→x, then for every ε > 0there exists N > 0such that for every n≥N, x−ε < f(n)< x +ε. That is Skf(n)∈(x−ε, x +ε)for each k≥Nand for all n∈N. Now, since by definition of mean inf nf(n)≤µ(f)≤sup n f(n) for each f∈`∞(N), we can write µ(f) = µ(`kf) = µ(Skf)to get x−ε < inf nSkf(n)≤µ(f)≤sup n Skf(n)< x +ε, for all k≥N. That is, for each ε > 0,µ(f)∈(x−ε, x +ε)yielding µ(f) = xi.e. 3. holds. On the other hand, let φbe a Banach limit. Then, since we have proved that every ϕ∈`∞(N)∗verifying 1.and 3.in the definition of Banach limit is a mean, φis a mean. Now, since S=`1and Sk=`k, we can apply induction on property 2.of a Banach limit to get φ(f) = φ(Sf) = µ(Skf) = µ(`kf), for every k∈N. That is, φis an invariant mean. Thus, we have shown that the set of all Banach limits on `∞(N)is exactly the set of all invariant means in `∞(N)∗. We now prove that free groups and semigroups of order two or greater defined in Section 3.1 are not amenable. Proposition 3.4.4. Free group of order two, or greater is neither left or right amenable. 36 Notice that, since all Ciare closed, all (Ci)care open subsets. Thus Fis a cover of X. Also, the fact that all finite subcollections of Fhave nonempty intersection yields that n [ j=1 (Cj)c6=X. That is, there is {(Ci)c}i∈Iis a covering of Xthat has no finite subcovering i.e. Xis not compact. Lastly, since this proof can be written backwards without any changes, the reciprocal statement is already proved. Theorem 4.1.5 (Markov-Kakutani).Let Kbe a nonempty convex compact subset of a topological vector space X. Let Fbe a family of continuous affine maps f:K→K such that all elements of Fcommute, that is, f1◦f2=f2◦f1for all f1, f2∈ F. Then there exists a point in Kwhich is fixed by all the elements of F. Proof. First, for each f∈ F, define Fix(f)as the set of all fixed points of fon K. More formally, Fix(f) = {x∈K:f(x) = x}. We want to show that \ f∈F Fix(f)6=∅. Every set Fix(f)is closed in X, since for every net (xi)i∈Ion Fix(f)converging to x∈Xwe have that, by continuity of f, f(x) = lim f(xi) = lim xi=x, thus x∈Fix(f). Now, since Kis compact, proving the theorem is equivalent to proving that the collection {Fix(f)}f∈F has the finite intersection property. First, note that the set Fix(f)is nonempty for all f∈ F by Lemma 4.1.3. It is compact since, as we proved above, it is a closed subset of the compact set K. Also, for all x, y ∈Fix(f)and for all α∈[0,1] it holds that, by linearity of f, f(αx + (1 −α)y) = αf(x) + (1 −α)f(y) = αx + (1 −α)y. Thus, Fix(f)is convex. 43 Now, let g∈ F and x∈Fix(f). Then the fact that fand gcommute yields that f(g(x)) = g(f(x)) = g(x), that is, g(x)∈Fix(f). Therefore, g|Fix(f):Fix(f)−→ Fix(f) is well defined and we can apply Lemma 4.1.3 to obtain a fixed point of fthat is fixed as well by gi.e. Fix(f)∩Fix(g)6=∅. By induction on the number of maps, we get that for any f1, . . . , fn∈ F, n \ i=1 Fix(fi)6=∅. To conclude the proof, apply the intersection property, since Kis compact, which implies that \ f∈F Fix(f)6=∅. That is, there exists a point in Kthat is fixed by all maps of F. We now proceed to prove that all abelian semigroups are two-sided amenable. We comment beforehand the fact that, if Sis abelian `s= rsfor all s∈S, since `sf(t) = f(st) = f(ts) = rsf(t), for all t∈Sand f∈`∞(S). Corollary 4.1.6. Let Sbe an abelian semigroup. Then Sis two-sided amenable. Proof. By Corollary 3.3.5, it suffices to prove that there exists a mean in `∞(S)∗ which is left invariant. As a consequence, since Sis abelian, Sis two-sided amenable. First, we recall from Proposition 2.4.1 that the set of means M(S)is a ω∗-compact and convex subset of `∞(S)∗. Also, we recall that, by Proposition 3.2.4, left actions 44 on `∞(S)∗are linear and continuous maps whose images of means are as well means. Then, if Sis abelian, we have that `u`vf(t) = `uf(vt) = f(uvt) = f(vut) = `vf(ut) = `v`uf(t), for all f∈`∞(S)and for all t∈S. This translates to the adjoint actions as the equality `∗ u`∗ vµ(f) = `∗ uµ(`vf) = µ(`u`vf) = µ(`v`uf) = `∗ vµ(`uf) = `∗ v`∗ uµ(f), for all µ∈`∞(S)∗and for all f∈`∞(S). We need to prove that the set of left actions has a common fixed point on M(S) i.e. there exists a mean µsuch that `∗ sµ=µfor all s∈S. Apply the Markov-Kakutani theorem with X=`∞(S)∗,K=M(S)and F= {`∗ s:s∈S}. Thus we obtain a mean that is a fixed point for all `∗ sand r∗ s. That is, the mean µsatisfies `∗ sµ= r∗ sµ=µ, for all s∈S. That is, `∗ sµ(f) = µ(`sf) = µ(f), for all f∈`∞(S)and for all s∈Si.e. µis left invariant. Finally, as discussed above, if Sis abelian `s= rsand thus µis two-sided invariant. 4.2 Day’s Fixed Point Theorem As it could be checked in the previous section, fixed point theorems can provide the existence of left invariant means and, in particular, can determine when a semigroup is amenable. In this section, we will prove Day’s Fixed Point Theorem that characterizes amenability through a common fixed point theorem. The proof was taken from Day’s article [7]. Theorem 4.2.1. Let Sbe a left amenable semigroup. Then every representation of Sas continuous affine maps from a compact convex set Kin a locally convex vector space (X, τ)into it-self has a common fixed point. 