Strongly amenable semigroups and nonlinear fixed point properties

Nonlinear Analysis 74 (2011) 5815–5832

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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Strongly amenable semigroups and nonlinear fixed point properties Nicolas Bouffard Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB, T6G 2G1, Canada

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Article history: Received 24 February 2011 Accepted 28 March 2011 Communicated by Ravi Agarwal Keywords: Amenable semigroups Semigroup compactifications Minimal ideals Left ideal groups Non-expansive maps Fixed point properties Uniformly convex spaces Left uniformly continuous functions Ultimately non-expansive maps

abstract A semi-topological semigroup is strongly left amenable if there is a compact left ideal group in the spectrum of its LUC-compactification. In this paper, we want to study those objects, and study some fixed point property related to non-expansive mapping and other similar kind of mapping. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Amenability traces its origins back to the work of Lebesgue on finitely additive measures in 1904 [1], and to the work of Banach and Tarski on their well-known paradox [2,3]. The first study of this class of group was done by von Neumann in 1929 [4] as the class of non-paradoxical group. This class was later named amenable by Day which started the modern period of amenability in the 1940s, when amenability shifted from the study of invariant measures to the study of invariant means [5]. For a more detailed account of the history of amenability, see [6]. Extremely amenable semigroups were introduced by Mitchell [7,8] and Granirer [9] in the 60s. This is a very strong nonlinear property which is never possible for locally compact groups [10,11] except in the case of the trivial group. More recent development in the theory of extremely left amenable groups is presented in [12,13]. A weaker version of extreme amenability, namely n-extreme amenability, was introduced in 1970 by Lau [14,15]. The following year, Lau and Granirer proved that all locally compact n-extremely left amenable (n-ELA) group are finite groups, which was a major drawback since no non-trivial examples of n-ELA group were known. The first example of extremely left amenable topological group was given by Herer and Christensen in 1975 [16]. It is now known that extremely left amenable topological groups are quite common and many examples are known [12]. The concept of strong left amenability, which is our main interest in this paper, sits somewhere between n-extreme left amenability and left amenability. In the first part of the paper (Sections 3–6) we study the structure of SLA semigroups, more specifically in Section 3, we give the definition of strong left amenability (SLA) and present some elementary results which act as the foundation to the following sections. Section 4 gives a characterization of SLA semigroups together with a justification for the use of the space LUC (S ) in our definition through the characterization of strong left amenability for other algebras. In Section 5, we show some ways to construct SLA semigroups and give a few examples. Finally, in Section 6, we characterize strong left amenability for some particular classes of semigroups which include discrete semigroups, and compact semigroups.

E-mail address: [email protected]. 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.03.052

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Amenability is particularly interesting due to its closed relationship with fixed point properties. The second part of the paper (Sections 7–9) study fixed point properties related to strong left amenability. In Section 7, we characterize strong left amenability in terms of a fixed compact set, and we get as corollary our first fixed point theorem. When K is a non-empty bounded closed convex subset of a Banach space E, we say that K has the fixed point property if for every non-expansive mapping T : K → K , K contains a fixed point for T . We say that a Banach space E has the weak fixed point property if every weakly compact convex subset of E has the fixed point property. It is known [17] that if E has the weak fixed point property, then any weakly compact convex subset K has the common fixed point property for any commutative semigroup acting on K . Every uniformly convex Banach space has the weak fixed point property. This was proved by Browder [18] in 1965, and improved by Kirk [19], who proved that every weakly compact subset of a Banach space with normal structure has the fixed point property. Many examples of Banach spaces having the weak fixed point property are known [20]. These include c0 , l1 and the Fourier algebra of a compact group. Notice that not all Banach spaces have the weak fixed point property. As was proved by Alspach [21], there exists a weakly compact convex subset K of L1 [0, 1] without any fixed point for some non-expensive map on K . Representation as non-expansive mappings are interesting to us since in many cases they characterize the existence of left invariant means on some C*-subalgebra of l∞ (S ) [22]. In 2007, Kang [23] studied fixed point for representation of a semitopological semigroup S as strongly continuous non-expensive mapping on a weakly compact convex subset of a Banach space, where CB(S ) is n-ELA. In Section 8, we generalize some of the work by Kang [23] to SLA semigroups. Fixed point properties for a semigroup S acting on a Banach space X have been widely studied. Many results are known [24,22,25–30], in particular, in the case where S is commutative, amenable or reversible. It is well known that every commutative semigroup is amenable, and every left amenable semigroup is left reversible. Papers investigating fixed point properties for left reversible semigroups include [31–33]. Similarly, fixed point properties for semigroup of non-expansive mappings have been studied extensively. In 1965, Kirk [19] proved that if C is a weakly compact convex subset of a Banach space with normal structure, then every nonexpansive mapping T on C has a fixed point. As well known, not every non-expansive action of a semigroup on a subset of a Banach space has a fixed point [21]. Many generalizations of this concept have also been investigated. For example, Holmes and Narayanaswami [24] introduced the concept of asymptotically non-expansive mapping and Kiang [34] studied eventually non-expansive mappings. In 1972, Goebel and Kirk [35] proved that if X is a uniformly convex Banach space, and C is a weakly compact convex subset of X , then every asymptotically non-expansive mapping on C has a fixed point. In 1982, Edelstein and Kiang [36] introduced the concept of ultimately non-expansive mappings. They proved in particular that if S is a commutative semigroup of ultimately non-expansive mappings on a reflexive locally uniformly convex Banach space X such that S (x) is precompact for some x ∈ X S , then there is a fixed point in X . Here X S denotes the S-closure points of S. In 1984, Edelstein [37] generalized this result to the case S generated by a single map, replacing the precompactness condition by the existence of an S-closure point whose orbit is bounded. Finally, in 1985, Edelstein and Kiang [38] extended this result for general commutative semigroups. In the last section of this paper, we extend this result to amenable semigroups, and notice in Corollary 9.12 that one of the conditions of Theorem 9.11 is always satisfied when the semigroup is SLA. 2. Preliminaries We give in this section some properties of the structure of semigroups. For a more complete treatment of the subject, see [39]. A semigroup is a set together with a binary operation which is associative. An element z ∈ S is a right zero, if sz = z for any s ∈ S. A subset I ⊆ S is a left ideal [resp. right ideal] if SI ⊆ I [IS ⊆ I]. A left ideal [resp. right ideal] is minimal, if it does not contain any proper left ideal [resp. right ideal]. A semigroup does not need to contain any minimal left or right ideal, but if it does, every left ideal [resp. right ideal] contains a minimal left ideal [resp. minimal right ideal]. Proposition 2.1 ([39, Page 16, Proposition 2.4]). Let S be a semigroup. Then a left ideal L is minimal if and only if Ls = Ss = L for all s ∈ L. Also, if S has a minimal left ideal L, then {Ls : s ∈ S } is the family of all minimal left ideal in S. Notice that the same result holds for right ideal if we change the order of the product. An element e ∈ S is an idempotent if e2 = e. An idempotent e is minimal, if it respects any one of the following equivalent properties. Proposition 2.2 ([39, Page 17, Theorem 2.8]). If e ∈ S is an idempotent, then the following are equivalent: 1. Se is a minimal left ideal, 2. eS is a minimal right ideal, 3. eSe is a group. Proposition 2.3 ([39, Page 19, Corollary 2.13]). Let S be a semigroup with minimal idempotent. Then the minimal left ideals of S are groups if and only if S has a unique minimal right ideal. Now, let S be a semigroup with a Hausdorff topology. We say S is right-topological semigroup [resp. left-topological semigroup] if for any net sα → s in S, and t ∈ S, we have that sα t → st [resp. tsα → ts]. A semigroup with a Hausdorff

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topology is a semi-topological semigroup if it is both right- and left-topological (i.e. the product is separately continuous). Finally, a semigroup with a Hausdorff topology is a topological semigroup if for any net sα → s and tβ → t in S, we have that sα tβ → st (i.e. the product is jointly continuous). Proposition 2.4 ([39, page 31, Theorem 3.11]). If S is a compact, Hausdorff, right-topological semigroup, then S has a minimal idempotent. Moreover, all minimal left ideal of S are closed and pairwise homeomorphic. If S is a right-topological semigroup, then we define the topological center of S to be the set Λ(S ) = {s ∈ S : the map t → st is continuous}. It is known that every compact right-topological groups admits a left Haar measure, which is not necessarily unique unless some extra assumptions are satisfied [39,40]. Notice that such a Haar measure is left invariant by members of the topological center, but not necessarily all elements of the group. For some recent development on compact right-topological semigroups, see [41]. Let S be a semi-topological semigroup, and let l∞ (S ) be the C*-algebra of all bounded complex-valued functions on S with the sup-norm topology, that is, f ∈ l∞ (S ) ⇐⇒ ‖f ‖ = sup |f (s)| < ∞. s∈S

For any f ∈ l∞ (S ) we define the left translation operator ls by ls f (t ) = f (st ), and the right translation operator rs by rs f (t ) = f (ts), for any s, t ∈ S. We also define the following C*-subalgebras of l∞ (S ): 1. CB(S ) is the C*-subalgebra of l∞ (S ) consisting of all bounded norm continuous complex-valued functions. 2. LUC (S ) is the C*-subalgebra of CB(S ) consisting of all functions f ∈ CB(S ) for which the map s → ls f is continuous. Similarly we can define RUC (S ) by replacing ls by rs . 3. AP (S ) is the C*-subalgebra of CB(S ) consisting of all functions f ∈ CB(S ) for which the set LO(f ) = {ls f : s ∈ S } is norm relatively compact in CB(S ). 4. WAP (S ) is the C*-subalgebra of CB(S ) consisting of all functions f ∈ CB(S ) for which the set LO(f ) is weakly relatively compact in CB(S ). For any semi-topological semigroup S, we have AP (S ) ⊆ WAP (S ) and AP (S ) ⊆ LUC (S ) ∩ RUC (S ). Also, we have AP (S ) = AP (Sd ) ∩ CB(S ) and WAP (S ) = WAP (Sd ) ∩ CB(S ), where Sd is the semigroup S with the discrete topology. If S is a compact topological semigroup, then AP (S ) = CB(S ), and if S is a compact semi-topological semigroup, then WAP (S ) = CB(S ) and AP (S ) = LUC (S ) = RUC (S ). Let A be a left translation invariant C*-subalgebra of l∞ (S ) containing the constants. A mean on A is a linear functional µ : A → C such that ‖µ‖ = 1 and µ(1S ) = 1, where 1S is the constant one function on S. A mean µ is left invariant if l∗s µ = µ for all s ∈ S, where l∗s µ(f ) = µ(ls f ). A mean µ is multiplicative if µ(fg ) = µ(f )µ(g ) for all f , g ∈ A. A is said to be [extremely] left amenable if there exists a [multiplicative] left invariant mean on A. S is said to be [extremely] left amenable, if LUC (S ) is [extremely] left amenable. Note that for a compact semi-topological semigroup S, CB(S ) is left amenable if and only if S has a unique minimal right ideal. Now, let S A denote the set of all multiplicative means on A with the weak*-topology, and ∆(S ) = S LUC (S ) . We define the left introversion operator determined by µ ∈ CB(S )∗ by Tµ : A → l∞ (S ), (Tµ f )(s) = µ(ls f ) for all f ∈ A, s ∈ S, and we say A is m-left introverted if Tµ A ⊆ A for all µ ∈ S A . If A is m-left introverted, we define the Arens product ⊙ on S A by

