# Fixed point properties for semigroups of non-expansive mappings in conjugate Banach spaces.

1 IntroductionLet K be a non-empty subset of Banach space E.We say that amapping T : K [right arrow] K is non-expansive if

[parallel]T(x) - T(y)[parallel] [less than or equal to] [parallel]x - y[parallel], for all x, y [member of] K.

A bounded closed convex subset C [subset or equal to] E is said to have normal structure, if for all closed convex subset W of C such that [delta](W) > 0 (i.e., with positive diameter), there is x [member of] W such that :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The notion of normal structure was introduced by Brodskii and Milman (see [4]), when they studied fixed points of isometries. During the 60's, DeMarr in [7] showed that norm compact convex subsets of Banach spaces possess normal structure. This result played a crucial key on DeMarr's proof on the existence of a common fixed point for commuting families of non-expansive mappings on non-empty compact convex sets in Banach spaces. Takahashi in [29], extended DeMarr's result by considering left amenable discrete semigroups of non-expansive mappings. Mitchell in [25], improved Takahashi's result by showing that it still holds even for left reversible semigroups of non-expansive mappings (note that a left amenable discrete semigroup is always left reversible). A generalization ofMitchell's result to left reversible semi-topological semigroups was established by Lim in [22] to separated locally convex spaces, and by Lau and Holmes in [12] with some continuity assumption. A related result was proved by Hsu [13], where he showed that every weakly continuous non-expansive action of a left reversible discrete semigroup on a non-emptyweakly compact convex subset of a Banach space possesses a common fixed point. When we deal with weak topologies, one cannot in general avoid the use of normal structure for non-expansive mappings without additional assumption. Indeed, it was proved by Alspach [1] in 1981 that there is a non-expansive mapping on a non-void weakly compact convex set into itself that is fixed point free. In 1965, Kirk [16] proved a surprising result showing that a non-expansive self-map on a weakly compact convex set with normal structure has a fixed point. Hsu's result shows that if we restrict to the subclass of weakly continuous non-expansive mappings, then normal structure can be avoided. Amodification of Kirk's proof shows that his result can be extended to conjugate Banach spaces with the weak* topology. Karlovitz [15] proved that in [l.sup.1], any non-expansive mapping on a bounded weak* closed convex subset has a fixed point. Lim [23] extended this result to left reversible semi-topological semigroups of non-expansive mappings.

A semigroup S is said to be a semi-topological semigroup, if it has a Hausdorff topology such that, for all s [member of] S, the following mappings : t [??] s.t and t [??] t.s from S into itself, are continuous. A non-expansive action of S on K, is a mapping S : S x K [right arrow] K such that for all s, t [member of] S and for all x [member of] K we have :

S(st, x) = S(s,S(t, x))

and x [??] S(s, x) : K [right arrow] K is non-expansive. For short, we shall denote the value S(s, x) of the mapping S at the point (s, x), by the symbol "s.x" or sometimes by "s(x)".

Given an action of S on K, an element x [member of] K is said to be a common fixed point for S if it is subject to the condition bs(x) = x for all s [member of] S. The collection of all such x in K is called the fixed point set of S and denoted by F(S).

When S is semi-topological, let S : S x X [right arrow] X be a representation of S on a topological space X. S is said to be jointly continuous, if it is continuous whenever S x X is given the product topology. A subset C of X is called S-invariant, if s.C := S(s, C) [subset or equal to] C for all s [member of] S.

Given a semi-topological semigroup S, we shall denote by [C.sub.b](S) the Banach algebra of all bounded continuous real-valued functions on S, with the sup norm. Let F be a closed subspace of [C.sub.b](S). We say that F is left translation invariant, if it has the following property :

For all f [member of] [PHI] and s [member of] S, we have [l.sub.s] f [member of] [PHI]

where the operator [l.sub.s] : [C.sub.b](S) [right arrow] [C.sub.b](S) is defined by the formula [l.sub.s] f (t) = f (st), for all t [member of] S. The element "[l.sub.s] f " is called the left translate of f by s.Analogously, we define the right translation operator rs and the right translate [r.sub.s] f of f by [r.sub.s] f (t) := f (ts). If [PHI] is a left translation invariant subspace of [C.sub.b](S) containing the constant functions on S, a member m of [[PHI].sup.*.sub. (topological dual of [PHI]) is called a mean on [PHI], if m(e) = 1 = [parallel]m[parallel].

A mean m on [PHI] is called left invariant if it satisfies the following equation :

m([l.sub.s] f) = m(f) for all f [member of] [PHI] and for all s [member of] S.

We say that the subspace [PHI] is left amenable, if it possesses a left invariant mean. For short, we write " [PHI] has a LIM". A multiplicative mean on [PHI], is a mean m such that

m(f.g) = m(f).m(g), for all f , g [member of] [PHI].