45 Proof. Let Sbe a semigroup acting on a compact convex set Kin a locally convex vector space X. As the action of Son Kis continuous and affine by hypothesis, the mappings s: (K, τ)−→ (K, τ) x7−→ sx are continuous for the weak topology as well, by Proposition 1.1.6. Fix a point p∈K and, for each x∗∈X∗, define fx∗:S−→ R s7−→ hx∗, spi, Note that fx∗∈`∞(S), since sp ∈Kfor every s∈Sand x∗is bounded on C. We recall that the means δsare defined as δs(f) = f(s), for each f∈`∞(S) (see Example after definition 2.3.1). Let s1, . . . , sn∈Sand λ1, . . . , λn≥0such that Pn i=1 λi= 1. Define the mean µ∈ M1(S)as µ= n X i=1 λiδsi. and define xµas xµ= n X i=1 λisip, (4.2) which, by convexity of K, is a point in K. Notice that µ(fx∗) = n X i=1 λiδsi(fx∗) = n X i=1 λifx∗(si) = = n X i=1 λihx∗, sipi=hx∗, n X i=1 λisipi=hx∗, xµi. Also, for every t∈Swe have that the action of ton Kis affine and thus µ(`tfx∗) = n X i=1 λiδsi(`tfx∗) = n X i=1 λifx∗(tsi) = = n X i=1 λihx∗, tsipi=hx∗, n X i=1 λitsipi=hx∗, t n X i=1 λisip!i=hx∗, txµi. 46 Since Sis left amenable, we can choose a left invariant mean µ0. Now, by Proposition 3.3.7, there exists a net {µα} ⊂ M1(S)of finite means verifying that µα ω∗ −→ µ0 and that {µα}is convergent to left invariance, yielding lim αµα(`tf)−µα(f) = 0,(4.3) for all f∈`∞(S)and for all t∈S. Finally, let xµαdefined as in (4.2). Then, since Kis τcompact, there is a convergent subnet (µβ)⊂(µα). Let x0∈Kbe the limit of (xµβ). Let t∈S, it only remains to prove that tx0=x0. Note that, since Xis locally convex, it suffices to show that hx∗, tx0i=hx∗, x0i, for every x∗∈X∗. Fix x∗∈X∗and consider the corresponding function fx∗. We showed at the beginning of the proof that µβ(fx∗) = hx∗, xµβiand µβ(`tfx∗) = hx∗, txµβi. Thus, since tis ω-continuous, µβ(fx∗) = hx∗, xµβi → hx∗, x0i, µβ(`tfx∗) = hx∗, txµβi → hx∗, tx0i. Finally, apply (4.3) to fx∗to yield the desired result 0 = lim αµα(`tfx∗)−µα(fx∗) = lim βµβ(`tfx∗)−µβ(fx∗) = hx∗, tx0i−hx∗, x0i. This result is indeed a generalization of the Markov-Kakutani Theorem 4.1.5 swapping the commutativity with the condition that the family of maps forms (or is contained) in a left amenable semigroup. In fact, it immediately provides a fixed point characterization of amenability. Corollary 4.2.2. Let Sbe a semigroup. The following assertions are equivalent: a) Sis left amenable. b) Every representation of Sas continuous affine maps from a compact convex set Kin a locally convex space (X, τ)into it-self has a common fixed point. 47 Proof. First implication comes from last Theorem. Thus assume that every representation of Sas continuous affine maps from a compact convex set in a locally convex space into it-self has a common fixed point. Now let M(S)be the compact convex set on the Banach space `∞(S)∗with the ω∗-topology. Naturally, {`∗ s:s∈S}is a representation of Sacting on the set of means M(S)which, by Proposition 3.2.4, are affine and continuous. Then, there exists a common fixed point of the actions of S on M(S)i.e. there is a mean µ∈ M(S)verifying `∗ sµ=µfor all s∈S. Hence, by definition, Sis left amenable. 48 Chapter 5 Amenability by Examples. Paradoxical Decompositions In this chapter we will show different examples of amenable and nonamenable semigroups. Moreover, we will showcase a collection of processes through which one can get new amenable semigroups from any given amenable one. In order to do so, we will make use of all the results provided in previous chapters. Afterwards, we will relate amenability theory with paradoxical decompositions. We will discuss the main statement that connects amenable groups with paradoxical results called the Tarski Theorem, which establishes that a group is paradoxical if and only if it is not amenable. 5.1 Amenability Properties Throughout this segment we will provide several examples of amenable groups and semigroups. The main references for this section are: [4, Sections 4.5-4.6], [5, Section II.4], [33, Section 12] or [16, Chapter 3]. We first discuss a notable difference between amenability in groups and semigroups. For groups, left and right amenability coincide with two-sided amenability, as shown in next proposition. The proof was taken from [33, p. 221] and afterwards, in the particular case of groups, both-sided amenability and left [right] amenability will be treated indistinctly, referring to them simply as amenable groups. 49 Proposition 5.1.1. Let Gbe a left amenable group, then Gis two-sided amenable. Proof. Let mlbe a left invariant measure on PM(G)and define the measure mr∈ PM(G)as mr(A) = ml(A−1). Notice that mris right invariant, since mrg(A) = mr(Ag−1) = ml(gA−1) = g−1ml(A−1) = ml(A−1) = mr(A). Now define the right invariant mean associated to mrby µ= Φ(mr), where Φis the isometry in Theorem 2.3.3. For each A⊆G, define fAby fA(g) = ml(Ag−1). Notice that fAis bounded by 1 and thus fA∈`∞(G), for each A⊆G. Now define the measure m0by m0(A) = µ(fA). It is easy to check that m0(G) = 1 and, since mlis finitely additive and µis linear, fA∪B=fA+fBif A∩B=∅. Hence m0is finitely additive meaning m0∈ PM(G). Now we only need to prove that m0is both-sided invariant. Let A⊆Gand h∈G. Then, apply left invariance of mlto get fh−1A(g) = ml(h−1Ag−1) = hml(Ag−1) = ml(Ag−1) = fA(g), fAh−1(g) = ml(Ah−1g−1) = fA(gh) = rhfA(g). Now, by right invariance of µ, hm0(A) = m0(h−1A) = µ(fh−1A) = µ(fA) = m0(A), m0h(A) = m0(Ah−1) = µ(fAh−1) = µ(rhfA) = µ(fA) = m0(A). Hence, m0is two-sided invariant and Gis amenable. On the other hand, semigroups do not check the statement above, although semigroups do verify that left and right amenable semigroups are two-sided amenable. This result is not covered in this manuscript and it requires the definition of a product operation on the space of means. For a complete view on this result see [5, Theorem II.2.12]. For an example on a semigroup that is left amenable but not right amenable consider S={s, t}such that st =t=tt and ts =s=ss. Then, left maps are neutral, meaning `sf=`tf=ffor all f∈`∞(S), hence every mean in `∞(S)∗is left 50 invariant. Now define the function fas f(s) = 1 and f(t) = 