µ ⊙ ν(f ) = µ(νl (f )), ∀f ∈ A νl (f ) = ν(ls f ), s ∈ S . For any semi-topological semigroup, AP (S ), WAP (S ) and LUC (S ) are all translation invariant, left introverted C*-subalgebra of CB(S ) containing the constant functions. Also, with this product, ∆(S ) is a compact right-topological semigroup and it is the largest compactification among all, and S AP (S ) is a compact topological semigroup. For every s ∈ S, we define the point measure δs : A → C by δs (f ) = f (s) for all f ∈ A. The point measures are dense in S A , and define an embedding of S in S A . As is well known, a semigroup S is extremely left amenable (ELA) if and only if there exists a right zero in ∆(S ) (see [7,8]). Also, when S is a discrete semigroup, then S is ELA if and only if every two elements of S have a common right zero (i.e. for all s, t ∈ S, there exists z ∈ S such that sz = tz = z; see [9]). For any n ∈ N, we say A is n-ELA if there exists a set F ⊆ ∆(S ) such that |F | = n and which is minimal with respect to the property l∗s F = F for all s ∈ S. A semigroup S is n-ELA if and only if there exists a left ideal group of order n in ∆(S ) (see [14,15]). When S is a discrete semigroup with finite intersection property for right ideals, we define the equivalence relation (r ) by a(r )b ⇔ ac = bc for some c ∈ S. We denote by S /(r ) the semigroup S with this equivalence relation, and we have the following characterization due to Lau (see [14]). 1. If S is n-ELA and F0 is a coset representative of S /(r ), then for each finite subset σ of S, there exists tσ ∈ S, depending on σ , such that aF0 tσ = F0 tσ for all a ∈ σ . 2. If for any finite subset σ of S, there exists Fσ ⊆ S, |Fσ | = n, such that aFσ = Fσ for all a ∈ S, then S is m-ELA for some m ≤ n, m dividing n. 3. Strong left amenability Definition 3.1. A semi-topological semigroup S is strongly left amenable (SLA) if there exists a left ideal group in ∆(S ).

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Notice that all n-ELA semigroups are SLA. Also, all compact groups are SLA, this is because if G is a compact group, then G = ∆(G). Theorem 3.2. If S is a semi-topological semigroup with left ideal group K in ∆(S ), then the following holds. 1. K is a minimal left ideal. 2. K is compact. 3. K is a right-topological group. 4. l∗s K = K for all s ∈ S. 5. For any µ, ν ∈ K and s ∈ S, l∗s µ = l∗s ν ⇒ µ = ν . 6. K ⊙ µ is a compact left ideal group for all µ ∈ ∆(S ). Proof. 1. Let M ⊆ K be another left ideal in ∆(S ) and let m ∈ M, then K = K ⊙ m ⊆ M. Therefore K = M, and K is a minimal left ideal. 2. It suffices to show that K is closed since ∆(S ) is compact. Take a k ∈ K . Then ∆(S ) ⊙ k is a left ideal of ∆(S ) that sits in K . It is closed since ∆(S ) is compact and the multiplication is right continuous. But K is a minimal left ideal from (1). Therefore, K = ∆(S ) ⊙ k and thus is closed. 3. By assumption, K is a group. Since ∆(S ) is right-topological then so is K . 4. Let s ∈ S, µ ∈ K and δs ⊙ µ = ν . Then since K is a left ideal, we have that δs ⊙ K ⊆ K , also, since K is a group, we have ν ⊙ K = µ ⊙ K = K . Therefore, we have that l∗s K = δs ⊙ K = δs ⊙ (µ ⊙ K ) = (δs ⊙ µ) ⊙ K = ν ⊙ K = K . 5. Let ψ be the identity in K , then we have l∗s µ = l∗s ν ⇒ δs ⊙ µ = δs ⊙ ν ⇒ δs ⊙ (ψ ⊙ µ) = δs ⊙ (ψ ⊙ ν) ⇒ (δs ⊙ ψ) ⊙ µ = (δs ⊙ ψ) ⊙ ν ⇒ µ = ν, where the last implication follows from the fact that K is a left ideal, and therefore δs ⊙ ψ is in the group K . 6. Consider the map fa : K → ∆(S ) defined by fa (µ) = µ ⊙ a. This map is continuous by continuity of the Arens product in the first variable, and therefore K ⊙ a is a continuous image of a compact set, and is therefore compact. Since K is a minimal left ideal, then by Proposition 2.1, K ⊙ a is also a minimal left ideal. We only need to prove the existence of an identity and the existence of an inverse for any element in K ⊙ a to complete the proof. The identity is given by ea = x ⊙ (a ⊙ x)−1 ⊙ a for any x ∈ K , where the inverse is taken in the group K . This ea is independent of the choice of x since if y ∈ K , we have y ⊙ (a ⊙ y)−1 ⊙ a = (x ⊙ x−1 ) ⊙ y ⊙ (a ⊙ y)−1 ⊙ a = x ⊙ (y−1 ⊙ x)−1 ⊙ (a ⊙ y)−1 ⊙ a

= x ⊙ (a ⊙ y ⊙ y−1 ⊙ x)−1 ⊙ a = x ⊙ (a ⊙ x)−1 ⊙ a = ea . Also, for any y ⊙ a ∈ K ⊙ a, its inverse in K ⊙ a is given by y ⊙ (a ⊙ y)−1 ⊙ (a ⊙ y)−1 ⊙ a. This follows readily by direct verification.  Theorem 3.3. Let S be a strongly left amenable semigroup and let K1 and K2 be two left ideal groups in ∆(S ). Then K1 and K2 are isomorphic (as right-topological group). Proof. We define the map f : K1 → K2 by f (x) = x ⊙ e2 , and the map g : K2 → K1 by g (y) = y ⊙ e1 , where e1 is the identity in K1 , and e2 is the identity in K2 . This is well defined since K1 and K2 are left ideal. We first show that those maps are onto. To see that, first notice that f (K1 ) is a left ideal contains in K2 . Let y be any element in ∆(S ), and x be any element of K1 . Then we have y ⊙ f (x) = y ⊙ (x ⊙ e2 ) = (y ⊙ x) ⊙ e2 ∈ K1 ⊙ e2 = f (K1 ), and therefore f (K1 ) is a left ideal in K2 , so by minimality of K2 this implies that f (K1 ) = K2 . It follows that f is onto. We now want to show that those maps are homomorphisms: f (x ⊙ y) = (x ⊙ y) ⊙ e2 = x ⊙ (y ⊙ e2 ) = x ⊙ [e2 ⊙ (y ⊙ e2 )]

= (x ⊙ e2 ) ⊙ (y ⊙ e2 ) = f (x) ⊙ f (y). Therefore, f is a (semigroup) homomorphism. The same proof replacing e2 by e1 shows that g is also a homomorphism. To see that they are actually group homomorphism, we need to show that ei ⊙ ej = ej . To prove that, let y ∈ K2 . Since f is onto, there exists x ∈ K1 such that f (x) = y. It follows that y ⊙ (e1 ⊙ e2 ) = f (x) ⊙ (e1 ⊙ e2 ) = f (x) ⊙ f (e1 ) = f (x ⊙ e1 ) = f (x) = y

(e1 ⊙ e2 ) ⊙ y = (e1 ⊙ e2 ) ⊙ f (x) = f (e1 ) ⊙ f (x) = f (e1 ⊙ x) = f (x) = y. Therefore, e1 ⊙ e2 is the identity of K2 , and therefore e1 ⊙ e2 = e2 . The same proof can be used to show ei ⊙ ej = ej , where i, j ∈ {1, 2}. And to finish the proof that f is a (group) homomorphism, we have [f (x−1 )]⊙[f (x)] = f (x−1 ⊙ x) = f (e1 ) = e2 . Since this is true for any x ∈ K1 , it follows that [f (x)]−1 = f (x−1 ). Therefore, it is a group homomorphism. To finish the proof of this theorem, we only have left to prove that f is continuous. Let xα be a net in K1 which converges to x ∈ K1 . Then, it follows that f (xα ) = xα ⊙ e2 → x ⊙ e2 = f (x). To finish the proof, we need to show that f ◦ g = id and g ◦ f = id.

(f ◦ g )(y) = f (y ⊙ e1 ) = y ⊙ e1 ⊙ e2 = y ⊙ e2 = y, ∀y ∈ K2 , (g ◦ f )(x) = g (x ⊙ e2 ) = x ⊙ e2 ⊙ e1 = x ⊙ e1 = x, ∀x ∈ K1 , and therefore, K1 and K2 are isomorphic as right-topological group.



