Normal Structure and Common Fixed Point Properties for Semigroups of Nonexpansive Mappings in Banach Spaces

In 1965, Kirk proved that if is a nonempty weakly compact convex subset of a Banach space with normal structure, then every nonexpansive mapping has a fixed point. The purpose of this paper is to outline various generalizations of Kirk's fixed point theorem to semigroup of nonexpansive mappings and for Banach spaces associated to a locally compact group.

A closed convex subset of a Banach space has normal structure if for each bounded closed convex subset of which contains more than one point, there is a point which is not a diametral point of that is, where the diameter of

The set is said to have fixed point property (FPP) if every nonexpansive mapping has a fixed point. In [1], Kirk proved the following important celebrated result.

Theorem 1.1 (Kirk [1]).

Let be a Banach space, and a nonempty closed convex subset of If is weakly compact and has normal structure, then has the FPP.

As well known, compact convex subset of a Banach space always has normal structure (see [2]). It was an open problem for over 15 years whether every weakly compact convex subset of has normal structure. This problem was answered negatively by Alspach [3] when he showed that there is a weakly compact convex subset of which does not have the fixed point property. In particular, cannot have normal structure.

It is the purpose of this paper to outline the relation of normal structure and fixed point property for semigroup of nonexpansive mappings. This paper is organized as follows. In Section 3, we will focus on generalizations of Kirk's fixed point theorem to semigroups of nonexpansive mappings. In Section 4, we will discuss about fixed point properties and normal structure on Banach spaces associated to a locally compact group.

All topologies in this paper are assumed to be Hausdorff. If is a Banach space and then and will denote the closure of and the closed convex hull of in respectively.

Let be a Banach space and let a subset of A mapping from into itself is said to be nonexpansive if for each A Banach space is said to be uniformly convex if for each there exists such that for each satisfying and

Let be a semigroup, the Banach space of bounded real valued functions on with the supremum norm. Then a subspace of is left (resp., right) translation invariant if (resp., for all where and ,

A semitoplogical semigroup is a semigroup with Hausdorff topology such that for each the mappings and from into are continuous. Examples of semitopological semigroups include all topological groups, the set of all matrices with complex entries, matrix multiplication, and the usual topology, the unit ball of with weak-topology and pointwise multiplication, or the space of bounded linear operators on a Hilbert space with the weak-topology and composition.

If is a semitopological semigroup, we denote the closed subalgebra of consisting of continuous functions. Let (resp., be the space of left (resp., right) uniformly continuous functions on that is, all such that the mapping from into defined by (resp., is continuous when has the sup norm topology. Then as is known (see [4]), and are left and right translation invariant closed subalgebras of containing constants. Also let (resp., denote the space of almost periodic (resp., weakly almost periodic) functions in that is, all such that is relatively compact in the norm (resp., weak) topology of or equivalently is relatively compact in the norm (resp., weak) topology of Then as is known [4, page 164], and When is a locally compact group, then (see [4, page 167]).

A semitopological semigroup is left reversible if any two closed right ideals of have nonvoid intersection.

The class of all left reversible semitopological semigroups includes trivially all semitopological semigroups which are algebraically groups, and all commuting semigroups.

The class is closed under the following operations.

(a)If and is a continuous homomorphic image of then

(b)Let , and be the semitopological semigroup consisting of the set of all functions on such that , the binary operation defined by for all and and the product topology. Then

(c)Let be a semitopological semigroup and , semitopological sub-semigroups of with the property that and, if , then there exists such that If for each then

Let be a nonempty set and a translation invariant subspace of containing constants. Then is called a mean on if As well known, is a mean on if and only if

for each

Also is called a left (resp., right) invariant mean if (resp., for all

Lemma 2.1.

Let be a semitopological semigroup and a left translation invariant subspace of containing constants and which separates closed subsets of If has a left invariant mean, then is left reversible.

Proof.

Let be a left invariant mean of , and disjoint nonempty closed right ideals of By assumption, there exists such that on and on Now if then So,

But if then So which is impossible.

Corollary 2.2.

If is normal and has a left invariant mean, then is left reversible.

See [5] for details.

A discrete semigroup is called left amenable [6] if has a left invariant mean. In particular every left amenable discrete semigroup is left reversible by Corollary 2.2. The semigroup is amenable if it is both left and right amenable. In this case, there is always an invariant mean on

Remark 2.3.