Let LUC(S) be the subspace of [C.sub.b](S) of those functions f such that the mapping t [??] [l.sub.t] f : S [right arrow] [C.sub.b](S) is continuous when [C.sub.b](S) is given the sup norm topology. The elements of LUC(S) are called left uniformly continuous functions on S. It is well-known that LUC(S) is a translation invariant (i.e. left and right invariant) closed sub-algebra of [C.sub.b](S) and contains constants functions (see [2] or [24]). The Banach algebra LUC(S) was introduced jointly by Mitchell and Itzkowitz (see [14]). By misuse of language, we shall say that a semi-topological semigroup S is left amenable, if LUC(S) is.

Example 1. When S is a topological group, then LUC(S) is the set of all uniformly continuous functions on S with respect to the right uniformity of S i.e., f [member of] LUC(S) [??] [for all] [epsilon] > 0, [there exists] U neighborhood of the identity of S such that [s.sup.-1]t [member of] U [??] [absolute value of f (s) - f (t)] [less than or equal to] [epsilon]. See [11] for more details.

The following properties are well-known (see [2]):

* If S is a discrete semigroup, then LUC(S) = [C.sub.b](S) = [l.sup.[infinity]](S).

* If S is a compact topological semigroup (i.e., the operation of S is jointly continuous), then LUC(S) = [C.sub.b](S).

A semi-topological semigroup S is called left reversible, if any two closed right ideals intersect i.e.,

[bar.a.S] [intersection] [bar.b.S] [not equal to] [empty set], for all a, b [member of] S.

When S is discrete, then S left amenable implies S left reversible. Note that the converse is not true in general e.g., just consider a non-amenable group (free group on two generators see [5] and [6]). If we consider the topological case, a left amenable semi-topological semigroup need not be left reversible. Indeed, Hewitt[10] has constructed a regular Hausdorff topological space S such that the only continuous real-valued functions on it are constant functions; in [9], Granirer defined a semi-topological semigroup structure on S by letting a.b = a for all a, b [member of] S. Then it is easy to see that for all a [member of] S the point mass f [??] [[delta].sub.a](f) = f (a) defines a left invariant mean on [C.sub.b](S). However, S is not left reversible since, if a [not equal to] b, ({a} = [bar.a.S]) [intersection] ([bar.b.S] = {b}) = [empty set]. See also [12].

2 Main Results

In this section, we shall present our main results. We first give the following definitions which will be used in the sequel. Let S be a semi-topological semigroup.

* S is said to be sequentially left amenable, if there is a left invariant mean m [member of]LUC(S)* and a sequence [([m.sub.n]).sub.n[member of]N] of finite means such that m = weak*-[lim.sub.n] [m.sub.n]. In this case, we shall write "S is seq-LA".

The class Seq-LA of all sequentially left amenable semi-topological semigroups was introduced by the author (see [28]). It contains all countable left amenable discrete semigroups (see [28, theorem 2.3]), all compact metrizable left amenable semi-topological semigroups.

* S is said to be [sigma]-left amenable, if there is a family [([S.sub.[gamma]]).sub.[gamma][member of][GAMMA]] of sub- semi-topological semigroups subject to the following conditions:

1. S = [[union].sub.[gamma]] [S.sub.[gamma]];

2. For all [gamma], [gamma]' [member of] [GAMMA], there is [gamma]'' [member of] [GAMMA] such that [S.sub.[gamma]] [union] [S.sub.[gamma]'] [subset or equal to] [S.sub.[gamma]"] ;

3. For all [gamma] [member of] [GAMMA], [S.sub.[gamma]] is separable;

4. For all [gamma] [member of] [GAMMA], LUC([S.sub.[gamma]]) has a LIM.

The class [GAMMA]-LA of all [sigma]-left amenable semi-topological semigroups contains trivially all separable left amenable semi-topological semigroups and all amenable locally compact topological groups (due to the fact that each closed subgroup is amenable).

Example 1. Discrete left amenable semigroups are in [GAMMA]-LA. Indeed, given a discrete left amenable discrete semigroup S, we know that each countable subsemigroup of S is contained in some countable left amenable one (see [8]). Define

S := {Z [subset or equal to] S ; Z is a left amenable countable sub-semigroup}.

Note that S is non-void because if we fix s [member of] S, S contains the commutative semigroup <s> generated by s which is countable an amenable. We order S by letting Z [less than or equal to] Z' [??] Z [subset or equal to] Z' . Then it is clear that S = [[union],.sub.Z[member of]S] Z and given Z, Z' [member of] S, there is Z" [member of] S such that Z [union] Z' [subset or equal to] Z". Because Z and Z' being countable, it follows that hZ [union] Z'i is countable too, and we choose Z" [member of] S such that Z" [contains] <Z [union] Z'> using [8, theorem E1].

* S is said to be strongly left reversible, if there is a family [([S.sub.[gamma]]).sub.[gamma][member of][GAMMA]] of countable left reversible sub-semigroups of S satisfying the conditions 1 and 2 of the previous definition.

The class of all strongly left reversible semi-topological semigroups was introduced in [19] by Lau and Zhang. It includes all discrete left reversible semigroups (see [13]), all separable left reversible semi-topological semigroups and allmetrizable left reversible semi-topological semigroups see [19].