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Theorem 3.4. Every strongly left amenable semigroups is left amenable. Proof. This proof is inspired by a similar proof in [42]. Since K is a compact right-topological group, it admits a left Haar measure ν [39]. Then we can define the following function:

∫

λ(f ) =

⟨f , µ⟩dν(µ). K

We want to show that this is a left invariant mean on LUC (S ). First, if 1 is the constant one function on S, then we have

∫

λ(1) =

⟨1, µ⟩dν(µ) = K

∫

1dν(µ) = 1. K

Also, we have

‖λ‖ = sup |λ(f )| ≤ sup sup |µ(f )| ≤ sup sup ‖µ‖‖f ‖ = sup ‖f ‖ ≤ 1, ‖f ‖≤1 µ∈K

‖f ‖≤1

‖f ‖≤1 µ∈K

‖f ‖≤1

and therefore, λ is a mean on LUC (S ). Now, we want to show that this mean is left invariant. Let s ∈ S, and ψ be the identity of K . Then, we have ls λ(f ) = λ(ls f ) = ∗

∫

∫

⟨ls f , µ⟩dν(µ) = ⟨f , l∗s µ⟩dν(µ) ∫ ∫ K ∗ = ⟨f , ls (ψ ⊙ µ)⟩dν(µ) = ⟨f , (l∗s ψ) ⊙ µ⟩dν(µ) K ∫K = ⟨f , µ⟩dν(µ) = λ(f ). K

K

Notice that the 4th and 6th equalities follow from the fact that the Haar measure is left invariant by elements of the topological center Λ(K ), and both ψ and l∗s ψ are in Λ(K ). It follows that λ is left invariant, and therefore if S is strongly left amenable, then S is left amenable.  4. Characterization of SLA semigroups We are now ready for our characterization theorem for SLA semigroup. Recall first that since ∆(S ) is a compact righttopological semigroup, it follows from Proposition 2.4 that ∆(S ) has a minimal idempotent. Therefore by Proposition 2.2, there exist a minimal left ideal and a minimal right ideal in ∆(S ). Theorem 4.1. Let S be a semi-topological semigroup. Then the following are equivalent. 1. 2. 3. 4. 5. 6.

There exists a left ideal group in ∆(S ). There exists a compact left ideal group in ∆(S ). All minimal left ideals of ∆(S ) are compact groups. There exists a ∈ ∆(S ) for which ∆(S ) ⊙ a is a group. There exists a compact group K ⊆ ∆(S ) such that l∗s K = K for all s ∈ S. There exists a unique minimal right ideal in ∆(S ).

Proof. (1) ⇒ (2): This follows from Theorem 3.2. (2) ⇒ (1): Trivial. (3) ⇔ (6): Since ∆(S ) is a compact right-topological semigroup, it follows by Proposition 2.4 that ∆(S ) has a minimal idempotent, so we can apply Proposition 2.3, which gives us that all minimal left ideals of ∆(S ) are groups if and only if there is a unique minimal right ideal in ∆(S ). (3) ⇒ (2): By Proposition 2.4, we know that ∆(S ) has (at least one) a minimal left ideal. Since all minimal left ideals of ∆(S ) are groups, it follows there exists a compact left ideal group in ∆(S ). (2) ⇒ (3): Let K be a compact left ideal group in ∆(S ). Then {K ⊙ µ : µ ∈ ∆(S )} is the collection of all minimal left ideal in ∆(S ) by Proposition 2.1. Also, we proved in Theorem 3.2 that K ⊙ µ is a group for all µ ∈ ∆(S ), which completes the proof. (4) ⇒ (1): ∆(S ) ⊙ a is a left ideal since if µ ∈ ∆(S ) we have µ ⊙ (∆(S ) ⊙ a) = (µ ⊙ ∆(S )) ⊙ a ⊆ ∆(S ) ⊙ a. Therefore, ∆(S ) ⊙ a is a left ideal group in ∆(S ). (2) ⇒ (4): Let K be a compact left ideal group in ∆(S ). Then, K ⊙ µ = ∆(S ) ⊙ µ = K for all µ ∈ K . Therefore, ∆(S ) ⊙ µ is a group. (1) ⇒ (5): If K is a left ideal group in ∆(S ), then µ ⊙ K ⊆ K for any µ ∈ ∆(S ), but since K is also a group, then for any x ∈ K we have that K = K ⊙ x = x ⊙ K , which implies for any µ ∈ ∆(S ),

µ ⊙ K = µ ⊙ (x ⊙ K ) = (µ ⊙ x) ⊙ K = ν ⊙ K = K , for some ν ∈ K . Therefore, for any s ∈ S, we have that l∗s K = δs ⊙ K = K . (5) ⇒ (1): By continuity of the Arens product in the first variable, l∗s K = δs ⊙ K is a left ideal. By assumption this is also a group. 

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If S is a semi-topological semigroup, and A is a left translation invariant C ∗ -subalgebra of CB(S ) containing the constants, then we say that A is m-admissible if the function s → (Tµ f )(s) = µ(ls f ) is in A for all f ∈ A and µ ∈ S A [39]. In this case, S A is a semigroup with the Arens product. A closed invariant (left and right) subalgebra of WAP (S ) containing the constants is always m-admissible [43]. Also, m-admissible left translation invariant C∗ -subalgebras are always right translation invariant. If A is an m-admissible subalgebra of LUC (S ), then we say A is strongly left amenable, if S A is strongly left amenable. Theorem 4.2. Let A be an m-admissible subalgebra of WAP (S ). If A is left amenable, then A is strongly left amenable. Proof. We define the following maps:

ϵ : S → SA,

ϵ(s) = δs , ∀s ∈ S ϵ : CB(S ) → A, ϵ ∗ (f˜ )(s) = f˜ (ϵ(s)), ∀f˜ ∈ CB(S A ), ∀s ∈ S ∗∗ ∗ A ∗ ϵ : A → CB(S ) , ϵ ∗∗ (µ)(f˜ ) = µ(ϵ ∗ (f˜ )), ∀µ ∈ A∗ , ∀f˜ ∈ CB(S A ). ∗

A

Notice that if λ is a mean invariant, then so is µ. Let s, t ϵ ∗ (lϵ(s) f˜ )(t ) = lϵ(s) (f˜ )(ϵ(t )) = have

on A, then µ = ϵ ∗∗ (λ) is a mean on CB(S A ). We want to prove that if λ is also left ∈ S and f˜ ∈ CB(S A ). We define f ∈ CB(S ) to be the restriction of f˜ to S. Then we have ˜f (ϵ(st )) = f (st ) = ls f (t ), and also we have that ϵ ∗ (f˜ )(t ) = f˜ (ϵ(t )) = f (t ). Therefore, we

lϵ(s) µ(f˜ ) = µ(lϵ(s) f˜ ) = ϵ ∗∗ (λ)(lϵ(s) f˜ ) = λ(ϵ ∗ (lϵ(s) f˜ )) = λ(ls f )

= λ(f ) = λ(ϵ ∗ (f˜ )) = ϵ ∗∗ (λ)(f˜ ) = µ(f˜ ). Now, using the continuity of the map s → l∗s µ, we get that l∗x µ = µ for all x ∈ S A , which shows that µ is a left invariant mean on CB(S A ). Now, we want to show that S A has a unique minimal right ideal. Suppose that R1 and R2 are two distinct minimal right ideals in S A . Since A is a subalgebra of WAP (S ), S A is a semi-topological semigroup, and therefore minimal right ideals of S A are closed. Therefore, we can apply Urysohn’s lemma [44] to construct a function g which is 0 on R1 and 1 on R2 . If we take x ∈ R1 and y ∈ R2 , then we get that µ(lx g ) = 0 and µ(ly g ) = 1, which contradict the left invariance of µ. Therefore, there is unique minimal right ideal. Now, pick any minimal left ideal L in S A , then RL is a left ideal contained in L, therefore by minimality of L, we have that RL = L. But we also know that the product of a right ideal by a left ideal is a group. It follows that L is a left ideal group in S A , and therefore S A is strongly left amenable.  Notice that the only place in the proof where we use the fact that A is an m-admissible subalgebra of WAP (S ) is to prove that minimal right ideals are closed. Also, the previous theorem justifies why we defined strong left amenability of a semitopological semigroup S using LUC (S ), and centered almost all our work on this algebra. Corollary 4.3. Let S be a semi-topological semigroup. If LUC (S ) is left amenable and if the minimal right ideals of S LUC are closed, then LUC (S ) is strongly left amenable. Proof. If the minimal right ideals of S LUC are closed, then we can apply the same proof as in the theorem.



Corollary 4.4. Let S be a left amenable locally compact, non-compact group. Then the minimal right ideals of S LUC are not closed. Proof. If the minimal right ideals of S LUC are closed, then S is strongly left amenable. But, we know that the only locally compact strongly left amenable groups are the compact groups. Therefore, it is impossible.  Theorem 4.5. If S is a semi-topological semigroup, then the following are equivalent. 1. AP (S ) is left amenable. 2. AP (S ) is strongly left amenable. 3. There exists a compact set K ⊆ S AP (S ) which is minimal with respect to the property l∗s K = K for all s ∈ S. Proof. (1) ⇒ (2): AP (S ) is an m-admissible subalgebra of WAP (S ), therefore we can apply Theorem 4.2. (2) ⇒ (1): This is the same proof as in the LUC (S ) case. (2) ⇒ (3): See Theorem 3.2. (3) ⇒ (2): Since l∗s K = K for all s ∈ S, it follows that µ ⊙ K ⊆ K for all µ ∈ S AP (S ) , and therefore K is a left ideal. Let µ ∈ K , then K ⊙ µ is also a left ideal in S AP (S ) , and K ⊙ µ ⊆ K . Therefore, it follows from the minimality of K that K = K ⊙ µ. Now, we want to show µ ⊙ K = K . Let ψ ∈ K , and sα be a net in S such that δsα → µ, then since l∗sα K = K for all α , there is a net ϕα in K such that δsα ⊙ ϕα = ψ for all α . Now since K is compact, there exists a subnet ϕβ of ϕα such that ϕβ converges to some ϕ ∈ K . It follows that δsβ ⊙ ϕβ = ψ for all β , and δsβ ⊙ ϕβ → µ ⊙ ϕ . Therefore µ ⊙ ϕ = ψ , and it follows that µ ⊙ K = K . Now since K is both left and right simple, it must be a group, and AP (S ) is strongly left amenable. 