0. Then, rtf=f(t) = 0 and rsf=f(s) = 1. Hence, any invariant mean µ∈`∞(S)∗must verify µ(f) = µ(rtf) = 0, µ(f) = µ(rsf) = 1, yielding a contradiction. Then, the semigroup Sdoes not admit a right invariant mean. At this time, we recall that a first positive result on amenability for abelian semigroups was given as a consequence of the Markov-Kakutani Theorem in Corollary 4.1.6. Likewise, as an application of the characterization of left amenability for semigroups by using Day’s Fixed Point Theorem we derive the following: Corollary 5.1.2. An homomorphic image of a left amenable semigroup is a left amenable semigroup. Proof. Let h:S→Tbe a semigroup homomorphism where Sis left amenable. Consider the surjective homomorphism h:S→h(S). It is trivial that h(S)is a subsemigroup of T. Let h(S)be represented as continuous affine maps from a compact convex set C⊂Xonto itself where Xis a locally convex space. Then h(s)∈Im(h)is a representation of s∈Sas continuous affine map from Conto itself. Hence, there is a common fixed point x0∈Csuch that h(s)x0=x0for all s∈S. Thus, the semigroup h(S)fixes x0which by Theorem 4.2.2 is equivalent to left amenability. Recall the definition of a quotient group from Section 3.1. Given a normal subgroup Hof a group G, define G/H as the quotient G/ ∼Hwhere ∼His the equivalence relation defined as x∼Hyif xy−1∈H. As a consequence of last Corollary, the next result is immediate. Corollary 5.1.3. Every quotient group of an amenable group is amenable. Proof. Let Hbe a normal subgroup of an amenable group G. Applying last Corollary to the surjective canonical homomorphism Q:G−→ G  H g7−→ [g], we deduce that G/H is amenable. 51 Next result does not require any use of means as the whole proof consists on an explicit construction of an invariant measure. To avoid overload, we discussed in Section 3.1 the definition and existence of a right coset. Proposition 5.1.4. Every subgroup Hof an amenable group Gis amenable. Proof. Let m be an invariant probability measure on Gand let Rbe a right coset for Hin G, that is a subset of Gsuch that for every g∈Gthere exist a unique pair (h, r)verifying h∈H,r∈Rand hr =g. We prove that the following map verifies the three properties of an invariant measure in PM(H). For each A⊆Hdefine ˜m(A) = m [ x∈R Ax!. The first property is checked immediately by definition of right coset, ˜m(H) = m [ x∈R Hx!=m(G) = 1. To prove finite additivity we first prove that, if A, B ⊆Hare disjoint, then Ax∩By = ∅for all x, y ∈R. Assume there exist an element g∈Ax ∩By for some x, y ∈R. Then, there exist an element h1∈Aand an element h2∈Hsuch that h1x=g=h2y, but, since Ris a right coset, the pair (h, r)such that hr =gis unique i.e. h1=h2 and x=y, contradicting A∩B=∅. Hence, ˜m(A∪B) = m [ x∈R (A∪B)x!= =m [ x∈R Ax!+m [ x∈R Bx!= ˜m(A) + ˜m(B). Finally, since m is left invariant, for all h∈H h˜m(A) = ˜m(h−1A) = m [ x∈R h−1Ax!= =m h−1[ x∈R Ax!=m [ x∈R Ax!= ˜m(A). 52 To show 3-b, decompose ωonce more in the following possibilities ω=             φ±1ρ±1v ρ±1φ±1v φ±1φ±1v ρ±1ρ±1v . This time distinguish when the first two elements are the same (not regarding inverses) or not and discard the cases were the syllable is trivial. If the elements are different, we have b=b0∓2a0or b=b0±2c0where either 3|a0or 3|c0, since it comes from the first component of (5.2). In both cases, 3-b0implies 3-b. For the two remaining cases, applying the induction hypothesis, we now there are some integers a00, b00 yc00 such that v(1,0,0)T= (a00, b00√2, c00)T. where 3-b. Respectively in each case, substitute a0=a00 ∓4b00 and ±2a00 =b0−b00, or c0=c00 ∓4b00 and ±2c00 =b0−b00, to obtain b=b0±2(a00 ∓4b00) = b0+b00 ±2a00 −9b00 = 2b0−9b00. Hence, 3-b0implies 3-b= 2b0−9b00. Finally, we have shown that ω(1,0,0)T= (a, b√2, c)T/3kwith a, b, c ∈Zand b6≡ 0mod 3which implies that φand ρare independent. Example. We reference from [4, Example 4.5.3 and Lemma 2.3.2] that the matrix group generated by the matrices, 1 0 2 1!, 1 2 0 1! is isomorphic to F2. Thus the group SL(2,Z)of 2×2matrices of determinant 1with coefficients in Zis not amenable by Proposition 5.2.1 as it contains a free subgroup of two generators. As a consequence of Corollary 5.1.8 we can prove that G1and G2are amenable as they are solvable, since it is easy to check that the consecutive quotients in T1⊂G1 and T2⊂SG2⊂G2are all abelian. 59 For the last examples on amenable groups we need the definition of alternating groups. We use the notation in [4, Example 4.6.4]. Definition 5.3.2. An alternating group is the group of even permutations of a finite set. The alternating group on a set of nelements is called the alternating group of degree nand it is denoted by Sym+ n. Thus, the alternating group Sym+ 5is finite and hence amenable. This is an example of a non solvable group that is amenable (see [4, Example 4.6.2 e)]. For an example of an infinite amenable group which is not solvable, apply Corollary 5.1.7 to Z×Sym+ 5 (recall that Zis amenable since it is abelian). It is worth remark that nilpotent groups are solvable (see [4, Proposition 4.6.6] and hence nilpotent groups are amenable by Proposition 5.1.8. A nilpotent group G is a group that has an upper central series that terminates with Gi.e. it admits a finite chain of normal subgroups starting on {e}and terminating in G. An example of a nilpotent group is the Heisenberg group over a ring R. This group is the group of matrices of the form    1x z 0 1 y 0 0 1    where x, y, z ∈R. Thus, the Heisenberg group is amenable. 