2.1 Common fixed point properties in dual spaces

Given a Banach space E, let [B.sub.E**] denote the unit closed ball of the second dual E**; and let Ext([B.sub.E**]) be the set of all extreme points of [B.sub.E**] (which is of course nonvoid by virtue of the Krein-Milman theorem). Consider on the dual E* the locally convex topology [tau] defined by the family of semi-norms Q := {[p.sub.e] ; e [member of] Ext([B.sub.E**])} where, [p.sub.e](f) = [absolute value of e(f)]; then using the Krein-Milman theorem, it is easy to see that [tau] is separated. On the other hand, by construction [tau] is weaker than the weak topology [sigma](E*, E**).

In this section, the notations [[bar.A].sup.[tau]] and [[bar.co].sup.[tau]](A) will stand respectively for the closure and closed convex hull of a subset A [subset or equal to] E* with respect to a locally convex topology [tau] on E*.

Theorem 1. Let S be a semi-topological semigroup. Assume that it satisfies either one of the following conditions :

1. S is [sigma]-LA;

2. S is seq-LA;

3. S is strongly left reversible.

Then S possesses the following fixed point property : ([[PHI].sup.*.sub.[tau]]) : Whenever S x K [right arrow] K is a weak* jointly continuous non-expansive action on a non-empty weak* compact convex subset K of a dual E* of a Banach space E, such that for all non-void weak* closed and S-invariant subset B of K with the property s.B = B for all s [member of] S, there is x [member of] B whose orbit [O.sub.x] is relatively countably [tau]-compact, then there is in K a common fixed point for S.

In order to prove this result, the following lemmas are needed :

Lemma 1. Whenever S defines a jointly continuous action on a compact topological space M, then for all x [member of] M and f [member of] C(M) the mapping [[theta].sup.f.sub.x]: S [right arrow] R, s [??] f (s.x) lies in LUC(S).

Proof. See [25, proof of theorem 1].

Lemma 2. For each non-void weak* compact and S-invariant subset K* of K, there is a minimal non-empty weak* compact set [OMEGA]* [subset or equal to] K* such that s.[OMEGA]* = [OMEGA]*, for all s [member of] S.

Proof. See [28, lemma 2.12].

Lemma 3. Let [OMEGA]* be as in the previous lemma. Then the following facts hold :

1. For all x [member of] [OMEGA]*, the orbit [O.sub.x] := {s.x ; s [member of] S} of x, is weak* dense in [OMEGA]*.

2. [OMEGA]* is [sigma](E*, E**)-compact.

For the proof of this lemma, we shall need the following well-known characterization of weak relative compactness in Banach space theory (see [27]) :

Lemma 3.1. Let B be a Banach space and C be a non-empty bounded subset of B. C is relatively weakly compact if and only if, for all sequence [([x.sub.n]).sub.n] in C, there is a sequence [([y.sub.n]).sub.n] [member of] E such that [y.sub.n] [member of] co([x.sub.i]; i [greater than or equal to] n) for all n that is weakly convergent.

Proof of Lemma 3. For part 1, clearly orbits are S-invariant; and since for all s in S, the mapping x [??] s.x is weak*-weak* continuous (due to the continuity of the action) then weak* closures of orbits are also S-invariant. Hence, byminimality, it follows that 1 holds. For part 2, from lemma 2 we have that [OMEGA]* is non-empty with the property s.[OMEGA]* = [OMEGA]* for all s [member of] S. Let us fix x [member of] [OMEGA]* with relatively [tau]- compact orbit; and let [([z.sub.n]).sub.n] be a sequence in [O.sub.x]. Since the orbit [O.sub.x] is bounded (as a subset of K), then so is [([z.sub.n]).sub.n]; therefore, by [26, corollary 0.2], there is a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all n. Then {[F.sub.n] ; n [member of] N} is a decreasing sequence of [tau]-closed non-empty subsets of the countably [tau]-compact space [[bar.O.sub.x].sup.[tau]]; therefore, [[intersection].sup.[infinity].sub.n=1] [F.sub.n] [not equal to] [empty set]; and this implies a fortiori, the existence of [xi] [member of] [[intersection].sup.[infinity].sub.n=1] [bar.co]([z.sub.i] ; i [greater than or equal to] n). Thus it follows the existence of a sequence [([[xi].sub.n]).sub.n] [member of] E* such that [[xi].sub.n] [member of] co([z.sub.i] ; i [greater than or equal to] n) for all n and [parallel][[xi].sub.n] - [xi][parallel] [right arrow] 0. Hence by lemma 3, it follows that [[bar.[O.sub.x]].sup.wk] is weakly compact; and therefore weakly* closed and togetherwith the first part,we deduce that

[OMEGA]* = [[bar.[O.sub.x]].sup.wk*] = [[bar.[O.sub.x]].sup.wk].

Hence, [OMEGA]* is a weakly compact space.