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5. Construction and examples In the next theorem, we want to establish that every homomorphic image of an SLA semigroup is also SLA. Let S , T be semi-topological semigroups, and let f : S → T be a continuous surjective homomorphism. We define f˜ : LUC (T ) → LUC (S )

˜

˜

by [f˜ (g )](s) = g [f (s)] for all s ∈ S and we define f˜ : ∆(S ) → ∆(T ) by [f˜ (µ)](g ) = µ[f˜ (g )] for all g ∈ LUC (T ). We first want to show that those maps are properly defined. Let g ∈ LUC (T ), we want to show f˜ (g ) ∈ LUC (S ). Let sα be a net in S which converges to s ∈ S. Then we have f (sα ) → f (s) since f is continuous, therefore since g is also continuous, g (f (sα )) → g (f (s)). It follows that

[f˜ (g )](sα ) = g (f (sα )) → g (f (s)) = [f˜ (g )](s). Therefore, f˜ (g ) is continuous. To see it is actually left uniformly continuous, let t ∈ S. Then we have sup |lsα (f˜ (g ))(t ) − ls (f˜ (g ))(t )| = sup |f˜ (g )(sα t ) − f˜ (g )(st )| t ∈S

t ∈S

= sup |g (f (sα t )) − g (f (st ))| = sup |g (f (sα )f (t )) − g (f (s)f (t ))| t ∈S

t ∈S

= sup |lf (sα ) g (f (t )) − lf (s) g (f (t ))| → 0, t ∈S

since g ∈ LUC (T ) and f (sα ) → f (s). Therefore f˜ (g ) ∈ LUC (S ).

˜

Now, we want to show that f˜ (µ) ∈ ∆(T ). First, notice that if e : T → C, defined by e(t ) = 1 for all t ∈ T , then

[f˜ (e)](s) = e(f (s)) = 1. Therefore, [f˜˜ (µ)](e) = µ(f˜ (e)) = 1 for all µ ∈ ∆(S ). Now, let g ∈ LUC (T ) be such that ‖g ‖ ≤ 1. Then

‖f˜ (g )‖ = sup |[f˜ (g )](s)| = sup |g (f (s))| ≤ sup |g (t )| ≤ 1. s∈S

s∈S

t ∈T

It follows that

‖f˜˜ (µ)‖ = sup |f˜˜ (µ)(g )| = sup |µ(f˜ (g ))| ≤ 1, ‖g ‖≤1

‖g ‖≤1

˜

and therefore f˜ (µ) ∈ ∆(T ) for all µ ∈ ∆(S ). Finally, we define the map π : S → ∆(S ) by π (s) = δs and we define the map ρ : T → ∆(T ) by ρ(t ) = δt . Now, let K

˜

be a compact left ideal group in ∆(S ), and let K ′ = f˜ (K ). We will show that K ′ is a compact left ideal group in ∆(T ), but first we need a lemma, whose proof being direct verification, will be omitted. Lemma 5.1. The following is true for the maps we just defined. 1. ls f˜ (g ) = f˜ (lf (s) g ) for all s ∈ S and g ∈ LUC (T ).

˜ ˜ 2. l∗f (s) f˜ (µ) = f˜ (l∗s µ) for all s ∈ S and µ ∈ ∆(S ). ˜

3. f˜ (δs ) = δf (s) for all s ∈ S.

˜

4. ρ ◦ f = f˜ ◦ π . 5. νl (f˜ (g )) = f˜ (νl′ (g )) for all g ∈ LUC (T ) and ν ∈ ∆(S ).

˜ 6. f˜ and f˜ are homomorphisms. 7. 8. 9. 10.

˜

f˜ and f˜ are continuous. l∗t K ′ = K ′ for all t ∈ T . K ′ is a group. K ′ is compact.

˜

11. f˜ is onto. Theorem 5.2. Every homomorphic image of a strongly left amenable semigroup is strongly left amenable. Proof. Let S be an SLA semigroup, and f : S → T be a homomorphism onto a semi-topological semigroup T . If S is an SLA,

˜

with K a compact left ideal group in ∆(S ), then f˜ (K ) = K ′ is a compact left ideal group in ∆(T ). That K ′ is a compact group follows from Lemma 5.1. To see that is also a left ideal, let µ ∈ ∆(T ). Then there is a net tβ in T such that δtβ → µ. Since l∗tβ K ′ = K ′ for all β , it follows from the compactness of K ′ that for any ν ∈ K ′ , l∗tβ ν → µ ⊙ ν ∈ K ′ . Therefore, µ ⊙ K ′ = K ′ for all µ ∈ ∆(T ). So K ′ is a left ideal.



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We already know that every compact group, ELA semigroup and n-ELA semigroup are strongly left amenable. Now, we want to prove that the product of an extremely left amenable semigroup by a compact group is also strongly left amenable, which will prove, using Pestov [12], that the unitary group U (M ) of an injective von Neumann algebra M with the s(M , M∗ )-topology is strongly left amenable. Let S be an extremely left amenable and G be a compact group. We define S × G using the cartesian product, with the product topology. If µ ∈ ∆(S ) and ν ∈ ∆(G) = {δg : g ∈ G}, then we define µ × ν ∈ ∆(S × G) by

(µ × ν)(f ) = ν(h),

h(t ) = µ(πt f ),

πt f (s) = f (s, t ),

for any f ∈ LUC (S ×G). Notice that whenever ‖f ‖ ≤ 1, then ‖πt f ‖ ≤ 1. Therefore, we have ‖µ×ν‖ = sup‖f ‖≤1 |(µ×ν)(f )| ≤ 1. Also, since (µ × ν)(1) = 1, then ‖µ × ν‖ = 1. Therefore, µ × ν ∈ ∆(S × G). Lemma 5.3. Let S be an extremely left amenable semigroup with µ ∈ ∆(S ) be a right zero, and let ϕ ∈ ∆(S ). Let G be a compact group, and let ψ, ν ∈ ∆(G), with ψ = δg and ν = δg ′ . Then the following holds. 1. 2. 3. 4. 5. 6.

δs × δg = δ(s,g ) . (µ × δg )(f ) = µ(πg f ). πg (l(s,g ′ ) f ) = ls (πg ′ g f ). πg (µ × δg ′ )l (f ) = µl (πgg ′ f ). (ϕ × δg ) ⊙ (µ × δg ′ ) = µ × δgg ′ . δ(s,g ) ⊙ (µ × δg ′ ) = µ × δgg ′ .

The proof of the previous lemma is direct calculation and will be omitted. Theorem 5.4. The product of an extremely left amenable semigroup by a compact group is strongly left amenable. Proof. This follows from Lemma 5.3. Let µ ∈ ∆(S ) be a right zero, then we want to show that K = µ × {δg : g ∈ G} is a left ideal group in ∆(S × G). Since the point measures are dense in ∆(S × G), it is enough to show that K is a group, and l∗(s,g ) K ⊆ K . We have l∗(s,g ) K = δ(s,g ) ⊙ K ⊆ K by property (6) of Lemma 5.3. Also, by property (6) of Lemma 5.3, it is clear that K is a group. Therefore, S × G is strongly left amenable.  From the previous theorem, we get the following examples of strongly left amenable semigroups as given in the following proposition. Proposition 5.5. 1. The unitary group U (M ) of an injective von Neumann algebra M with the s(M , M∗ ) topology is strongly left amenable. 2. The unitary group of the group von Neumann algebra VN (G) of an amenable locally compact group G is strongly left amenable. Proof. (1): By Pestov [12], we known that U (M ) is the product of an extremely left amenable semigroup by a compact group. Therefore, the result follows from Theorem 5.4. (2): The group von Neumann algebra VN (G) of an amenable locally compact group G is an injective von Neumann algebra [45], and therefore its unitary group is strongly left amenable by (1).  6. Special classes of semigroups In this section, we want to characterize strong left amenability for some particular classes of semigroups that include compact semigroups, locally compact groups, discrete semigroups and connected semigroups. Theorem 6.1. If S is a compact semi-topological semigroup or a totally bounded topological group, then LUC (S ) is strongly left amenable if and only if LUC (S ) is left amenable. Proof. It is known [39, Proposition 4.4.8] that if S is either a compact semi-topological semigroup or a totally bounded topological group, then AP (S ) = LUC (S ) = RUC (S ). Therefore, we can apply Theorem 4.5, which gives the result.  In the particular case where the semigroup is indeed a group, it is known that the only locally compact groups which are strongly left amenable are compact. (See [46].) Let S be an SLA semigroup and K be a compact left ideal group in ∆(S ). We define an equivalence relation ∼ by s ∼ t if and only if l∗s µ = l∗t µ for all µ ∈ K . We denote this semigroup by S /K . On this semigroup, we use the topology induced by S. Let ψ ∈ K be the identity in K . We define the map f : S → K by f (s) = l∗s ψ , the map π : S → S /K by π (s) = s and the map fˆ : S /K → K by fˆ (s) = f (s). Then f is a homomorphism since if s, t ∈ S then f (st ) = l∗st ψ = l∗s l∗t ψ = δs ⊙ δt ⊙ ψ = δs ⊙ [ψ ⊙ (δt ⊙ ψ)]

= (δs ⊙ ψ) ⊙ (δt ⊙ ψ) = f (s) ⊙ f (t ). Also, f is continuous since if sα is a net in S which converges to some s ∈ S, then f (sα ) = l∗sα ψ = δsα ⊙ ψ → δs ⊙ ψ = l∗s ψ = f (s). Also, the map fˆ is well defined, since if s, t ∈ S such that s = t then fˆ (s) = l∗s ψ = l∗t ψ = fˆ (t ).

Lemma 6.2. The map fˆ is a continuous, one-to-one, homomorphism. Also, fˆ (S /K ) is dense in K . Therefore, |S /K | ≤ |K | and if |K | = n we have that S /K is a group of order n.

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Proof. The continuity and the fact that fˆ is a homomorphism is a direct consequence of the fact that f is a continuous homomorphism. Now to show that it is injective, let s, t ∈ S /K such that fˆ (s) = fˆ (t ). For any µ ∈ K we have fˆ (s) = fˆ (t ) ⇒ l∗s ψ = l∗t ψ ⇒ δs ⊙ ψ = δt ⊙ ψ ⇒ δs ⊙ ψ ⊙ µ = δt ⊙ ψ ⊙ µ ⇒ δs ⊙ µ = δt ⊙ µ. Since this is true for all µ ∈ K , it follows that s = t and the map fˆ is injective. Now, to show that f˜ (S /K ) is dense in K , let µ ∈ K , then there is a net {sα } in S which converges to µ, then fˆ (sα ) = δsα ⊙ ψ → µ ⊙ ψ = µ. By injectivity of fˆ , we have |S /K | ≤ |K |. Finally if |K | = n, then K is discrete, and since fˆ (S /K ) is dense in a discrete group, |S /K | = |K |.  Lemma 6.3. If S is an SLA semigroup with a compact left ideal group K in ∆(S ), then S /K is a cancellative semigroup. Proof. Let r , s, t ∈ S /K and suppose that st = rt. Then we have st = rt ⇒ f˜ (st ) = f˜ (rt )

⇒ f˜ (s) ⊙ f˜ (t ) = f˜ (r ) ⊙ f˜ (t ) since f˜ is a homomorphism ⇒ f˜ (s) = f˜ (r ) since K is a group ⇒ s = r since f˜ is one-to-one. A similar proof shows that st = sr ⇒ t = r, and therefore S /K is a cancellative semigroup.