Lemma 2.1 is not true without normality. Let be a topological space which is regular and Hausdorff and consists of constant functions only [7]. Define on the multiplication for all Let be fixed. Define for all Then is a left invariant mean on but is not left reversible.

By a (nonlinear) submean on we will mean a real-valued function on satisfying the following properties:

(1) for every

(2) for every and

(3)for implies

(4) for every constant function

Clearly every mean is a submean. See [8] for details.

If is a semigroup and is left translation invariant, a submean on is left subinvariant if for each and

Let be a semitopological semigroup, a nonempty subset of a Banach space then a representation of as mappings from into is continuous if defined by is continuous when has the product topology. It is called separately continuous if for each and the maps from into and the map from into are continuous.

Theorem 3.1.

Let be a semitopological semigroup, let a nonempty weakly compact convex subset of a Banach space which has normal structure and let a continuous representation of as nonexpansive self-mappings on Suppose that has a left subinvariant submean. Then has a common fixed point in

Corollary 3.2.

Let be a left reversible semitopological semigroup. Let be a nonempty weakly compact convex subset of a Banach space which has normal structure and let a continuous representation of as nonexpansive self-mappings on Then has a fixed point in

Proof.

If is left reversible, define Then the proof of Lemma in [9] shows that is a submean on such that for all and that is, is left subinvariant.

Note that since every compact convex set has normal structure, Corollary 3.2 implies the following.

Corollary 3.3 (DeMarr [10]).

Let be a Banach space and a nonempty compact convex subset of If is a commuting family of nonexpansive mappings of into then the family has a common fixed point in

Remark 3.4.

Theorem 3.1 is proved by Lau and Takahashi in [11]. Mitchell [12] generalized the theorems of DeMarr [10, page 1139] and Takahaski [13, page 384] by showing that if is a nonempty compact convex subset of a Banach space and is a left-reversible discrete semigroup of nonexpansive mappings from into then contains a common fixed point for Belluce and Kirk [14] also improved DeMarr's result in [10] and proved that if is a nonempty weakly compact convex subset of a Banach space and if has complete normal structure, then every family of commuting nonexpansive self-maps on has a common fixed point.

This result was extended to the class of left reversible semitopological semigroup by Holmes and Lau in [15]. Corollary 3.2 is due to Lim [16] who showed that normal structure and complete normal structure are equivalent.

The following related theorem was also established in [15].

Theorem 3.5.

Let be a left reversible semitopological semigroup, let a nonempty, bounded, closed convex subset of a Banach space and let a separately continuous representation of as nonexpansive self-maps on If there is a nonempty compact subset and such that commutes with all elements of and for each the closure of the set contains a point of then contains a common fixed point of

Let be a semitopological semigroup and is a nonempty subset of a Banach space and a separately continuous representation of as mappings from into We say that the representation is asymptotically nonexpansive if for each there is a left ideal such that for all

We also say that the representation has property (B) if for each whenever a net converges to then the net also converges to for each

Clearly condition (B) is automatically satisfied when is commutative.

The semitopological semigroup is right reversible if

The following theorem is proved in [17].

Theorem 3.6.

Let be a nonempty compact convex subset of a Banach space and a right reversible semitopological semigroup. If is a separately continuous asymptotically nonexpansive representation of as mappings from into with property (B), then contains a common fixed point for

The following example from [17] shows a simple situation where our fixed point theorem applies, but DeMarr's fixed point theorem does not.

Let be the closed unit disc in with polar coordinates and the usual Euclidean norm. Define continuous mappings from into by

Then the semigroup of continuous mappings from to generated by and under usual composition is commutative and asymptotically nonexpansive. However, the action of (or any ideal of on is not nonexpansive.

Open Problem 1.

Can right reversibility of and property (B) in Theorem 3.6 be replaced by amenability of

Let be a nonempty closed convex subset of a Banach space Then has the fixed point property for nonexpansive mappings if every nonexpansive mapping has a fixed point; has the onlyconditional fixed point property for nonexpansive mappings if every nonexpansive mapping satisfies either has no fixed point in or has a fixed point in every nonempty bounded closed convex -invariant subset of For commuting family of nonexpansive mappings, the following is a remarkable common fixed point property due to Bruck [18].

Theorem 3.7.