5.4 Amenability, paradoxical decompositions and extension of measures Along the manuscript we have already studied different characterizations of amenability in terms of the existence of left invariant means and in terms of a common fixed point theorem. This section is devoted to a new and surprising characterization of amenability in groups that is related to the notion paradoxical decompositions. Proofs in this section are just referenced or sketched since they deviate from the initial scope of this master thesis. We will finish with some comments related to the measure problem. We start by defining paradoxical decompositions. 60 Definition 5.4.1. Let Gbe a group acting on a set E. A G-paradoxical decomposition of Eis a triplet (K, (Ak)k∈K,(Bk)k∈K)where Kis a finite subset of Gand, (Ak)k∈K,(Bk)k∈Kare disjoint families of subsets of Eindexed by Ksuch that E= G k∈K kAk!t G k∈K kBk!= G k∈K Ak!= G k∈K Bk! Each Akand Bkis called a piece of the decomposition. Note that each Akor Bk may be empty sets. This definition can naturally be considered for a group acting on itself. That is, if the set Eis also the group G. We give the following definition. Definition 5.4.2. A group Gis paradoxical if it admits a paradoxical decomposition. This concept plays an essential role is the Banach-Tarski Paradox, where the paradoxical set is the unit sphere in R3and the action group Gis the group of rotations G3. Banach and Tarski showed that any subset of R3with non-empty interior is paradoxical (it can be split into disjoint pieces that can be rearranged by rigid motions to form the original set twice), thus showing that there is no finitely additive extension to all of P(R3)of the Lebesgue measure. This is known as the measure problem and we will discuss it further at the end of this section for cases n= 1,2. As seen in last section (Lemma 5.3.1), the group of isometries is not amenable for n≥3. This idea is key in the understanding of the construction of the Banach-Tarski paradox (see [33, Theorem 3.10]). We now relate amenability with paradoxical decompositions. We first define the so called Følner’s condition. Definition 5.4.3. A group Gis said to verify the Følner condition if for every finite subset K⊂Gand every real number ε > 0, there exists a nonempty finite subset F⊂Gsuch that |F\kF| |F|< ε, for all k∈K, where |·| represents the cardinal measure. For more details on Følner’s condition see [4, Section 4.7] and [33, Section 12.4]. The main purpose of this definition is that it provides a new characterization on groups and is strongly used when proving Tarski’s theorem. 61 Theorem 5.4.4 (Tarski-Følner).Let Gbe a group. The following properties are equivalent: a) Gis amenable. b) Gsatisfies Følner’s condition. c) Gdoes not admit a paradoxical decomposition. Proof. See [4, Theorem 4.9.2]. The Følner condition is also a necessity for semigroups to be amenable. More formally there is the following theorem. Theorem 5.4.5. Let Sbe a left amenable semigroup. Then, Ssatisfies the Følner’s condition. Proof. See [22, Theorem 3.5]. As a remark, the Følner condition is not sufficient for a semigroup to be left amenable. One can prove that finite semigroups verify Følner’s condition although the semigroup S={s, t}with st =s=ss and ts =t=tt is not amenable. To prove it, define the function f(s) = 0 and f(t) = 1. Then, consider h= (f−`tf)−(f−`sf) = −1but then any left invariant mean on `∞(S)∗must verify µ(h) = −1by definition and, by linearity and left invariance, µ(h) = µ((f−`tf)−(f−`sf)) = µ(f)−µ(`tf)−µ(f)+µ(`sf) = 2µ(f)−2µ(f) = 0. To finish this chapter, we take one last look at the measure problem. That is, the existence of an extension of the Lebesgue measure on Rand R2. We proved that the isometry groups G1and G2on Rand R2respectively are amenable since they are solvable. This property allows the construction of a finitely additive extension of the Lebesgue measure on Rand R2. This is known as the Banach Theorem. The proof of Banach’s Theorem we showcase here is an application of [33, Theorem 12.11]. The notation in [33] was given in terms of measures and integrals. For a detailed view on this result see [33, Another proof of Corollary 12.9 p.235] or [30, Theorem 3.3.4]. Here we exhibit a sketch of the proof where means play the essential role and the Hahn-Banach extension Theorem [27, Theorem 3.2] is used. 62 Theorem 5.4.6 (Hahn-Banach).Let Xbe a real vector space and p:X→Ra functional verifying: p(x+y)≤p(x) + p(y), p(αx) = αp(x), for every x, y ∈Xand for ever α≥0. Let Ybe a subspace of X. Then for every linear functional ψ:Y→Rsuch that ψ≤p, there exists a linear functional ψ:X→Rthat extends ψi.e. ψ|Y=ψand verifies ψ≤pon X. Theorem 5.4.7 (Banach).There exists a finitely additive extension of the Lebesgue measure on Rthat is invariant with regards to isometries. This statement also holds true for R2. Sketch of the Proof: Let λdenote the Lebesgue integral on R(analogous for R2). Let G=G1. Let V0be the set of all Lebesgue integrable functions in `∞(R). Define the set V⊂`∞(R)as V={f∈`∞(R) : ∃f0∈V0such that f0≥f}. It is a simple verification that V0⊂V⊂`∞(R)is a chain of vectorial subspaces. Define the operator p:V→Ras, p(f) = inf ZR f0:f0∈V0, f0≥f, for each f∈`∞(R). This functional is well defined on Vas for every f∈Vthere is at least one function in V0that majores f. It is an easy check that this functional is well defined and that verifies p(f1+f2)≤p(f1) + p(f2), p(αf1) = αp(f1), for all f1, f2∈Vand for all α≥0. Then, apply Hahn-Banach’s extension Theorem to obtain a linear functional λ0that extends λto Vand is bounded by the functional p. Now, since G1is amenable, there exist an invariant mean µon `∞(G1). 