Corollary 6.4. If K is finite or if S is compact, then S /K is isomorphic to K . Proof. In either case, using Lemma 6.2, it follows that the map f is onto. Therefore, fˆ is a continuous bijective homomorphism; hence, it is an isomorphism between S /K and K .  Theorem 6.5. If S is a discrete semigroup, then S is strongly left amenable if and only if S is n-extremely left amenable for some n ∈ N, that is, if S is a discrete semigroup, then the Stone-Čech compactification β S never contains a compact left ideal group of infinite order. Proof. We know that if S is an SLA, then S /K is a cancellative SLA semigroup. Therefore β(S /K ) has a unique minimal right ideal. In [47, Proposition 6.23], it is proved that if T is an infinite cancellative semigroup, then β T has at least 2c minimal right ideal. Therefore, it follows that S /K is finite. Now, since fˆ (S /K ) is dense in K , it follows that K is also finite, and therefore S is n-ELA for some n ∈ N.  There is a characterization of n-ELA discrete semigroups and of SLA discrete semigroups in [48]. In view of our last theorem, those two characterization actually describe the same objects. Let S be strongly left amenable. If |K | = ∞ we will say S is ∞-ELA. Otherwise it is n-ELA for some n ∈ N. Theorem 6.6. If S is a connected SLA semigroup, then S is either ELA or ∞-ELA. Proof. We know that every connected n-ELA semigroup is ELA (see [14]). Also every SLA semigroups are either n-ELA or ∞-ELA. So we only need to see that a connected SLA semigroup can be ∞-ELA. For example, if we take the circle group T with its usual topology induced by R2 . By compactness of T, we can take K = T, which show that S is ∞-ELA.  7. Fixed point properties We define an anti-action of S on l∞ (S ) by S × l∞ (S ) → l∞ (S ),

(s, f ) = s · f = ls f , ls f (t ) = f (st ), ∀s, t ∈ S , ∀f ∈ l∞ (S ). This map restricted to CB(S ) define an anti-action on CB(S ), and similarly for LUC (S ). We define the following action of S on ∆(S ): S × ∆(S ) → ∆(S ),

(s, µ) = s · µ = l∗s µ, l∗s µ(f ) = µ(ls f ). When we consider S as a subset of ∆(S ), this map coincides with the Arens product already defined: s · µ = δs ⊙ µ. If X is a Hausdorff topological space, we define CB(X ) to be the set of all continuous, bounded, complex-valued functions on X , and

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LUC (S , X ) to be the subalgebra of CB(X ) for which the map S → CB(X ) defined by s → ls f is continuous (see [49]). As for LUC (S ), we define an anti-action of S on LUC (S , X ) by S × LUC (S , X ) → LUC (S , X ),

(s, f ) = s · f = ls f . Fix x ∈ X . We define the following map:

Φ : LUC (S , X ) → LUC (S ), Φ (f ) = Tx (f ), Tx f (s) = f (s · x). Notice that this map commutes with the action of S, i.e. s · Φ (f ) = Φ (s · f ) for all s ∈ S, and for all f ∈ LUC (S , X ). To see that, let t ∈ S then we have s · (Φ (f ))(t ) = (Φ (f ))(st ) = (Tx f )(st ) = f (st · x) = (s · f )(t · x)

= (Tx (s · f ))(t ) = (Φ (s · f ))(t ). Now, let ∆(LUC (S , X )) denote the set of all multiplicative means on LUC (S , X ). We define the action of S on ∆(LUC (S , X )) by S × ∆(LUC (S , X )) → ∆(LUC (S , X )),

(s, µ) = s · µ = l∗s µ, and we define the following map:

Ψ : ∆(S ) → ∆(LUC (S , X )), Ψ (µ) = µ′ , µ′ (f ) = µ(Φ (f )). Once again, this map commutes with the action of S. If f ∈ LUC (S , X ), we have

(s · (Ψ (µ)))(f ) = (Ψ (µ))(s · f ) = µ(Φ (s · f )) = µ(s · Φ (f )) = (s · µ)(Φ (f )) = (Ψ (s · µ))(f ), which proves that s · (Ψ (µ)) = Ψ (s · µ) for all s ∈ S and for all µ ∈ ∆(S ). Now, if we associate X with the point measures on ∆(LUC (S , X )), then notice that the actions of S on X and δX correspond. That is, for any f ∈ LUC (S , X ), we have

δs·x (f ) = f (s · x) = (s · f )(x) = δx (s · f ) = (s · δx )(f ), and therefore we have δs·x = s · δx for all s ∈ S and for all x ∈ X . Now we want to prove that Ψ is continuous. Let µα → µ be a net in ∆(S ). Then we have

(Ψ (µα ))(f ) = µα (Tx f ) → µ(Tx f ) = (Ψ (µ))(f ), which proves the continuity of Ψ . Theorem 7.1. If S is a semi-topological semigroup, then the following are equivalent. 1. There exists a compact set K ⊆ ∆(S ) such that l∗s K = K . 2. Whenever S acts on a compact Hausdorff space X , where the action is jointly continuous, there exists a compact set K ′ ⊆ X such that s · K ′ = K ′ . Also, |K ′ | ≤ |K |. Moreover, if S is an SLA semigroup, then the two conditions hold. Proof. (1) ⇒ (2): If X is compact, then LUC (S , X ) = C (X ) since the action of S on X is jointly continuous. So ∆(LUC (S , X )) = X . Let K ′ = Ψ (K ). Then K ′ is compact and s · K ′ = s · Ψ (K ) = Ψ (s · K ) = Ψ (K ) = K ′ . (2) ⇒ (1): Since ∆(S ) is a compact Hausdorff space, and the action of S on ∆(S ) defined by (s, µ) = l∗s µ is jointly continuous, there exists K ⊆ ∆(S ) such that s · K = l∗s K = K for all s ∈ S. (SLA) ⇒ (1): This is trivial using Theorem 4.1.  Corollary 7.2. Let S be a semi-topological semigroup acting on a compact subset X of a complete metric space. Suppose that the action is jointly continuous and for at least one s ∈ S, the map s : X → X is contractive, then there is a common fixed point for S in X . Proof. Since s : X → X is a contractive map, there exists a point x ∈ X such that sn y → x for all y ∈ X . Therefore, diam(sn X ) → 0 as n → ∞. Now, by the theorem, we know that there exists K ⊆ X such that t · K = K for all t ∈ S. Now, since K ⊆ sn X for all n, we have that K = {x}, and x ∈ X is a common fixed point. 

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Now, if S is a semi-topological semigroup which acts on a compact Hausdorff space X , where the action is jointly continuous, then we can extend the action S × X → X , to an action ∆(S ) × X → X , which is continuous in the first variable in the following way:

µ · x = Ψx (µ),

µ ∈ ∆(S ), x ∈ X .

Lemma 7.3. Let S be a semi-topological semigroup, and let K ⊆ ∆(S ) such that µ ⊙ K = K for all µ ∈ ∆(S ). Then there is a compact left ideal group in ∆(S ), i.e., S is strongly left amenable. Proof. First, notice that K is a left ideal, but we do not know if K is minimal. Let ϕ ∈ K and define J = K ⊙ ϕ . Therefore for any µ ∈ ∆(S ),

µ ⊙ J = µ ⊙ (K ⊙ ϕ) = (µ ⊙ K ) ⊙ ϕ = K ⊙ ϕ = J . Also, since ∆(S ) is a compact right-topological semigroup, ∆(S ) contains a minimal left ideal, and therefore any left ideal of ∆(S ) contains a minimal left ideal. Let J = K ⊙ ϕ ⊆ ∆(S ) ⊙ ϕ which is a minimal left ideal (see [39]). Therefore, J is left and right simple, which implies that J is a left ideal group in ∆(S ). It follows that S is strongly left amenable.  Theorem 7.4. If S is a semi-topological semigroup, then the following are equivalent. 1. S is strongly left amenable. 2. Whenever S acts on a compact Hausdorff space X , where the action is jointly continuous, there exists a compact set K ′ ⊆ X such that µ · K ′ = K ′ for all µ ∈ ∆(S ), where µ · K ′ denotes the extension of the action to an (right continuous) action of ∆(S ) on X . Proof. (1) ⇒ (2) If K is a compact left ideal group in ∆(S ), let K ′ = Ψ (K ). For all µ ∈ ∆(S ), we have

µ · K ′ = µ · Ψ (K ) = µ · (K · x) = (µ ⊙ K ) · x = K · x = Ψ (K ) = K ′ . (2) ⇒ (1) Let X = ∆(S ), then we have K ⊆ ∆(S ) such that µ ⊙ K = K for all µ ∈ ∆(S ). Now, we can apply Lemma 7.3 to prove that S is strongly left amenable.  8. Non-expansive mappings In this section, E will denote a Banach space, and C a weakly compact, bounded convex subset of E. Also, S will denote a semi-topological semigroup, and S will be a strongly continuous representation of S as non-expensive mapping on C , that is, S = {Ts : s ∈ S } such that 1. 2. 3. 4.

Ts : C → C , ∀s ∈ S, Tst = Ts Tt , ∀s, t ∈ S, ‖Ts x − Ts y‖ ≤ ‖x − y‖, ∀s ∈ S , ∀x, y ∈ C , s → Ts x is continuous on S , ∀x ∈ C .