Lemma 4. Let [OMEGA]* be as in the previous lemma. If S is a separable or a seq-LA semi-topological semigroup, then [OMEGA]* is compact in the norm topology.

Proof. We first show that [OMEGA]* is separable in the norm topology. Note that on [OMEGA]* weak and weak* topologies agree (see lemma 3).

* If S is seq-LA, let us pick x [member of] [OMEGA]* and consider [([m.sub.n]).sub.n] be a sequence of finite means converging pointwise to a LIM m on LUC(S). Define [psi] : C([OMEGA]*) [right arrow] R by [psi](f) := m([f.sub.x]), with [f.sub.x](s) = f (s.x) ([psi] is well-defined by lemma 1). As readily checked, [psi] is a non-zero non-negative linear functional. By the Riesz representation theorem, there is a regular Borel measure [union] on [OMEGA]* such that [psi](f) = [[integral].sub.[OMEGA]*] f d[union] which is moreover a probability ([mu]([OMEGA]*) =1). Let [[omega].sub.*] be the support of [union]. It is easy to see that [[omega].sub.*] is characterized by :

x [member of] [[omega].sub.*] [??] [for all] V [member of] [V.sub.wk](x), [mu](V [intersection] [[omega].sub.*]) > 0.

Where [V.sub.wk](x) denotes the collection of all weak neighborhoods of x in E. For all n [member of] N, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then for n fixed, [[psi].sup.n.sub.i] : f [??] f ([s.sup.n.sub.i] .x) is a non-zero multiplicative linear functional on C([OMEGA]*); therefore there is [x.sup.n.sub.i] [member of] [OMEGA]* such that [[psi].sup.n.sub.i] (f) = f ([x.sup.n.sub.i]). We claim that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(the closure being taken with respect to the weak topology). Indeed, if y is a point outside this (weakly) closed set, then by Urysohn's lemma, there is f [member of] C([OMEGA]*) such that f [greater than or equal to] 0, f (y)=1 and f ([x.sup.n.sub.i]) = 0 for all n and i=1, ... , [[alpha].sub.n]. Then V := { f > 0} is a neighborhood of y (in the weak topology) and furthermore we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore y [not member of] [[omega].sub.*], since [mu](V [intersection] [[omega].sub.*] = 0; which proves our claim. Hence [[omega].sub.*] as a subset of a weakly separable set (therefore norm separable) is a fortiori norm separable. On the other hand, by [28], [[omega].sub.*] satisfies s.[[omega].sub.*] = [[omega].sub.*] for all s [member of] S. Thus by minimality, [OMEGA]* = [[omega].sub.*] is norm separable.

* Now we assume that S is separable. Let D [subset or equal to] S be a dense subset and fix [x.sub.o] [member of] [OMEGA]*. We shall show that the countable set {d.[x.sub.o]; d [member of] D} is dense in [OMEGA]* in the weak topology. Let y [member of] [OMEGA]* and V [member of] V *(0) (fundamental system of convex neighborhoods of "0" in the weak* topology). We choose s [member of] S such that s.[x.sub.o] [member of] y + 1/2V. Then by continuity of z [??] z.[x.sub.o], there is a neighborhood [U.sub.s] of s in S such that z [member of] [U.sub.s] [??] z.[x.sub.o] [member of] s.[x.sub.o] + 1/2V. Now let us fix d [member of] D [intersection] [U.sub.s]. Then,

d.[x.sub.o] [member of] s.[x.sub.o] + 1/2V [subset or equal to] y + 1/2V +1/2V = y + V.

Therefore we have :

(y + V) [intersection] {d.[x.sub.o] ; d [member of] D} [not equal to] [empty set] for all V [member of] V *(0).

Thus, y [member of] [[bar.{d.[x.sub.o] ; d [member of] D}].sup.wk*]. Since y is arbitrary, it follows that [OMEGA]* = [[bar.{d.[x.sub.o] ; d [member of] D}].sup.wk*] is weak* separable. But, we know from lemma 3 that [OMEGA]* is weakly compact. So weak and weak* topologies must coincide; and this fact implies the weak separability of [OMEGA]*. Next, we justify that [OMEGA]* is separable in the norm topology. Let M := [bar.span](d.[x.sub.o] ; d [member of] D) (norm closed linear manifold generated by the d.[x.sub.o]'s, d [member of] D). The subset D := spanQ(d.[x.sub.o] ; d [member of] D) (linear manifold generated over Q) is clearly countable and it is norm dense in M. Therefore M is norm separable and a fortiori [OMEGA]* (as a subspace of a separable metric space).