Let be a Banach space and a nonempty closed convex subset of If has both the fixed point property and the conditional fixed point property for nonexpansive mappings, then for any commuting family of nonexpansive mappings of into there is a common fixed point for

Theorem 3.7 was proved by Belluce and Kirk [19] when is finite and is weakly compact and has normal structure, by Belluce and Kirk [14] when is weakly compact and has complete normal structure, Browder [20] when is uniformly convex and is bounded, Lau and Holmes [15] when is left reversible and is compact, and finally by Lim [16] when is left reversible and is weakly compact and has normal structure.

Open Problem 2 (Bruck [18]).

Can commutativity of be replaced by left reversibility?

The answer to Problem 2 is not known even when the semigroup is left amenable.

Let be a compact right topological semigroup, that is, a semigroup and a compact Hausdorff topological space such that for each the mapping from into is continuous. In this case, must contain minimal left ideals. Any minimal left ideal in is closed and any two minimal left ideals of are homeomorphic and algebraically isomorphic.

Let be a nonempty weakly compact convex subset of a Banach space Let be a representation of a semigroup as nonexpansive and weak-weak continuous mappings from into Let be the closure of in the product space weak) Then is a compact right topological semigroup consisting of nonexpansive mappings from into Further, for any there exists a sequence of convex combination of operators from such that for every See [21] for details.

is called the enveloping semigroup of

Theorem 3.8.

Let be a nonempty weakly compact convex subset of a Banach space, and has normal structure. Let be a representation of a semigroup as norm nonexpansive and weakly continuous mappings from into and let be the enveloping semigroup of Let be a minimal left ideal of and let a minimal -invariant closed convex subset of Then there exists a nonempty weakly closed subset of such that is constant on

Corollary 3.9.

Let and as in Theorem 3.8. Then there exist and such that for every

Proof.

Pick and of the above theorem.

Remark 3.10.

If is commutative, then for any and that is, is in fact a common fixed point for (and, hence, for Note that if is norm compact, the weak and norm topology agree on Hence every nonexpansive mapping from into must be weakly continuous. Therefore, Corollary 3.9 improves the fixed point theorem of DeMarr [10] for commuting semigroups of nonexpansive mappings on compact convex sets.

The above theorem proved in [21] provides a new approach using enveloping semigroups in the study of common fixed point of a semigroup of nonexpansive mappings on a weakly compact convex subset of a Banach space.

Open Problem 3.

Can the above technique applied to give a proof of Lim's fixed point theorem for left reversible semigroup in [16].

The following generalization of DeMarr's fixed point theorem was proved in [22].

Theorem 3.11.

Let a be semitopological semigroup.

If has a left invariant mean, then has the following fixed point property. Whenever is a separately continuous representation of as nonexpansive self-mappings on a compact convex subset of a Banach space, then contains a common fixed point for

Quite recently the Lau and Zhang [23] are able to establish the following related fixed point property.

Theorem 3.12.

Let be a separable semitopological semigroup. If has a left invariant mean, then has the following fixed point property:

Whenever is a continuous representation of as nonexpansive self-mappings on a weakly compact convex subset of a Banach space such that the closure of in with the product of weak topology consists entirely of continuous functions, then contains a common fixed point of

Remark 3.13.

(a) The converse of Theorem 3.12 also holds when has an identity by considering the semigroup of right translations, on the weakly compact convex sets for each (see [24]).

(b) When is a discrete semigroup, the following implication diagram is known: The implication " is left reversible has a " for any semitopological semigroup was established in [22]. During the 1984 Richmond, Virginia conference on analysis on semigroups, T. Mitchell [12] gave two examples to show that for discrete semigroups " has " " is left reversible" (see [25] or [23]). The implication " is left reversible has " for discrete semigroups was proved by Hsu [26]. Recently, it is shown in [23] that if is the bicyclic semigroup generated by such that is the unit of and and then has a but is not left reversible. Also if is the bicyclic semigroup generated by where is the unit element and then has a but does not have a

The following is proved in [5] (see also [27]).

Theorem 3.14.

Let be a left reversible discrete semigroup. Then has the following fixed point property.

Whenever is a representation of as norm nonexpansive weak-weak continuous mappings of a norm-separable weak-compact convex subset of a dual Banach space into then contains a common fixed point for

It can be shown that the following fixed point property on a discrete semigroup implies that is left amenable.

(G) Whenever is a representation of as norm nonexpansive weak-weak continuous mappings of a weak-compact convex subset of a dual Banach space into then contains a common fixed point for

Open Problem 4.