In most of this section, we will be using S CB(S ) instead of S LUC (S ) , but notice that S CB(S ) is not in general a semigroup, since it is not necessarily closed under the Arens product. Let µ be any multiplicative mean on CB(S ). Define the equivalence relation ∼µ on S by s ∼µ t if and only if l∗s µ = l∗t µ, and we denote by S /µ the quotient of S with this equivalence relation. Also, we define the set Sµ = {s ∈ S : l∗s µ = µ}. Notice that this section is mostly based on [50,23]. Now, for any µ ∈ S CB(S ) and z ∈ C , we define Tµ z weakly by ⟨Tµ z , x∗ ⟩ = µs ⟨Ts z , x∗ ⟩, for all x∗ ∈ E ∗ . This map is well ∗ defined, since if {sα } is a net in S such that δsα →w µ, then ⟨Tsα z , x∗ ⟩ = δsα ⟨Ts z , x∗ ⟩ → µs ⟨Ts z , x∗ ⟩ = ⟨Tµ z , x∗ ⟩. It follows w that Tsα z → Tµ z. By weak compactness, we then have that Tµ z ∈ C . Finally, if z ∈ C and x∗ ∈ E ∗ are fixed, we define fx∗ (s) = ⟨Tµ z , x∗ ⟩ for all s ∈ S. The map fx∗ is in CB(S ), but not necessarily in LUC (S ), which is why we need to work with CB(S ) instead of LUC (S ). Lemma 8.1. Let S be a semi-topological semigroup. Let {sα } be a net in S such that δsα →w µ ∈ S CB(S ) , let s ∈ Sµ , and let r , t ∈ S such that r ∼µ t. Also, let z ∈ C . Then, 1. Tssα z − Tsα z →w 0, 2. Trsα z − Ttsα z →w 0. ∗

Proof. 1. If x∗ ∈ E ∗ , then we have

⟨Tssα z − Tsα z , x∗ ⟩ = = = → = = =

fx∗ (ssα ) − fx∗ (sα )

(ls fx∗ − fx∗ )(sα ) δsα (ls fx∗ − fx∗ ) µ(ls fx∗ − fx∗ ) (l∗s µ − µ)(fx∗ ) (µ − µ)(fx∗ ) 0

and therefore, it follows that Tssα z − Tsα z →w 0.

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2. If x∗ ∈ E ∗ , then we have

⟨Trsα z − Ttsα z , x∗ ⟩ = = = → = =

fx∗ (rsα ) − fx∗ (tsα )

(lr fx∗ − lt fx∗ )(sα ) δsα (lr fx∗ − lt fx∗ ) µ(lr fx∗ − lt fx∗ ) (l∗r µ − l∗t µ)(fx∗ ) 0.

Therefore, it follows that Trsα z − Ttsα z →w 0.



Theorem 8.2. Let E be a Banach space, and C be a compact convex subset of E. Let S be a semi-topological semigroup and let µ ∈ S CB(S ) . Assume that Sµ is non-empty, S = {Ts : s ∈ S } is a strongly continuous representation of S as non-expansive mappings, and Tµ z = z for some z ∈ C . 1. Then Ts z = z for all s ∈ Sµ . 2. If s ∈ S is such that Ts z = z. Then Tt z = z for all t ∼µ s in S /µ. 3. If there exists an element s in each equivalence class of S /µ such that Ts z = z, then Tt z = z for all t ∈ S. Proof. 1. Since C is compact, the weak and norm topologies coincide. Therefore, if sα is a net in S such that δsα →w µ, then we have ‖Tsα z − Tµ z ‖ → 0. Also, ∗

‖Ts z − Tsα z ‖ = ≤ ≤ →

‖Ts z − Tssα z + Tssα z − Tsα z ‖ ‖Ts z − Tssα z ‖ + ‖Tssα z − Tsα z ‖ ‖z − Tsα z ‖ + ‖Tssα z − Tsα z ‖ 0 + 0 = 0.

Notice that for the last inequality, we used the fact that Tµ z = z and Lemma 8.1(1). Therefore, Tsα z → Tµ z = z and also Tsα z → Ts z. By unicity of the limit in a Hausdorff space, it follows that Ts z = z. 2. Let {sα } be a net in S such that δsα → µ. We know that Tsα z → Tµ z = z and therefore, by the continuity of Ts and Tt , we have Ts (Tsα z ) → Ts z and Tt (Tsα z ) → Tt z. Therefore, using Lemma 8.1(2), we have 0 ← ‖Tssα z − Ttsα z ‖ = ‖Ts (Tsα z ) − Tt (Tsα z )‖ → ‖Ts z − Tt z ‖. It follows, by the unicity of the limit, that Ts z = Tt z = z. 3. We apply (2) for each equivalence class.  Theorem 8.3. Let E be a Banach space, and C be a weakly compact convex subset of E. Let S be a semi-topological semigroup and let µ ∈ S CB(S ) . Assume that Sµ is non-empty, S = {Ts : s ∈ S } is a strongly continuous representation of S as non-expansive mappings, Ts is weak-to-weak continuous for all s ∈ S, and Tµ z = z for some z ∈ C . 1. Then Ts z = z for all s ∈ Sµ . 2. If s ∈ S is such that Ts z = z. Then Tt z = z for all t ∼µ s. 3. If every equivalence class of S /µ contains an element s ∈ S such that Ts z = z, then Tt z = z for all t ∈ S. Proof. 1. Let sα be a net in S such that δsα → µ, then by Lemma 8.1, Tsα z → Tµ z. By the weak-to-weak continuity of Ts , Tssα z = Ts (Tsα z ) →w Ts (Tµ z ) = Ts z. Also, by Lemma 8.1, Tsα z − Tssα z →w 0. Hence, Tssα z − Ts z →w Ts z − Ts z = 0. Therefore, Tsα z − Ts z = (Tsα z − Tssα z ) + (Tssα z − Ts z ) → 0. Thus Tµ z − Ts z = 0 and Ts z − Tµ z = z. 2. Let {sα } be a net in S such that δsα → µ. We have that Ts z − Tt z w ← Tssα z − Ttsα z →w 0. Therefore, it follows that Ts z = Tt z = z. 3. We apply (2) for each equivalence class.  Notice that most of the results at the beginning of this section are based on the requirement that the map fx∗ (s) = ⟨Ts z , x∗ ⟩ is in CB(S ). If we add the extra assumption that the semigroup S is such that the map fx∗ is in LUC (S ), the last two theorems are still valid if CB(S ) is replaced by LUC (S ). This is the case in particular if we assume S to be discrete. Also, some of the theorems require the assumption that for some multiplicative mean µ ∈ ∆(S ), the set Sµ is non-empty. It would be interesting to know when this set is non-empty. Theorem 8.4. Let S be a strongly left amenable semi-topological semigroup with a compact left ideal group K in ∆(S ). Let C be a non-empty compact convex subset of a Banach space E, and let S = {Ts : s ∈ S } be a strongly continuous representation of S as non-expansive mappings from C into C . 1. If S /K is a group, then Sµ is non-empty for all µ ∈ K . 2. If there exists a z ∈ C such that Tµ z = z for some µ ∈ K . Then Te z = z where e is the identity of K . Proof. 1. Let (sα ) be a net in S such that δsα → e. Then we have Te z = lim Tsα z = lim Tsα (Tµ z ) = lim Tl∗sα µ z = Te⊙µ z = Tµ z = z . α

α

α

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2. Recall that for any s ∈ S, we have l∗s K = K . Let e be the identity in S /K , and let e be any element of S in the equivalence class of e. Now for any element µ ∈ K , there exists ν ∈ K such that l∗s ν = µ. Therefore, we have l∗e l∗s ν = l∗s ν and l∗e µ = µ. Now, since this is true for any µ ∈ K , it follows that e ∈ S0 = {s ∈ S : l∗s µ = µ, ∀µ ∈ K }. Finally, we have S0 =



Sµ ,

µ∈K

and therefore e ∈ Sµ for any µ ∈ K .



Theorem 8.5. Let S be a semi-topological semigroup for which there exists a compact set K ⊆ S CB(S ) which is left invariant, that is l∗s K = K for all s ∈ S. Let C be a non-empty compact convex subset of a Banach space E, and let S = {Ts : s ∈ S } be a strongly continuous representation of S as non-expansive mappings from C into C . If z ∈ C , then the following are equivalent. 1. Ts z = z for all s ∈ S. 2. Tµ z = z for all µ ∈ K . Proof. (1) ⇒ (2) Suppose Ts z = z for all s ∈ S, and let µ be any element in K . Then we have

⟨Tµ z , x∗ ⟩ = µs ⟨Ts z , x∗ ⟩ = µs ⟨z , x∗ ⟩ = ⟨z , x∗ ⟩,

∀ x∗ ∈ E ∗ .

Therefore, it follows that Tµ z = z for all µ ∈ K . (2) ⇒ (1) Suppose Tµ z = z for all µ ∈ K . Then we have Ts z = Ts Tµ z = Tl∗s µ = Tν z = z , where ls µ = ν ∈ K . ∗

∀s ∈ S ,



Theorem 8.6. Let S be an SLA semi-topological semigroup, E be a Banach space, C be a compact subset of E, and let S = {Ts : s ∈ S } be a jointly continuous representation of S as non-expansive mappings on C . Then at least one of the following is true. 1. There exists (a unique) z ∈ C such that Ts z = z for all s ∈ S. 2. For each s ∈ S, there exists x, y ∈ C such that ‖Ts x − Ts y‖ = ‖x − y‖. Proof. Since the action of S on C is jointly continuous, by Theorem 7.1 there exists a compact set K ⊆ C such that Ts K = K for all s ∈ S. If |K | = 1, then (1) is true. Otherwise, fix s ∈ S. Since K is compact, there exists x′ , y′ ∈ K such that ‖x′ − y′ ‖ = diam(K ). Now, since Ts K = K , there is x, y ∈ K such that Ts x = x′ and Ts y = y′ . By non-expansiveness, we then have

‖x′ − y′ ‖ = ‖Ts x − Ts y‖ ≤ ‖x − y‖ ≤ ‖x′ − y′ ‖ = diam(C ). Therefore, ‖Ts x − Ts y‖ = ‖x − y‖ and (2) is satisfied.



9. Ultimately non-expansive mapping Let S be a (discrete) semigroup, we say that S is right reversible if for any a, b ∈ S we have that Sa ∩ Sb ̸= ø, that is for each a, b ∈ S there exists c , d ∈ S such that ca = db. Similarly, S is left reversible if aS ∩ bS ̸= ø, for all a, b ∈ S. In this paper, any semigroup could be considered semi-topological, as long as the semigroup is consider reversible in the discrete sense. Let S be a semigroup which acts on a Banach space X where the action is continuous in the second variable. We say that the action is ultimately non-expansive if for all u, v ∈ X and for all α > 0, there is a left ideal I ⊆ S such that

‖s · u − s · v‖ ≤ (1 + α)‖u − v‖,

∀s ∈ I .