63 For each function, f∈Vdefine the function h∈`∞(G1)as hf(g) = λ0(gf),(5.3) where gf(x) = f(gx)for each x∈R. Hence, we can define a functional λas λ(f) = µ(hf). It follows from definition (5.3), that for each s∈G, hsf (g) = λ0(gsf) = hf(gs) = rshf(g). Hence, by invariance of µ, λ(sf) = µ(hsf ) = µ(rshf) = µ(hf) = λ(f). That is, λis Ginvariant. Now define the measure m(A) = λ(χA)for each A⊂Rif χA∈Vand m(A) = +∞ otherwise. Then, this measure is G1-invariant, since gχE=χg−1E, and it agrees with the Lebesgue measure on measurable sets and that is well defined for all A∈ P(R). Thus, we have found an isometry invariant and finitely additive extension of the Lebesgue measure. 64 Chapter 6 Some “Kakutani-type" fixed point theorems for semigroups In this chapter we introduce the notion of a “Kakutani-type” fixed point theorem, term coined by Namioka in [23] strongly inspired by Kakutani theorem for compact flows, and that gathers the flavour of different common fixed point theorems under the action of semigroups and strongly inspired by Kakutani fixed point theorem for compact flows. So far, we have studied two results that lie within the above landscape: Markov- Kakutani’s theorem and Day’s theorem (see Chapter 4). We will state some other “Kakutani-type” fixed point theorems, among which we will exhibit a detailed proof of Ryll-Narzdewski’s theorem for noncontracting semigroups, that covers the previous cases. As a conclusion, some applications will be shown and further extensions to the linear and nonlinear case will be displayed. 6.1 Introduction and Preliminaries It is an easy consequence of Bolzano Theorem that every continuous function f: [a, b]→[a, b]has a fixed point. Brouwer fixed point theorem asserts that every continuous selfmapping defined on a closed convex bounded subset of Rnhas a fixed point and this result was extended to the setting of locally convex topological spaces by Schauder and Tychonoff: Every continuous mapping defined from a convex compact 65 subset of a locally convex topological space into itself has a fixed point [32]. The situation is completely different if we try to seek for a common fixed point for more than one continuous function defined on the same convex compact space, even if these mappings commute. For instance, in 1969, Boyce [3] and Huneke [15] independently published examples of two commuting continuous maps f, g of [0,1] into itself without a common fixed point. Note that every collection of self-mappings defined over the same domain generates a semigroup acting by composition and that a point is fixed for every element of the family if and only if it is fixed by all the mappings belonging to the generated semigroup. Thus, in search of a common fixed point (f.p. in short), we can always assume that we are within the framework of the action of a semigroup over the shared domain. In Chapter 4 we already proved Markov-Kakutani’s theorem that claims that in case that Sis a commutative semigroup and the mappings are continuous and affine, then there is a common fixed point whenever the domain is a convex compact subset of a locally convex space. Afterwards, Day’s fixed point Theorem (that characterizes amenability) shows that the previous conclusion can be extended replacing commutativity by amenability. During this last chapter of the manuscript, we will study sufficient conditions upon a semigroup of mappings acting over a convex compact domain so that there exists at least a common fixed point. Following [23] we introduce the following notation: A statement is said to be a Kakutani-type fixed point theorem if it is of the form: Given a group or semigroup Sacting on a compact convex subset Kof a locally convex vector space Xinto itself. Assume that each action s:K→Kis a continuous affine transformation. Then, under suitable conditions, Shas a common fixed point in K. These theorems are called “Kakutani-type” in reference to the fixed point theorems proved in Kakutani’s paper [17]. For the sake of context, we introduce some notions on topological dynamic systems. Definition 6.1.1. A flow or dynamical system is a pair (S, C)where Cis a nonempty subset of a topological space (X, τ)and Sis a semigroup acting on Cfrom the left where each action s:C→Cis a continuous map. Additionally, a flow (S, C)is said to be: 66 •a compact flow if Cis compact, •an affine flow if Cis convex and s:C→Cis an affine map, for each s∈S; or •an equicontinuous flow if, for each neighborhood Uof 0, there exists a neighborhood Vof 0such that x, y ∈Cand x−y∈Vimply sx −sy ∈Ufor each s∈S. •a distal flow if, it is a compact flow and, for each x, y ∈Csuch that limαsαx= limαsαyfor some net (sα)⊂S, then x=y. With this definitions, the Markov-Kakutani Theorem 4.1.5 can be enunciated as follows: Let (S, K)be a compact affine flow. If Sis commutative, then Sadmits a common f.p. in K. Now we state two more Kakutani-type theorems that we will prove to be weaker versions of the upcoming Ryll-Narzdewski’s Theorem 6.1.4, whose proof will be exhibited in detail in Section 6.3. Theorem 6.1.2 (Kakutani’s f.p. Theorem [17]).Let (S, K)be a compact affine flow. If Sis a group and equicontinuous on K, then Sadmits a common f.p. in K. Theorem 6.1.3 (Hahn’s f.p. Theorem [13]).Let (S, K)be a distal affine flow. Then Sadmits a common f.p. in K. It is in this context that Ryll-Narzdewski’s theorem appears, which is as well a Kakutani-type f.p. theorem. We now give the statement of this theorem, although its proof is given in Section 6.3 as we will prepare some tools in advance. Theorem 6.1.4 (Ryll-Nardzewski).Let Kbe a nonempty weakly compact convex subset of a locally convex space (X, τ)and let Sbe a semigroup of weakly continuous affine maps