The second part of the proof is devoted to showing the norm compactness of [OMEGA]*. We follow an argument of Hsu [13] or in [20] for non-expansive mappings in locally convex spaces. Note that as [OMEGA]* is norm closed, showing its compactness is equivalent to proving that it is totally bounded in the norm topology. So Let [epsilon] > 0 fixed. From the norm separability showed earlier, let [OMEGA]* = [??] (norm closure of [??]; a countable subset of [OMEGA]*). Then [OMEGA]* [subset or equal to] [[union].sub.S[sigma][member of][??]] B[[sigma], [epsilon]/2]. Since each closed ball B[[sigma], [epsilon]/2] is norm closed and convex, it is weakly closed. So {B[[sigma], [epsilon]/2] [intersection] [OMEGA]*; [sigma] [member of] [??]} is a countable weakly closed covering of [OMEGA]*. But [OMEGA]* being weakly compact, it is a Baire space. Therefore there is [??] [member of] [??] such that B[ [??], [epsilon]/2] [intersection] [OMEGA]* has non-void interior in the relative weak topology. So let [x.sub.[epsilon]] [member of] [OMEGA]* and [V.sub.[epsilon]] be a weak neighborhood of the origin such that ([x.sub.[epsilon]] +[V.sub.[epsilon]]) [intersection] [OMEGA]* [subset or equal to] B[ [??], [epsilon]/2] [intersection][OMEGA]*. Then ([x.sub.[epsilon]] + [V.sub.[epsilon]]) [intersection] [OMEGA]* [subset or equal to] B[[x.sub.[epsilon]], [epsilon]]. Indeed, if z [member of] ([x.sub.[epsilon]] + [V.sub.[epsilon]]) [intersection] [OMEGA]* then, z - [x.sub.[epsilon]] [member of] [V.sub.[epsilon]]. Thus [parallel][x.sub.[epsilon]] - z[parallel] [less than or equal to] [parallel][x.sub.[epsilon]] - [??][parallel] + [parallel][??] - z[parallel] [less than or equal to] [epsilon]. Now choose a weak neighborhood [V'.sub.[epsilon]] of "0" such that [V'.sub.[epsilon]] + [V'.sub.[epsilon]] [subset or equal to] [V.sub.[epsilon]]. Then let [[delta].sub.[epsilon]] > 0 such that B[0, [[delta].sub.[epsilon]]] [subset or equal to] [V'.sub.[epsilon]] (this can be done because the norm topology is finer than the weak topology). Then we have [OMEGA]* [subset or equal to] [[union].sub.x[member of][sigma]] B[x, [[delta].sub.[epsilon]]]. As [sigma](E*, E**) = [sigma](E*, E) on [OMEGA]*, then orbits are also weakly dense in [OMEGA]*. Since [??] is countable, let [sigma] := {[[sigma].sub.i] ; i = 1, 2, ... }. Then by induction the following implications hold

For n=1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [[??].sub.1]([[sigma].sub.1]) - [x.sub.[epsilon]] [member of] [V'.sub.[epsilon]].

For n=2, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [[??].sub.1]([[sigma].sub.2]) - [x.sub.[epsilon]] [member of] [V'.sub.[epsilon]].

For n=3, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that[??]([[sigma].sub.3]) - [x.sub.[epsilon]] [member of] [V'.sub.[epsilon]].

By induction for n=p, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] there is [s.sub.p] [member of] S such that [??]([[sigma].sub.p]) - [x.sub.[epsilon]] [member of] [V'.sub.[epsilon]].

Given n [member of] N, if x [member of] [??](B[[[sigma].sub.n], [[delta].sub.[epsilon]]] [intersection] [OMEGA]*), let x := [??]([[sigma].sub.n] + [z.sub.x]) for some zx [member of] B[0, [[delta].sub.[epsilon]]]. Then by non-expansiveness of the action, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This above inequality yields the following inclusions :

[??](B[[[sigma].sub.n], [[delta].sub.[epsilon]]] [intersection] [OMEGA]*) [subset or equal to] B[[??]([[sigma].sub.n]), [[delta].sub.[epsilon]]] [intersection] [OMEGA]* [subset or equal to] [x.sub.[epsilon]] + [V'.sub.[epsilon]] + [V'.sub.[epsilon]] [subset or equal to] [x.sub.[epsilon]] + [V.sub.[epsilon]].

Hence for all n [member of] N, we have [??](B[[[sigma].sub.n], [[delta].sub.[epsilon]]] [intersection] [OMEGA]*) [subset or equal to] [x.sub.[epsilon]] + [V.sub.[epsilon]]. So we have :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then {[??]([x.sub.[epsilon]] + [V.sub.[epsilon]]) [intersection] [OMEGA]* ; i = 1, 2, ... } is a weak open covering of the weakly compact set [OMEGA]*. Therefore there is m [member of] N such that [OMEGA]* = [[union].sup.m.sub.i=1] [??] ([x.sub.[epsilon]] + [V.sub.[epsilon]]) [intersection] [OMEGA]*. From lemma 2 we know that [??]([OMEGA]*) = [OMEGA]* therefore it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By non-expansiveness, we have [parallel] [??](y) - [??]([x.sub.[epsilon]])[parallel] [less than or equal to] [epsilon], for all y [member of] B[[x.sub.[epsilon]], [epsilon]]. Thus we have the following inclusion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore [OMEGA]* is norm totally bounded. On the other hand, since it is weakly closed, it is norm complete. Hence, these two facts imply [OMEGA]* is compact in the topology induced by the norm.