We can show that S is ultimately non-expansive if and only if for every u, v ∈ X and every α > 0 there is an s ∈ S such that for all t ∈ S:

‖ts · u − ts · v‖ ≤ (1 + α)‖u − v‖. Notice that this definition is slightly different from the one found in [38], but in the case S is abelian, the two definitions are equivalent. If the inequality still holds for α = 0, then we call the map asymptotically non-expansive. Example 9.1. Every non-expansive or asymptotically non-expansive mapping [51] is ultimately non-expansive. Example 9.2. Let K be the closed unit disc in R2 with Euclidean norm and polar coordinates. We define the following map on K : f (r , θ ) = (r /2, θ ), g (r , θ ) = (r , 2θ mod 2π ),

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then the semigroup of continuous mapping generated by f , g is asymptotically non-expansive, and therefore ultimately non-expansive [51]. Example 9.3. Let X be a normed linear space. For x ∈ X , define f : X → X by

f (x) =

x    4     (3/2)‖x‖ − 5/4 ‖x ‖       ‖ x ‖ − 1/2   x ‖x ‖

if ‖x‖ ≤ 1 x

if 1 ≤ ‖x‖ ≤ 3/2 if ‖x‖ ≥ 3/2.

Then f is an asymptotically non-expansive map on X , and hence the semigroup generated by f is also asymptotically nonexpansive [52]. A point x ∈ X is said to be S-left-recurrent if for any ε > 0 and any left ideal I ⊆ S there exists s ∈ I such that ‖s · x − x‖ < ε. This is the case if and only if for any ϵ > 0 and any s ∈ S there is a t ∈ S such that ‖ts(x) − x‖ < ϵ . We define right-recurrent by replacing left ideal by right ideal in the definition. Once again, this definition is slightly different from the one given in [38]. Example 9.4. If G is a group acting on a Banach space X , then e · x is left-recurrent for all x ∈ X , where e is the identity of G. If the action is a group action (that is a semigroup action with the extra condition that e · x = x for all x ∈ X , where e is the identity of G), then all points of X are left-recurrent. Example 9.5. If S is a strongly left amenable semigroup acting on a compact subset X of a Banach space where the action is jointly continuous, then there is a left-recurrent point in X . To see that, we first need to extend the action of S on X to an action of ∆(S ) on X . For a fix x ∈ X , let fx : S → X , fx (s) = s · x. Next, we define f˜x : LUC (S , X ) → LUC (S ) by f˜x (g )(s) = g (fx (s)),

˜

˜

and finally we define the map: f˜ x : ∆(S ) → ∆(LUC (S , X )) = X by f˜ x (µ)(g ) = µ(f˜x (g )). We define the action of a point

µ ∈ ∆(S ) on x ∈ X , by µ · x = f˜˜ x (µ). Direct calculation shows that this map is continuous in the first variable, and extends the action from S. Now, let x ∈ X and e ∈ K be the identity of the left ideal group K in ∆(S ). Define z = e · x. Then z is a left-recurrent point in X since if s ∈ S and µ = (se)−1 then ‖µs(z ) − z ‖ = 0. Now since µ ∈ ∆(S ), there exists a net {sα } ∗ in S such that sα →w µ and therefore sα s(z ) → µs(z ) = z. This means that for every n ∈ N there is an sn ∈ S such that 1 ‖sn s(z ) − z ‖ < n , which implies that z is left-recurrent. We say that x ∈ X has the approximate identity property for the orbit if for every net sα in S such that sα x → x, then sα y → y for all y ∈ Orb(x), where Orb(x) = {s · x : s ∈ S }. That is, if for every net sα in S such that sα x → x, then sα tx → tx for all t ∈ S. This last definition is similar to property (B) found in [51]. If X is a normed linear space, then X is locally uniformly convex [53] if and only if given ϵ > 0 and an element x ∈ X with ‖x‖ = 1, there exists δ(ϵ, x) > 0 such that

‖x + y‖

≤ 1 − δ(ϵ, x) whenever ‖x − y‖ ≥ ϵ and ‖y‖ = 1. 2 If δ depends only of ϵ , then X is uniformly convex [54]. Also, X is called strictly convex, if the unit sphere of X does not contain any line segment (i.e. each point of the unit sphere of X is an extreme point of the unit ball of X ). It is known that every uniformly convex space is locally uniformly convex, and every locally uniformly convex space is strictly convex [53]. If X is a locally uniformly convex space and (xn ) is a sequence in X such that ‖xn ‖ ≤ 1 for all n ∈ N, which converge weakly to some x ∈ X such that ‖x‖ = 1 then ‖xn − x‖ → 0 (the proof is similar to the one found in [55, page 32 Theorem 4(iii)]). Also, if X , Y are normed spaces, and Y is strictly convex, then every isometry π : X → Y is affine [56]. For more information on strict convexity, local uniform convexity and uniform convexity, we refer the reader to the books of Diestel [55], Carothers [57] and Day [58]. We will do the proof of the main result using 5 lemmas. These lemmas are inspired by those found in [38]. Lemma 9.6. Let S be an ultimately non-expansive right reversible semigroup of continuous mappings of a Banach space X into itself. Let u1 , u2 , . . . , un ∈ X and s ∈ S. Then to any positive integer k there is an sk in S with the property that, for any t ∈ S and each i, j ∈ {1, 2, . . . , n}, i ̸= j,

‖tsk s(ui ) − tsk s(uj )‖ ≤ (1 + 1/k)‖ui − uj ‖,

(1)

‖tsk s(ui ) − tsk s(uj )‖ ≤ (1 + 1/k)‖s(ui ) − s(uj )‖.

(2)

i ,j

Proof. For every i, j ∈ {1, 2, . . . , n}, there exists sk ∈ S such that

  1 ‖tsik,j (ui ) − tski,j (uj )‖ ≤ 1 + ‖ui − uj ‖. k

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i,j

Now let {t1 , t2 , . . . , tm } be all permutations of the product i,j sk . Now, by right reversibility, there exists c1 , c2 , . . . , cm ∈ S such that c1 t1 = c2 t2 = · · · = cm tm . Then we define sk = c1 t1 . Therefore, we have

∏

  1 ‖tsk (ui ) − tsk (uj )‖ ≤ 1 + ‖ui − uj ‖,

∀t ∈ S and i, j ∈ {1, 2, . . . , n}.

k

Now, let s ∈ S. Then by right reversibility, there exist sequences ck and ck′ such that ck sk s = ck′ ssk for all k. Then we have

  1 ‖tck sk s(ui ) − tck sk s(uj )‖ = ‖tck′ ssk (ui ) − tck′ ssk (uj )‖ ≤ 1 + ‖ui − uj ‖. k

′

Now, if we let sk = ck sk , then we have



1

‖tsk s(ui ) − tsk s(uj )‖ ≤ 1 + ′

′

 ‖ui − uj ‖.

k

Now, by replacing all ui by s(ui ), we can construct a sequence s′′k such that



‖tsk s(ui ) − tsk s(uj )‖ ≤ 1 + ′′

′′

1



‖s(ui ) − s(uj )‖.

k

Now, using right reversibility once again, we can define dk , d′k such that dk s′k s′′k = d′k s′′k s′k , and we define sk = dk s′k s′′k . Therefore

  1 ‖tsk s(ui ) − tsk s(uj )‖ ≤ 1 + ‖ui − uj ‖, k

and



‖tsk s(ui ) − tsk s(uj )‖ ≤ 1 + for all t ∈ S.

1 k



‖s(ui ) − s(uj )‖,



Lemma 9.7. Let S be an ultimately non-expansive right reversible semigroup of continuous mappings of a Banach space X into itself. Let z1 , z2 ∈ X , s ∈ S. Suppose there are sequences tk , tk′ in S such that lim tk′ sk s(zi ) = s(zi ) and

k→∞

lim tk sk s(zi ) = zi ,

i ∈ {1, 2},

k→∞

where sk is the sequence constructed in Lemma 9.6 with n = 2 and u1 = z1 , u2 = z2 . then ‖s(z1 ) − s(z2 )‖ = ‖z1 − z2 ‖. Proof. By Lemma 9.6, there exists a sequence sk such that

‖ssk t (z1 ) − ssk t (z2 )‖ ≤ (1 + 1/k)‖z1 − z2 ‖,

(3)

‖ssk t (z1 ) − ssk t (z2 )‖ ≤ (1 + 1/k)‖t (z1 ) − t (z2 )‖,

(4)

′

for all s ∈ S. By substituting tk and tk for s in Eqs. (3) and (4) respectively, we obtain

  1 ‖s(z1 ) − s(z2 )‖ = lim ‖tk′ sk t (z1 ) − tk′ sk t (z2 )‖ ≤ 1 + ‖z1 − z2 ‖, k

k→∞

and



‖z1 − z2 ‖ = lim ‖tk sk t (z1 ) − tk sk t (z2 )‖ ≤ 1 + k→∞

Therefore, it follows that ‖s(z1 ) − s(z2 )‖ = ‖z1 − z2 ‖.

1



k

‖s(z1 ) − s(z2 )‖.