Does left amenability of imply

Other related results for this section can also be found in [9, 28–38].

A Banach space has weak-normal structure if every nontrivial weakly compact convex subset has normal structure. If the Banach space is also a dual space then it has weak-normal structure if every nontrivial weak compact convex subset has normal structure. It is clear that a dual Banach space has weak-normal structure whenever it has weak-normal structure.

A [dual] Banach space is said to have the weak-fixed point property (weak-FPP) [(FPP if for every weakly [weak compact convex subset of and for every nonexpansive has a fixed point in Kirk proved that if has weak-normal structure then has property FPP [1]. Subsequently, Lim [39] proved that a dual Banach space has property FPP whenever it has weak-normal structure.

A Banach space is said to have the Kadec-Klee property (KK) if whenever is a sequence in the unit ball of that converges weakly to and where

then (see [40]).

For dual Banach spaces, we have the similar properties replacing weak converges by weak converges.

A Banach space is said to have the uniformly Kadec-Klee property (UKK) if for every there is a such that whenever is a sequence in the unit ball of converging weakly to and then This property was introduced by Huff [40] who showed that property UKK is strictly stronger than property KK. van Dulst and Sims showed that a Banach space with property UKK has property weak FPP [41].

It is natural to define a property similar to UKK by replacing the weak convergence by weak convergence in UKK and calling it UKK However, van Dulst and Sims found that the following definition is more useful.

A dual Banach space has property UKK if for every there is a such that whenever is a subset of the closed unit ball of containing a sequence with then there is an in weak-closure such that

They proved that a dual Banach space with property UKK has property FPP [41]. Moreover, they observed that if the dual unit ball is weak sequentially compact then property UKK as defined above, is equivalent to the condition obtained from UKK by replacing weak convergence by weak convergence.

We now summarize the various properties defined above by

(where n.s. = normal structure).

Let Then, as noted by Huff [40], is reflexive and has property KK but not UKK.

Let be a locally compact Hausdorff space, and the space of bounded continuous complex-valued functions defined on with the supremum norm. Let be the subspace of consisting of functions "vanishing at infinity," and be the space of bounded regular Borel measure on with the variation norm. Let be the subspace of consisting of the discrete measures on It is well known that the dual of can be identified with and that is isometrically isomorphic to

Lennard [42] proved the following theorem.

Theorem 4.1.

Let be a Hilbert space. Then the trace class operators on has the property UKK and has FPP when regarded as the dual space of the -algebra of compact operator on

Theorem 4.2.

Let be a locally compact group. Then the following statements are equivalent.

(1) is discrete.

(2) is isometrically isomorphic to

(3) has property UKK.

(4) has property KK.

(5)Weak convergence and weak convergence of sequences agree on the unit sphere of

(6) has weak normal structure.

(7) has property FPP.

Theorem 4.3.

Let be a locally compact group. Then the group algebra has the weak fixed point property for left reversible semigroups if and only if is discrete.

Theorem 4.4.

Let be a locally compact group. Let be a -subalgebra of containing and the constants. Then the following statements are equivalent.

(1) is finite.

(2) has property UKK

(3) has property KK

(4)Weak convergence and weak convergence for sequences agree on the unit sphere of

(5) has weak-normal structure.

Theorem 4.5.

Let be a locally compact group. Then

(1)Weak convergence and weak convergence for sequences agree on the unit sphere of if and only if is discrete.

(2) has weak-normal structure if and only if is finite.

Let be a locally compact group. We define the group -algebra of to be the completion of with respect to the norm

where the supremum is taken over all nondegenerate representations of as an algebra of bounded operator on a Hilbert space. Let be the Banach space of bounded continuous complex-valued function on with the supremum norm. Denote the set of continuous positive definite functions on by and the set of continuous functions on with compact support by Define the Fourier-Stieltjes algebra of denoted by to be the linear span of The Fourier algebra of denoted by is defined to be the closed linear span of Finally, let be the left regular representation of that is, for each is the bounded operator in defined on by (the convolution of and Then denote by to be the closure of in the weak operator topology in It is known that and Furthermore, if is amenable (e.g., when is compact), then

We refer the reader to [43] for more details on these spaces.

Notice that when is an abelian locally compact group, then and where is the dual group of It follows from Theorem 4.2 that was the weak-normal structure if and only if is discrete, or equivalently, is compact.

Theorem 4.6.

If is compact, then has weak-normal structure and hence the FPP.