Now we are ready to proceed to the proof of Theorem 1.

Proof. By a Zorn's lemma argument we fix a minimal non-void weak* compact convex subset K* of K. From lemma 2, there is a minimal non-empty, weak* compact set [OMEGA]* [subset or equal to] K* with the property that s.[OMEGA]* = [OMEGA]* for all s [member of] S.

* Step 1: S is a separable or seq-LA semi-topological semigroup. Then by lemma 4, [OMEGA]* is norm compact; and so is its closed convex hull (by Mazur's theorem). If it has a positive diameter then, by [7, lemma 1], there is u [member of] [bar.co]([OMEGA]*) such that :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As readily checked, the set [K.sub.*] := [[intersection].sub.x[member of][OMEGA]*] B[x, r] [intersection] K* is a non- void (because contains u) weak* compact (as a weak* closed subset of K*), convex (intersection of convex sets), proper (because of (*)) subset of K* which is also S-invariant. In fact, given s [member of] S if we fix x [member of] [OMEGA]* and y [member of] [K.sub.*] then, x = s.z for some z [member of] [OMEGA]* (because s.[OMEGA]* = [OMEGA]*). Therefore, [parallel]x - s.y[parallel] [less than or equal to] [parallel]z - y[parallel] [less than or equal to] r [??] s.y [member of] B[x, r] [intersection] K*. Hence s.y [member of] [K.sub.*]. Therefore by minimality of K*, [K.sub.*] = K* which is absurd since [K.sub.*] is a proper subset.

* Step 2: We assume that S is [sigma]-LA.

Let S = [[union].sub.[gamma][member of][GAMMA]] [S.sub.[gamma]] such that for all [gamma], [gamma]' [member of] [GAMMA], there is [gamma]" [member of] [GAMMA] such that [S.sub.[gamma]] [union][S.sub.[gamma]'] [subset or equal to] [S.sub.[gamma]"] where each [S.sub.[gamma]] is separable. Define

[gamma] [less than or equal to] [gamma]' if and only if [S.sub.[gamma]] [subset or equal to] [S.sub.[gamma]'] .

By the first step, for all [gamma] [member of] [GAMMA], the restriction [S.sub.[gamma]] x K [right arrow] K of the S- action on K possesses a common fixed point in K which we denote by [x.sub.[gamma]]. Since ([GAMMA],[less than or equal to]) is a directed set, then [([x.sub.[gamma]]).sub.[gamma][member of][GAMMA]] defines a net of elements of K such that s.[x.sub.[gamma]] = [x.sub.[gamma]] for all s [member of] [S.sub.[gamma]] and for all [gamma] [member of] [GAMMA]. Since K is weak* compact, there is a subnet [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is weak* convergent. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be its weak* limit. We shall show that [??] is a common fixed point for S in K. Let us fixed s [member of] S. Since S = [[union].sub.[gamma][member of][GAMMA]] [S.sub.[gamma]], let [[gamma].sub.s] [member of] [GAMMA] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Fix [t.sub.s] [member of] T such that [t.sub.s]Rt [??] [[gamma].sub.s] [less than or equal to] [[gamma].sub.t] (this implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Note that such a [t.sub.s] does exist because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a subnet of ([S.sub.[gamma]])[gamma]. On the other hand, byweak* continuity, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Since the norm of the dual is weak* lower semi-continuous, given [epsilon] > 0 there is [t.sub.[epsilon]] [member of] T such that lim [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Now let [t.sup.s.sub.[epsilon]] [member of] T such that [t.sub.[epsilon]]R[t.sup.s.sub.[epsilon]] and [t.sub.s] R[t.sup.s.sub.[epsilon]]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Therefore the following inequalities hold

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, [parallel]s. [??] - [??][parallel] [less than or equal to] [epsilon] for all [epsilon] > 0 which implies s [??] = [??] for all s [member of] S.

* Step 3 : We assume that S is strongly left reversible

Since a strongly left reversible semi-topological semigroup S is a direct union of a family of countable (left reversible) sub-semigroups [S.sub.[gamma]], if we consider the restriction of the S-action on each [S.sub.[gamma]] x K when [gamma] runs through G, and fix a minimal non-empty weak* compact convex and [S.sub.[gamma]]-invariant subset [K.sub.[gamma]] of K, then [20] guarantees the existence of a minimal non-void weak* compact subset [[OMEGA].sup.*.sub.[gamma]] [subset or equal to] [K.sup.*.sub.[gamma]] such that s.[[OMEGA].sup.*.sub.[gamma]] = [[OMEGA].sup.*.sub.[gamma]] for all s [member of] [S.sub.[gamma]]. Then using Step 1, the sub-semigroup [S.sub.[gamma]] possesses a common fixed point [x.sub.[gamma]] in K. By considering the family [([x.sub.[gamma]]).sub.[gamma]], a similar argument as in Step 2 shows that under a suitable pre-order on G, it becomes a net with a weak* convergent subnet converging to a common fixed point for S.