s:K→K. If Sis noncontracting on K, i.e., 0/∈ {sx −sy :s∈S}τfor distinct x, y ∈K, then Shas a common fixed point in K. Notice that, since for every locally convex space (X, τ)the topology is generated by a set of seminorms N, the assumption of noncontractiveness is equivalent to: for 67 each x6=ythere exists a seminorm ρ∈ N such that inf s∈Sρ(sx, sy)>0. In the next results we prove that Ryll-Narzdewski’s result is indeed an improvement of Theorems 6.1.2 and 6.1.3. First we prove that, when Sis a group, equicontinuous implies distal. Proposition 6.1.5. An affine compact flow (S, K)is distal if Sis a group and (S, K) is an equicontinuous flow. Proof. Denote as Xthe topological space containing K. Let x, y ∈Kbe two distinct points. Then, there exists a neighborhood Uof 0in Xsuch that x−y /∈U. By equicontinuity of Son K, there exists a neighborhood Vof 0in Xsuch that su−sv ∈ Ufor each s∈Swhenever u, v ∈Kand u−v∈V. Then for each s∈S,sx−sy /∈V as otherwise, sx −sy ∈Vfor some s∈Sand then x−y=s−1sx −s−1sy = s−1(sx)−s−1(sy)∈U, contradicting our choice of U. Consequently, there is no net (sα)⊂S, such that limαsαx= limαsαy. Hence (S, Q)is distal. We reference from [11] that there exist distal flows that are not equicontinuous. Proposition 6.1.6. Any distal affine flow (S, K)satisfies Ryll-Narzdewski’s Theorem 6.1.4 hypotheses. Proof. Any distal affine flow (S, K)verifies Kis weakly compact since it is τ-compact by definition. Also by τ-compactness, the assumption 0/∈ {sx −sy :s∈S}τ is equivalent to: no two τ-convergent nets sαx=sαyverifying x6=y. That is, a distal flow is noncontracting. Also, by Proposition 1.1.6, every τ-continuous affine mapping is weakly continuous. Hence, the mappings of Son Kare weakly continuous, concluding the proof. Ryll-Narzdewski Theorem was originally proved in [29]. The proof we show here is taken from Namioka’s article [23] that uses extreme points and its properties for convex compact sets. These tools will be detailed in the next section. 68 M. Hence for some s0∈S,s0u∈M∩W. Now let z= (x+y)/2. Once again, apply Milman’s theorem to K=co(Sz) = co(Szω)and thus u∈Szω. Then there is a net {sα}in Ssuch that sαz= (1/2)(sαx+sαy)ω −→ u (recall that all sα∈Sare affine actions on K). Since Kis weakly compact, taking subnets if necessary, there exists a, b ∈Ksuch that sαx→aand sαy→bweakly. Therefore u= (a+b)/2and thus a=u=bsince uis extreme in K. Since s0is weakly continuous, s0sαxω −→ s0a=s0u∈M∩W. Now recall that Mis S-stable and x∈M, thus s0sαx∈Mfor all α. Since M∩Wis a weakneighborhood of urelative to M,s0sαx∈M∩W⊂V+teventually. Similarly, s0sαy∈M∩W⊂V+teventually. Hence for some β, s0sβx−s0sβy∈(V+t)−(V+t) = V−V⊂U, contradicting our choice of U. This proves that Mmust be a singleton, and the theorem is proved. 6.4 Some applications Recall from the discussion in 6.1 that the Ryll-Nardzewski theorem is a generalization of Kakutani and Hahn fixed point theorems (6.1.2, 6.1.3) from two different approaches. Firstly, every group of equicontinuous mappings is distal (Proposition 6.1.5). Secondly, Ryll-Nardzewski theorem involves an interplay between the natural topology of a locally convex space (which is usually called the strong topology for emphasis) and its weak topology. If Xis in particular a normed infinite vector space, then its closed unit ball is never strong compact (or compact for the norm topology). As an application we deduce that if Kis a convex weakly compact subset of a normed space, then every group (or semigroup) of affine isometries (not necessarily onto) has at least one fixed point. We comment in this Section some further results that base on Ryll-Narzdewski’s work. Before displaying some applications of this theorem, we need to define a topological group. 75 Definition 6.4.1. Let Gbe a group and τa topology on G. The pair (G, τ)(or simply G) is a topological group if the group transformations on G,(x, y)7→ x·yand x7→ x−1, are τ-continuous mappings. Definition 6.4.2. The set C(G)is the space of all bounded and continuous real valued functions on G. As in the discrete case, we can consider the left transformations `g:C(G)→C(G) for each g∈G, defined as `gf(t) = f(gt). The first implication worth noting is the existence of an invariant measure on a compact group called Haar’s measure, where a compact group Gis a (G, τ)topological group that is also a τ-compact space. We first give the definition of a mean for a given topological group G. Let X= C(G), an element µ∈C(G)∗is a mean on C(G)if µ(1) = 1 and µ(f)≥0, for all f≥0. Recall that we also note the constant unit function on Gas 1. This set is clearly nonempty since the operators δgf=f(g)are means for every g∈G(see example after 2.3.1). Then, one can easily replicate Proposition 2.4.1 to prove that the set of means on a topological group G,M(G), is a ω∗-compact convex subset of C(G)∗. Similarly to means in discrete semigroups, we can define the mappings `s:C(G)→ C(G)as `sf(t) = f(st). Then, the adjoint mappings give a left action on means, `∗ sµ(f) = µ(`sf), for all f∈C(G). Hence, we can consider the compact affine flow (G, M(G)) where Gis represented as the mappings `∗ s. This flow was proved to be equicontinuous by Kakutani in [17, Lemma 3.3]. Now, apply Ryll-Narzdewski’s Theorem (or Kakutani’s Theorem 6.1.2) to obtain an element µ∈ M(G)that is a fixed point for every mapping `∗ si.e. µ(`sf) = µ(f). Thus, by the Riesz Representation Theorem there exists a measure λthat verifies µ(f) = ZG f(t)dλ(t) for every f∈C(G). Finally, since µis left invariant, we have that ZG f(t)dλ(t) = µ(f) = µ(`sf) = ZG `sf(t)dλ(t) = ZG f(st)dλ(t), for all f∈C(G). That is, we have proved that Gadmits an invariant measure. 