For a Banach space (resp., dual Banach space) we say that has the weak-FPP (weak-FPP) for left reversible semigroup if whenever is a left reversible semitopological semigroup and is a weak (resp., weak compact convex subset of and is a separately continuous representation of as nonexpansive mappings from into then there is a common fixed point in for

Theorem 4.7.

If is a separable compact group, then has the weak- FPP for left reversible semigroups.

Open Problem 5.

Can separability be dropped from Theorem 4.7?

A locally compact group is called an [IN]-group if there is a compact neighbourhood of the identity in which is invariant under the inner automorphisms. The class of [IN]-group contains all discrete groups, abelian groups and compact groups. Every [IN]-group is unimodular.

We now investigate the weak fixed point property for a semigroup. A group is said to an [AU]-group if the von Neumann algebra generated by every continuous unitary representation of is atomic (i.e., every nonzero projection in the van Neumann algebra majorizes a nonzero minimal projection). It is an [AR]-group if the von Neumann algebra is atomic. Since is the von Neumann algebra generated by the regular representation, it is clear that every [AU]-group is an [AR]-group. It was shown in [44, Lemma ] that if the predual of a von Neumann algebra has the Radon-Nikodym property, then has the weak fixed point property. In fact, since the property UKK is hereditary, the proof there actually showed that has property UKK and hence has weak normal structure. For the two preduals and we know from [45, Theorems and ] that the class of groups for which and have the Radon-Nikodym property is precisely the [AR]-groups and [AU]-groups, respectively. Thus by Lim's result [16, Theorem ] we have the following proposition

Proposition 4.8.

Let be a locally compact group.

(a)If is an [AR]-group, then has the weak fixed point property for left reversible semigroups.

(b)If is an [AU]-group, then has the weak fixed point property for left reversible semigroups.

Proposition 4.9.

Let be an [IN]-group. Then the following are equivalent.

(a)  is compact.

(b)  has property UKK.

(c)  has weak normal structure.

(d)  has the weak fixed point property for left reversible semigroups.

(e)  has the weak fixed point property.

(f)  has the Radon-Nikodym property.

(g)  has the Krein-Milman property.

A Banach space is said to have the fixed point property (FPP) if every bounded closed convex subset of has the fixed point property for nonexpansive mapping. As well known, every uniformly convex space has the FPP.

Theorem 4.10.

Let be a locally compact group. Then has the FPP if and only if is finite.

Remark 4.11.

(a) Theorems 4.1, 4.2, 4.4, 4.5 and 4.6 are proved by Lau and Mah in [46]; Theorems 4.3, 4.7, and Propositions 4.8 and 4.9 are proved by Lau and Mah in [47] and by Lau and Leinert in [48].

(b) Upon the completion of this paper, the author received a preprint from Professor Narcisse Randrianantoanina [49], where he answered an old question in [50] (see also [23]) and showed that for any Hilbert space (separable or not) the trace class opertors on has the weak-FPP for left reversible semigroups. He is also able to remove the separability condition in our Theorem 4.7, and show that for anylocally compact group G:

(i) has the weak FPP if and only if is an [AR]-group;

(ii) has the weak-FPP if and only if is an [AU]-group. In this case, even has the weak-FPP for left reversible semigroup.

We are grateful to Professor Randrianantoanina for sending us a copy of his work.

(c) An example of an [AU]-group which is not compact is the Fell group which is the semidirect product of the additive -adic number field and the multiplicative compact group of -adic units for a fixed prime So is solvable and hence amenable. We claim that cannot have property KK Indeed, the Fell group is separable. Hence is norm separable (see [29]). So the proof of [51] shows that there is a bounded approximate identity in consisting of a sequence positive definite with norm 1. The sequence converges to in in the weak-topology. Now if has property KK then and so In particular is compact. See [52] for a more general result.

(d) Theorem 4.10 is proved by Lau and Leinert in [48]. In a preprint of Hernandez Linares and Japon [53] sent to the author just recently, they have shown that if is compact and separable, then can be renormed to have the FPP. This generalizes an earlier result of Lin [54] who proves that can be renormed to have the FPP. Note that if the circle group, then is isometric isomorphic to We are grateful to Professor Japon for providing us with a preprint of their work.

(e) Other related results for this section can also be found in [55].

This research is supported by NSERC Grant A-7679 and is dedicated to Professor William A. Kirk with admiration and respect.

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