Lemma 9.8. Let X be a reflexive locally uniformly convex Banach space, and let S be an ultimately non-expansive right reversible semigroup of continuous mappings of X into itself. Suppose that p, q, z ∈ X and {sk } in S are such that z = λp + (1 − λ)q for some λ, 0 < λ < 1 and

‖ssk (p) − ssk (z )‖ ≤ (1 + 1/k)‖p − z ‖, ‖ssk (q) − ssk (z )‖ ≤ (1 + 1/k)‖q − z ‖, for all s ∈ S. Suppose further that a sequence {tk } in S exists such that {tk sk (p)} and {tk sk (q)} converge and limk→∞ tk sk (p) = p, limk→∞ tk sk (q) = q. Then {tk sk (z )} converges and lim tk sk (z ) = z .

k→∞

Proof. Replacing s by tk in both inequalities shows that tk sk (z ) is bounded since both tk sk (p) and tk sk (q) are bounded. Therefore, by reflexivity, we can apply the Eberlein–Smulian theorem [59], which gives us that there exists a subsequence

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tkj skj (z ) which is weakly convergent to some w ∈ X . Now, since norm closure is equivalent to weak closure for convex set, it follows that ‖p − w‖ ≤ ‖p − z ‖ and ‖q − w‖ ≤ ‖q − z ‖, and therefore

‖p − q‖ = ‖p − z ‖ + ‖q − z ‖ ≥ ‖p − w‖ + ‖q − w‖ ≥ ‖p − q‖. It follows that ‖p − w‖ = ‖p − z ‖ and ‖q − w‖ = ‖q − z ‖, and therefore using the strict convexity of X , z = w . Now, since this is true for any convergent subsequence, it follows that tk sk (z ) converges weakly to z. Now consider the following sequence tk sk (p) − tk sk (z ) . (1 + 1/k)‖p − z ‖ All those points are in the unit ball of X , and this sequence converges weakly to p−z

‖p − z ‖

.

Since X is locally uniformly convex, it follows that this sequence also converge in norm. Therefore, lim tk sk (p) − tk sk (z ) = p − z

and

k→∞

lim tk sk (z ) = z . 

k→∞

Lemma 9.9. Let X be a reflexive locally uniformly convex Banach space and S an ultimately non-expansive right reversible semigroup of continuous mappings of X into itself. Suppose x ∈ X is left-recurrent under S and x has the approximate identity property for the orbit. Let u1 , u2 ∈ Orb(x). Then the restriction of each member s ∈ S to the line segment [u1 , u2 ] is an affine isometry. Proof. Let z1 , z2 be points on the line segment [u1 , u2 ]. Since all isometries in a strictly convex Banach space are affine, we only need to prove that ‖s · z1 − s · z2 ‖ = ‖z1 − z2 ‖. By Lemma 9.6 there exists a sequence sk in S such that



1



‖u − v‖, ‖tsk s(u) − tsk s(v)‖ ≤ 1 + k   1 ‖s(u) − s(v)‖, ‖tsk s(u) − tsk s(v)‖ ≤ 1 + k

for all u, v in the set {u1 , u2 , z1 , z2 , s(u1 ), s(u2 ), s(z1 ), s(z2 )}. Now, since x is left-recurrent, we can construct a sequence tk such that limk→∞ tk sk s(x) = x. Now, since ui ∈ Orb(x), we have, using the approximate identity property for the orbit, limk→∞ tk sk s(ui ) = ui . Using Lemma 9.8 it follows limk→∞ tk sk s(zi ) = zi . Now, let tk′ = stk , then we have lim tk′ sk s(zi ) = lim stk sk s(zi ) = s lim tk sk s(zi ) = s(zi ),

k→∞

k→∞

k→∞

and therefore by Lemma 9.7, it follows that ‖sz1 − sz2 ‖ = ‖z1 − z2 ‖. Therefore, s is affine on the line segment [u1 , u2 ].



Lemma 9.10. Let X be a reflexive locally uniformly convex Banach space and S an ultimately non-expansive right reversible semigroup of continuous mappings of X into itself. If x ∈ X is left-recurrent under S, and x has the approximate identity property for the orbit, then the restriction of each s ∈ S to co(Orb(x)) is an affine isometry. Proof. We do the proof by induction. Consider the set C = co(u1 , u2 , . . . , un ), where each ui ∈ Orb(x). We want to show that the action of S on C is affine. If n = 2, then this is true by Lemma 9.9. Suppose it is true for any m < n. Let z1 , z2 be any two elements of C , and let p1 , p2 be the extreme points of the line segment defined by the intersection of the line passing by z1 , z2 , and the set C . Notice that p1 and p2 are on a facet of the polytope co(u1 , u2 , . . . ., un ), therefore p1 , p2 are affine combination of at most n − 1 elements of {u1 , u2 , . . . , un }. Now, let sk be a sequence such that

‖tsk s(u) − tsk s(v)‖ ≤ (1 + 1/k)‖u − v‖, ‖tsk s(u) − tsk s(v)‖ ≤ (1 + 1/k)‖s(u) − s(v)‖, for all u, v ∈ {p1 , p2 , z1 , z2 } and all t ∈ S. Now since x is left-recurrent, for each k there exists tk such that d(tk sk s(x), x) <

1 k

.

Therefore, limk→∞ tk sk s(x) = x. Now since x has the approximate identity property for the orbit, and each ui are in Orb(x), ∑n−1 ∑n−1 it follows that limk→∞ tk sk s(ui ) = ui . Since pi is the convex hull of n − 1 of the ui , say, pi = i=1 λi ui , with i=1 λi = 1, then we have

 lim tk sk s(pi ) = lim tk sk s

k→∞

k→∞

n −1 − i =1

 λi ui

=

n −1 − i =1

λi lim (tk sk sui ) = k→∞

n−1 − i=1

λi ui = pi .

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Using similar construction, we can find a sequence tk′ such that limk→∞ tk′ sk s(pi ) = s(pi ) and therefore applying Lemma 9.8 we get that limk→∞ tk′ sk s(zi ) = s(zi ), which finally give us, by Lemma 9.7, that ‖s(z1 ) − s(z2 )‖ = ‖z1 − z2 ‖. Since the same holds for any z1 , z2 ∈ co(u1 , u2 , . . . , un ), it follows that S acts on co(Orb(x)) isometrically, and since all isometries in a strictly convex Banach space are affine, the action of S on co(Orb(x)) is affine. Finally, by continuity we get the result.  In order to complete the proof of our main theorem, we need to apply the Ryll–Nardzewski fixed point theorem, which states that if X is a locally convex space, K is a weakly compact convex subset of X , and S is a noncontracting semigroup of weakly continuous affine maps of K into K , then there is a fixed point x0 in K such that s(x0 ) = x0 for every s ∈ S. Here noncontracting means that for every pair of distinct points x, y ∈ K : 0 ̸∈ {s(x) − s(y) : s ∈ S }.

Theorem 9.11. Let X be a reflexive locally uniformly convex Banach space and S an ultimately non-expansive right reversible semigroup of continuous self-maps of X . If an x ∈ X exists such that S (x) is bounded, x is a left-recurrent point under S and x has the approximate identity property for the orbit, then co(Orb(x)) contains a point ξ such that S (ξ ) = {ξ }. Proof. We know that the action of each s ∈ S on co(Orb(x)) is an affine isometry by Lemma 9.10. So first we need to show that the action is noncontracting. Let sα be a net in S, then for any x, y ∈ X , we have that ‖sα (x) − sα (y)‖ = ‖x − y‖, and therefore sα (x) − sα (y) cannot converge to 0, which means that it is noncontracting. Next, we need to show that Orb(x) is weakly compact. Since Orb(x) is bounded, co(Orb(x)) is also bounded; therefore it is contained in a closed ball, which is w∗ -compact by Alaoglu’s theorem. It follows that co(Orb(x)) is w ∗ -compact. Now since X is reflexive, the weak and weak∗ topologies coincide, which means that co(Orb(x)) is weakly compact. We can therefore apply the Ryll–Nardzewski fixed point theorem to give us a ξ ∈ co(Orb(x)) such that s(ξ ) = ξ for all s ∈ S.  Corollary 9.12. Let X be a compact subset of a reflexive locally uniformly convex Banach space and S an ultimately non-expansive right reversible strongly left amenable semigroup of continuous self-maps of X . If whenever sα x → x implies that sα tx → tx for all t ∈ S, there is a point ξ ∈ X such that S (ξ ) = {ξ }. Proof. This is a direct consequence of Theorem 9.11 and Example 9.5.



In this section we want to make use of the fact that the action of S on co(Orb(x)) as defined in the previous theorem is an isometry to prove that it is actually a bijective map on Orb(x). This is based on a similar result by Kiang (see [60]). But in order to prove our result, we will need a few more assumptions on S. Theorem 9.13. Let X be a reflexive locally uniformly convex Banach space and S an ultimately non-expansive left and right reversible semigroup of continuous self-maps of X . If an x ∈ X exists such that Orb(x) is bounded, x is a left and right-recurrent point under S and x has the approximate identity property for the orbit, then the action of S on Orb(x) is a bijective isometry. Proof. By Lemma 9.10, we know that the action of S is isometric. To show that it is injective, let y, z ∈ Orb(x) and s ∈ S be such that s(y) = s(z ). Then ‖y − z ‖ = ‖s(y) − s(z )‖ = 0, and therefore y = z. Finally, to show that it is surjective, let s ∈ S and y ∈ Orb(x). We want to find z ∈ Orb(x) such that s(z ) = y. Since y ∈ Orb(x), there exists a sequence sn in S such that

‖s n x − y ‖ <

1 n

,

∀n ∈ N.

Now, by left reversibility there are sequences {an } and {bn } in S such that ssn an = sn sbn for all n ∈ N. Also, by right-recurrence, we have a sequence {tn } in S such that

‖sbn tn x − x‖ ≤

1 n

∀n ∈ N.

It follows that

‖ssn an tn x − y‖ ≤ ‖ssn an tn x − sn x‖ + ‖sn x − y‖ = ‖sn sbn tn x − sn x‖ + ‖sn x − y‖ = ‖sbn tn x − x‖ + ‖sn x − y‖ ≤

1 n

+

1 n

=

2 n

.

It follows that {ssn an tn x} converges to y. We want to show that {sn an tn x} is also convergent.

‖sn an tn x − sm am tm x‖ = ‖ssn an tn x − ssm am tm x‖ ≤ ‖ssn an tn x − y‖ + ‖y − ssm am tm x‖ →m,n→∞ 0. Therefore, {sn an tn (x)} is a Cauchy sequence. Since X is complete, {sn an tn x} converges to some z ∈ Orb(x). Now, since {sn an tn x} → z and {ssn an tn x} → y, we conclude that s(z ) = y, which prove the surjectivity. 

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N. Bouffard / Nonlinear Analysis 74 (2011) 5815–5832

Acknowledgements This paper is part of the author’s Ph.D. thesis to be submitted at the University of Alberta, Canada, under the supervision of Professor Anthony To-Ming Lau, to whom the author could not be more grateful. The author would also like to thank Professors Peter Mah and Yong Zhang for their very helpful comments on the preliminary version of this paper. The referee is also gratefully acknowledged for his/her valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60]

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