From Theorem 1, we derive the following result which is more easy to handle in applications.

Theorem 2. Let S be a semi-topological semigroup satisfying the conditions of Theorem 1. Then it has the following property :

([F.sup.*.sub.weak]) : Whenever SxK [right arrow] K is a weak* jointly continuous non-expansive action on a non-empty weak* compact convex subset K of a dual E* of a Banach space E, such that for all non-void weak* closed and S-invariant subset B of K with the property s.B = B for all s [member of] S, there is x [member of] B whose orbit [O.sub.x] is relatively compact in the weak topology; then there is in K a common fixed point for S. Proof. Indeed, theweak topology [sigma](E*, E**) is finer than the locally convex topology generated by Ext([B.sub.E**]).

We know that discrete left amenable semigroups and separable left amenable semi-topological semigroups have the above fixed point property. A natural question to raise at this point is the following :

Question 1 : Do left amenable semi-topological semigroups possess the fixed point property ([F.sup.*.sub.weak]) ?

Remark 1. We point out that, theorem 1 is related to the following long-standing and difficult question raised by A. T. -M. Lau in 1976 which can be stated as follows :

Does any left amenable semi-topological semigroup S possess the following fixed point property :

(F*) : Whenever S defines a jointly weak* continuous non-expansive action on a non-void, weak* compact convex set K in a dual E* of a Banach space E, then S has a common fixed in K. The converse of this question is true, just by looking at the action of the adjoints of left translation operators on the set of all means on LUC (which is a non-void weak* compact convex set). If the answer to this question is affirmative, then (F*) will be a non-linear fixed point characterization of left amenable semi-topological semigroups. While an affine characterization was established by Mitchell [24].

Remark 2. In our best knowledge up to now, the answer of this question is affirmative for commutative semigroups [3]. Unfortunately, the proof for the commutative case does not use the fact that such semigroups are left amenable, but strongly the abelian property. Sowe still do not knowfor the general case,whether the answer is positive. On the other hand, ourwork shows thatwith an additional assumption on K, it is possible to guarantee the non-emptiness of the fixed point set. For instance, that is the case if K has normal structure [20]; or norm separable [21]. These partial results show that, the answer to this question is very related to the geometrical and topological structures of K. Up to now, some partial noncommutative answers have been established (see [18],[20],[21]).

Theorem 3. Let S be a semi-topological semigroup. If LUC(S) has a LIM, then S has the following fixed point property :

([F.sup.*.sub.isom]): Whenever S x K [right arrow] K is a jointly weak* continuous non-expansive action of S on a non-empty weak* compact convex subset K of a dual Banach space E* such that there is a weak* closed isometry from K into [l.sup.1], then there is in K a common fixed point for S.

Remark 3. In the above theorem, we consider [l.sup.1] as the dual of [c.sub.0] and point out that the isometry in the above theorem need not be linear, but only closed in the weak* topology (i.e., the direct image of a weak* closed subset of K is weak* closed in [l.sup.1]).

Proof. By [28, lemma 2.12], there is a non-void weak* compact subset [[omega].sub.wk*] of K with the property :

s.[[omega].sub.wk*] = [[omega].sub.wk*] for all s [member of] S. (*).

Let [phi] : K [right arrow] [l.sup.1] be the isometry whose existence is guaranteed by assumption. Then by [49], the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a non-void and norm compact set. Therefore its preimage [[phi].sup.-1](C) is a non-empty norm compact subset of E*. Now define analogously as C the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The set [??] is non-void because on the one hand, it can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and on the other hand, each set in the intersection is non-empty and weak* closed due to theweak* lower-semi-continuity of the dual normon E*. Hence, theweak* compactness of K forces [??] to be non-void. Next we show that [??] is norm compact. For that, it is enough to prove that it is a subset of [[phi].sup.-1](C). Fortunately, it does. In fact, given x [member of] [??] we have [phi](x) [member of] [phi](K) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore the inclusion holds and it follows that [??] is a non-empty norm compact convex subset of K. We prove that [??] is S-invariant. Let x [member of] [??] and s [member of] S fixed. Given y [member of] [[omega].sub.wk*], using the property (*) we let y = s.z for some z [member of] [[omega].sub.wk*]. Then using the non-expansiveness of the action, we get :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, s.x [member of] [??] ; i.e., [??] is invariant under S. It follows that the restriction Sx [??] [right arrow] [??] of the S-action on [??] is jointly continuous in the norm topology (since norm and weak* topologies agree on [??]). Using a Zorn's lemma argument if necessary, we may assume that [??] is minimal (i.e., there is no proper subset of [??] which is nonvoid, normcompact convex and S-invariant). Then [??] must be a singleton because otherwise, being norm compact and convex, it has normal structure (see [7]) and therefore a similar argument as in Step 1 in the proof of Theorem 1 leads us to a contradiction.

The following question appears natural from this result.