76 Note that these arguments are also used in [27, Theorem 5.14] where the Haar measure is constructed using Kakutani’s Theorem. For a second application, we introduce the concept of almost periodic functions. Definition 6.4.3. Let Gbe a topological group. A function f∈C(G)is called a weakly [strong] almost periodic function if the set co({`gf:g∈G}) is weakly [norm] compact. Notice that, since Gis a group, the sets {`gf:g∈G}and {`g(`hf) : g∈G}are equal for every h∈G. This implies that the set of weakly [strong] almost periodic functions is closed under the left transformations of G. In [10, Theorem 4.2] Eberlein proved that the set of weakly almost periodic functions is a closed subspace of C(G). Let W(G)denote the space of weakly almost periodic functions (w.a.p. in short). We give now an application of Ryll-Narzdewski’s Theorem to find a left invariant mean in W(G)that is, a left invariant functional µ∈W(G)∗that satisfies µ(1) = 1 and µ(f)≥0, for all f≥0for all f∈W(G). In the particular case that Gis amenable, this was proved using Day’s fixed point Theorem by Eberlein in [10] and it was raised the question whether or not W(F2)admits a left invariant mean. This problem was completely solved for every topological group by Ryll-Narzdewski using his fixed point theorem. In order to finish this section, we give here a sketch of the proof. Theorem 6.4.4. Let Gbe a topological group and f∈C(G)be w.a.p. function. Then, there is a constant function, which we denote by M(f), in co({`gf:g∈G}). Proof. In Ryll-Narzdewski’s Theorem 6.1.4, let X=C(G),K=co({`gf:g∈G}), which is weakly compact since fis w.a.p., and Grepresented as sf(t) = f(s−1t). This defines an equicontinuous flow (as k`sfk=kfkfor all g∈G) on a weakly compact convex subset of C(G). Thus, we can apply Ryll-Narzdewski’s to obtain a common fixed point in co({`gf:g∈G})i.e. a G-invariant function M(f)in co({`gf:g∈G}). That is, gM(f)(t) = M(f)(g−1t) = M(f)(t), for all g, t ∈G. This, since Gis a group, implies that M(f)(t) = M(f)(s)for all t, s ∈Gi.e. M(f)is constant. 77 Note that neither Kakutani’s Theorem 6.1.2 nor Hahn’s Theorem 6.1.3 can be applied above since the domain is not strong compact. The same happens in the next theorem. Theorem 6.4.5. Let (G, X)be an equicontinuous linear flow where Gis a group. We define OG(x) = co({gx :g∈G}), called the convex G-orbit of an element x∈X. Then, 1. for each x∈Xsuch that OG(x)is weakly compact, there exists a unique G- invariant element in OG(x)denoted as Mx. Moreover, if we define X0as the elements x∈Xsuch that OG(x)is weakly compact and Mis the mapping defined on X0as x→Mx, then 2. the set X0is a closed subspace of X, and 3. the operator Mis linear and g(Mx) = M(gx) = M(Mx) = Mx, for all g∈G and for all x∈X0. Proof. We proof each statement separately: 1. Let x∈X0. Since, (G, X)is equicontinuous, (G, OG(x)) is an affine equicontinuous flow with OG(x)weakly compact. Thus, we can apply Theorem 6.1.4 to yield the existence of a common fixed point in OG(x)for the action of G. To prove uniqueness, for each x∗∈X∗and for each y∈OG(x)we define the function Fy:G−→ R g7−→ x∗(gy). Then, in [29, Theorem 5] it is stated that Fis w.a.p. and thus, by Theorem 6.4.4, we can obtain an invariant mean value of F, which we called M(Fy). That is, for all s∈G M(Fy) = M(`sFy). Hence, for every pair y1, y2∈OG(x),y1=gy2for some g∈Gand hence M(Fy1) = M(Fy2). 78 On the other hand, for every x0∈OG(x)G-invariant element, we have that Fx0(g) = x∗(gx0) = x∗(x0)for all g∈G, that is Fx0is constant and thus M(F) = x∗(x0). Then we have that, for any pair x0, y0∈OG(x)of G-invariant elements x∗(x0) = M(Fx0) = M(Fy0) = x∗(y0). Since this equality holds for every x∗∈X∗and Xis locally convex, x0=y0. 2. By linearity of the action of G, OG(αx +y) = αOG(x) + OG(y), for all x, y ∈X0and for every scalar α. To prove that X0is closed see [29, Proposition 1 and Theorem 5]. 3. The fact that M(Mx) = Mx is immediate since Mx is the unique fixed point in OG(x). Also, since Mx is G-invariant for every x∈X0,g(Mx) = Mx is obvious. Lastly, for every OG(x) = OG(gx), for every g∈G, hence M(gx) = M(x)for every x∈X0. 6.5 Notes on further results In this section we aim to display some other classical and recent common fixed point results in the setting of Kakutani-type theorems. Firstly, we remark that some extensions of Ryll-Narzdewski’s Theorem are given in [12, Theorem 1.6]. Furstenberg fixed point theorem [23, Theorem 4.1] is another example of Kakutanitype fixed point theorem: Theorem 6.5.1. Let (S, Q)be a compact affine flow. Suppose there exists a nonempty compact S-stable subset K(i.e., sK ⊂Kfor each s∈S) of Qsuch that (S, K)is distal. Then there is a common f.p. of Sin Q. In this result, the distal hypothesis is restricted to some (not necessarily convex) closed S-stable subset. 79 Some non-linear counterparts of the Ryll-Nardzewski Theorem have been recently obtained by Wiśnicki in [34]. The assumption of linearity is changed by considering nonexpansive mappings. In the case of a normed space a mapping s:C→Cis nonexpansive if ksx −syk ≤ kx−yk, for all x, y ∈C. For the case of locally convex vector spaces, as every locally convex space is determined by a family of seminorms N, we have the following definition: the semigroup Sacting on Cis non-expansive if p(sx −sy)≤p(x−y), for all x, y ∈Cand for every p∈ N. The theorem proved by Wiśnicki for locally convex spaces [34, Theorem B] states as follows. Theorem 6.5.2. Let Kbe a nonempty weakly compact convex subset of a locally convex space (X, τ)and let (S, K)be a nonexpansive and τ−distal flow. Then, there is a common fixed point of Sin K. Moreover, the set of fixed points is a nonexpansive retract of K. We conclude by stating a theorem that connects once more amenability with Kakutani-type theorems this time for nonexpansive semigroups, given by Takahashi in [31]. Theorem 6.5.3. Let Sbe a left amenable semigroup. Then, every representation of Sas non-expansive mappings from a non-empty compact convex subset Kof a Banach space into itself has a common fixed point for Sin K. Generalizations of this theorem in the context of convex weakly compact sets of Banach spaces for different types of nonexpansive semigroups and strongly connected to the geometry of the space can be found in [19] and [20]. 