Question 2 : Can we remove or weaken the weak* closeness assumption of the isometry in Theorem 3 ?

Corollary 1. Any [sigma]-LA semi-topological semigroup S possesses the fixed point property ([F.sup.*.sub.isom]).

Proof. In fact, this follows using a similar argument as in Step 2 in the proof of Theorem 1.

When S is left reversible as a semi-topological semigroup, then (see [12] or [21]), S becomes a directed set if we let :

a [less than or equal to] b iff {b} [union] [bar.b.S] [subset or equal to] {a} [union] [bar.a.S]

Then if fix x [member of] K (i.e., whenever S defines an action as in theorem 3) we define [[OMEGA].sub.s] := [bar.s.S.x] for all s [member of] S. We obtain a decreasing net of non-void subsets of K whose asymptotic center AC([([[OMEGA].sub.s]).sub.s[member of]S]; K) is non-void, norm compact, convex and S-invariant. Hence, it follows:

Corollary 2. All left reversible semi-topological semigroups possess the fixed point property ([F.sup.*.sub.isom]).

2.2 A non-linear common fixed point property in Banach spaces

Using Theorem 1, we derive the following result which is a dual version of the fixed point property [F.sup.*.sub.[tau]].

Theorem 4. Let S be a semi-topological semigroup. Assume that it satisfies either one of the following conditions :

1. S is [sigma]-LA;

2. S is seq-LA;

3. S is strongly left reversible.

Then S possesses the following fixed point property :

([F.sub.weak]) : Whenever S x K [right arrow] K is a jointly weakly continuous non-expansive action of S on a non-empty weakly compact convex subset K of a Banach space E, then there is in K a common fixed point for S.

Proof.

* Step 1 : We first assume that S is separable or seq-LA.We embed E in its second dual E** through the canonical injection j : E [right arrow] E** which is an isomorphism from(E,wk) onto (j(E),wk*). Then [??] := j(K) is a non-voidweak* compact convex subset of E**. We carry the S-action on K to [??] by letting s * j(x) := j(s.x), for all s [member of] S and x [member of] K. As readily checked, the action S x [??] [right arrow] [??] is jointly weak* continuous and norm non-expansive. Let [??] be the locally convex topology on E** induced by the extreme points of [B.sub.E**]*[0,1] (the unit closed ball of the dual of E**). If [??] [subset or equal to] [??] is a non-void weak* compact subset such that s * [??] = [??] for all s [member of] S, using a Zorn's lemma argument if necessary together with [28, lemma 2.12] if S is left amenable or [20, corollary 3.7] if S is left reversible, wemay assume that [??] is minimal (in the sense that, if B is a non-void weak* compact S-invariant set contained in [??], then B = [??]). Then B := j-1([??]) is a minimal non-empty weakly compact S-invariant and separable subset of K with the property that s.B = B for all s [member of] S. Therefore using lemma 4, it follows that B is norm compact and a fortiori its image [??]. Thus, for all j(x) [member of] [??], the orbit [O.sub.j(x)] is relatively [??]-compact (since norm and [??] topologies agree on the norm closed orbit). Hence by Theorem 1, there is [??] [member of] K such that s * j([??]) = j([??]) for all s [member of] S. Hence, [??] is a common fixed point for S due to the injectivity of j.

* Step 2 : Now we assume that S is an arbitrary semi-topological semigroup with either one of the properties in the theorem. From Step 1 , a similar argument as in the proof of theorem 1, shows that if we consider the action of S carried on E**, then we have

F(S) := {x [member of] K; s * j(x) = j(x) for all s [member of] S} [not equal to] [empty set].

Hence using the injectivity of j, the non-empty set F(S) is contained in the fixed point set of S; which proves the existence of a common fixed point.

Remark 3. Theorem 4 extends Hsu [13, theorem 4], because discrete left reversible semigroups are strongly left reversible (nice proof due to Hsu [13, lemma 1]) and Mitchell [26]. On the other hand, it shows that joint weak continuity condition is a sufficient condition for avoiding the use of normal structure when dealing with non-expansive actions on weak compact convex sets.

Discrete left reversible semigroups and separable left reversible semi-topological semigroups possess the previous fixed point property. However, we may ask :

Question 3 : Do left reversible semi-topological semigroups possess the fixed point property ([F.sub.weak]) ?

Since a left amenable semi-topological semigroup S need not be [sigma]-LA, the following question is natural :

Question 4 : Do left amenable semi-topological semigroups have the fixed point property ([F.sub.weak])?

Acknowledgment This paper is partly contained in the PhD thesis of the author under the supervision of Professor Anthony To-Ming Lau.

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Received by the editors in May 2016.

Communicated by F. Bastin.

2010Mathematics Subject Classification : 47H10, 47H20.

Department of Mathematical and Statistical Sciences

University of Alberta, Edmonton, Alberta T6G 2G1, Canada

E-mail address : khadime@ualberta.ca

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Author: | Salame, Khadime |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Article Type: | Report |

Date: | Oct 1, 2016 |

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