80 Bibliography [1] S. Banach. “Sur le problème de la mesure”. In: Fundamenta Mathematicae 4.1 (1923), pp. 7–33. [2] S. Banach and A. Tarski. “Sur la décomposition des ensembles de points en parties respectivement congruentes”. In: Fundamenta Mathematicae 6 (1929), pp. 244–277. [3] W. M. Boyce. “Commuting functions with no common fixed point”. In: Transactions of the American Mathematical Society 137 (1969), pp. 77–92. [4] T. Ceccherini-Silberstein and M. Coornaert. “Cellular Automata and Groups”. In: Encyclopedia of Complexity and Systems Science. Springer New York, 2009. [5] H. Chan. “Amenable semigroups and ergodicity”. PhD thesis. 1972. url:https: //knowledgecommons.lakeheadu.ca/bitstream/handle/2453/2215/ChanH1972m- 1b.pdf?sequence=1&isAllowed=y. [6] M. M. Day. “Amenable semigroups”. In: Illinois Journal of Mathematics 1.4 (1957), pp. 509–544. url:https://doi.org/10.1215/ijm/1255380675. [7] M. M. Day. “Fixed-point theorems for compact convex sets”. In: Illinois Journal of Mathematics 5.4 (1961), pp. 585–590. url:https://doi.org/10.1215/ ijm/1255631582. [8] J. Diestel. Sequences and Series in Banach Spaces. Graduate Texts in Mathematics. Springer New York. [9] N. Dunford and J.T. Schwartz. Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert Space. Wiley Classics Library. Wiley, 1988. 81 [10] W. F. Eberlein. “Abstract ergodic theorems and weak almost periodic functions”. In: Transactions of the American Mathematical Society 67 (1949), pp. 217– 240. [11] H. Furstenberg. “The Structure of Distal Flows”. In: American Journal of Mathematics 85.3 (1963), pp. 477–515. [12] E. Glasner and M. Megrelishvili. “On fixed point theorems and nonsensitivity”. In: Israel Journal of Mathematics 190.10 (2012), pp. 289–305. url:https: //doi.org/10.1007/s11856-011-0192-4. [13] F. Hahn. “A fixed-point theorem”. In: Mathematical systems theory 1 (1967), pp. 55–57. [14] M. Hochster. “Subsemigroups of amenable groups”. In: Proceedings of the American Mathematical Society 21.2 (1969), pp. 363–364. [15] J. P. Huneke. “On common fixed points of commuting continuous functions on an interval”. In: Transactions of the American Mathematical Society 139 (1969), pp. 371–381. [16] K. Juschenko. Amenability of discrete groups by examples. Vol. 266. American Mathematical Society, 2022. [17] S. Kakutani. “Two fixed-point theorems concerning bicompact convex sets”. In: Proceedings of the Imperial Academy 14.7 (1938), pp. 242–245. [18] A. Kolpakov and A. Talambutsa. “On free semigroups of affine maps on the real line”. In: Proc. Amer. Math. Soc. 150 (2022), pp. 2301–2307. url:https: //doi.org/10.1090/proc/15832. [19] A. T-M. Lau. “Invariant means and fixed point properties of semigroup of nonexpansive mappings”. In: Taiwanese Journal of Mathematics 12.6 (2008), pp. 1525–1542. url:http://www.jstor.org/stable/43834611. [20] A. T-M. Lau and Y. Zhang. “Fixed point properties for semigroups of nonlinear mappings and amenability”. In: Journal of Functional Analysis 263.10 (2012), pp. 2949–2977. url:https://doi.org/10.1016/j.jfa.2012.07.013. [21] N. Monod. “Groups of piecewise projective homeomorphisms”. In: Proceedings of the National Academy of Sciences 110 (2012), pp. 4524–4527. url:https: //doi.org/10.1073/pnas.1218426110. 82 [22] I. Namioka. “Følner’s conditions for amenable semi-groups”. In: Mathematica Scandinavica 15.1 (1964), pp. 18–28. [23] I. Namioka. “Kakutani-type fixed point theorems: A survey”. In: Journal of Fixed Point Theory and Applications (2011). url:https://link.springer. com/article/10.1007/s11784-010-0036-6. [24] A. Y. Ol’shanskii. “On the problem of the existence of an invariant mean on a group”. In: Russian Mathematical Surveys 35.4 (1980), p. 180. [25] Alan LT Paterson. Amenability. 29. American Mathematical Soc., 1988. [26] D. JS Robinson. “An introduction to abstract algebra”. In: de Gruyter, 2008. [27] W. Rudin. Functional analysis. McGraw-Hill, Inc., New York, 1991. [28] Volker Runde. Amenable Banach Algebras: A Panorama. Springer Monographs in Mathematics, 2020. [29] C. Ryll-Nardzewski. “On fixed points of semigroups of endomorphisms of linear spaces”. In: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability. 1967, pp. 55–61. [30] J. Suarez Labat. Equidescomponibilidad, el problema de la medida y la paradoja de Banach-Tarski. Trabajo de Fin de Grado, Universidad de Sevilla, 2022. url: https://idus.us.es/handle/11441/142910. [31] W. Takahashi. “Fixed point theorem for amenable semigroup of nonexpansive mappings”. In: Kodai Mathematical Seminar Reports 21.4 (1969), pp. 383–386. url:https://doi.org/10.2996/kmj/1138845984. [32] A. Tychonoff. “Ein Fixpunktsatz”. In: Mathematische Annalen 111 (1935), pp. 767– 776. [33] S. Wagon. The Banach-Tarski Paradox. 24. Cambridge University Press, 1993. [34] A. Wisnicki. “Around the nonlinear Ryll-Nardzewski theorem”. In: Mathematische Annalen 377 (2020), pp. 267–279. url:https://doi.org/10.1007/ s00208-020-01985-8. 83