Chapter 1

FIXED POINT AND NON-LINEAR ERGODIC THEOREMS FOR SEMIGROUPS OF NON-LINEAR MAPPINGS

Anthony To-Ming Lau^{∗}

Department of Mathematical Sciences University of Alberta Edmonton, Alberta, Canada T6G-2G1

tlau@math.ualberta.ca

Wataru Takahashi

Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ohokayama, Meguro-ku, Tokyo 152-8552, Japan

wataru@is.titech.ac.jp

1. Introduction

Let S be a semigroup, ℓ^{∞}(S) be the Banach space of bounded real valued functions on S with the supremum norm. There is a strong relation between the existence of an invariant mean (or submean) on an invariant subspace of ℓ^{∞}(S) and fixed point or ergodic properties of S when S is represented as a semigroup of nonexpansive mappings on a closed convex subset of a Banach space. It is the purpose of this chapter of the Handbook to exhibit on some recent results on such relations. Since this handbook is intended for researchers and graduate students, detailed proofs for central results, historical remarks, open problems and many references will be included. It is our hope that our effort will generate further research in this direction of non-linear analysis which depends on the ideal theory of S, and existence of an invariant mean (or submean) on a subspace of ℓ^{∞}(S) of a semigroup S.

This chapter is organized as follows: In section 3, we study the notion of non-linear “mean” (called submean) on a subspace of ℓ^{∞}(S), and its relation with left reversibility of S (i.e. S has finite intersection property with respect to closed right ideals of S). In section 4, we prove among other things that (Theorem 4.4) if ℓ^{∞}(S) has a submean,left invariant μ, and S = {T_{s} : s ∈ S} is a representation of S as nonexpansive maps on a weakly compact convex subset C of a Banach space E with more than one point, and C has normal structure, then the set {z ∈ C; μ_{t}∥T_{t}x − z∥ = inf{μ_{t}∥T_{t}x − y∥; y ∈ C}} is a proper subset of C. Theorem 4.4 is then used to obtain an improvement of a

∗This research is supported by NSERA7679.

classical fixed point theorem of Lim [?] in section 5 (Theorem 5.1) for left reversible semigroups of nonexpansive maps. In section 6, we study the relation between left ideal orbitand the fixed point set F (S) when S is arepresentation of S as nonexpansive mappingsmapping,nonexpansive on a closed convex subset C of E into C. The most important result of this section is Theorem 6.7 which shows that if the norm of E is Fréchet differentiable, then for each x ∈ C, F (S) ∩ Q(x) consists of at most one point, where Q(x) is the intersection of all left ideal orbits of x. Theorem 6.7 is crucial for some of the ergodic theorems (Theorem 7.4 and Theorem 7.10) which we study in section 7. In section 8, we shall discuss some related results.

2. Some preliminaries

All topologies in this paper are assumed to be Hausdorff. If E is a Banach space and A ⊆ E, then \overline{A} and \overline{co}A will denote the closure of A and the closed convex hull of A in E, respectively.

Given a non-empty set S, we denote by ℓ^{∞}(S) the Banach space of bounded realvalued functions on S with the supremum norm. Let S be a semigroup. Then a subspace X of ℓ^{∞}(S) is left (resp. right) translation invariant if ℓ_{a}(X) ⊆ X(resp. r_{a}(X) ⊆ X) for all a ∈ S, where (ℓ_{a}f)(s) = f(as) and (r_{a}f)(s) = f(sa), s ∈ S.

A semitopological semigroup S is a semigroup with Hausdorff topology such that for each a ∈ S, the mappings s ↦ a·s and s ↦ s·a from S into S is continuous. Examples of semitopological semigroups include all topological groups, the set M(n, ℂ) of all n×n matrices with complex entries, with matrix multiplication and the usual topology, the unit ball of ℓ^{∞} with weak^{∗}-topology and pointwise multiplication, or B(H) ( = the space of bounded linear operators on a Hilbert space H) with the weak^{∗}-topology and composition.

If S is a semitopological semigroup, we denote by CB(S) the closed subalgebra of ℓ^{∞}(S) consisting of continuous functions. Let LUC(S) (resp. RUC(S)) be the space of left (resp. right) uniformly continuous functions on S, i.e. all f ∈ CB(S) such that the mapping from S into CB(S) defined by s → ℓ_{s}f (resp. s → r_{s}f) is continuous when CB(S) has the sup norm topology. Then as is known (see [?]), LUC(S) and RUC(S) are left and right translation invariant closed subalgebras of CB(S) containing constants. Note that when S is a topological group, then LUC(S) is precisely the space of right uniformly continuous functions on S defined in [?]. Also let AP (S) (resp. W AP (S)) denote the space of almost periodic (resp. weakly almost periodic) functions f in CB(S), i.e. all f ∈ CB(S) such that {ℓ_{a}f; a ∈ S} is relatively compact in the norm (resp. weak) topology of CB(S), or equivalently {r_{a}f; a ∈ S} is relatively compact in the norm (resp. weak) topology of CB(S). Then as is known [?, p.164], AP (S) ⊆ LUC(S) ∩ RUC(S), and AP (S) ⊆ W AP (S). When S is a group, then W AP (S) ⊆ LUC(S) ∩ RUC(S) (see [?, p.167]).

Let E be a real Banach space and let E^{∗} be its topological dual. Then, the value of x^{∗} ∈ E^{∗} at x ∈ E will be denoted by ⟨x, x^{∗}⟩ or x^{∗}(x). With each x ∈ E, we associate the set

J(x) = {x^{∗} ∈ E^{∗}; ⟨x, x^{∗}⟩ = ∥x∥^{2} = ∥x^{∗}∥^{2}} .

Using the Hahn-Banach theorem, it is immediately clear that J(x) ≠ ϕ for each x ∈ E. The multivalued operator J : E → E^{∗} is called the duality mapping of E. Let S(E) = {x ∈ E; ∥x∥ = 1} be the unit sphere of E. Then the norm of E is said to be Gâteaux differentiable (and E is said to be smooth) if for each x, y ∈ S(E),

lim_{λ→0} \frac{∥x + λy∥ − ∥x∥}{λ}

exists. It is said to be Fréchet differentiable if for each x ∈ S(E), this limit is attained uniformly for y ∈ S(E). It is said to be uniformly Gâteaux differentiable if for each y ∈ S(E), this limit is attained uniformly for x ∈ S(E). It is well known that if E has a Fréchet differentiable norm, then J is single value and norm to norm continuous; see [?] or [?] for more details.

3. Submean and reversibility

Let S be a non-empty set and X be a subspace of ℓ^{∞}(S) containing constants. Then μ ∈ X^{∗} is called a mean on X if ∥μ∥ = μ(1) = 1. As is well known, μ is a mean on X if and only if

inf_{s∈S} f(s) ≤ μ(f) ≤ sup_{s∈S} f(s)

for each f ∈ X.

By a submean on X, we shall mean a real-valued function μ on X satisfying the following properties :

(1) μ(f + g) ≤ μ(f) + μ(g) for every f, g ∈ X;

(2) μ(αf) = αμ(f) for every f ∈ X and α ≥ 0;

(3) for f, g ∈ X, f ≤ q implies μ(f) ≤ μ(g);

(4) μ(c) = c for every constant function c.

If S is a semigroup, and X ⊆ ℓ^{∞}(S) is a left translation invariant subspace of ℓ^{∞}(S) containing contants, a submean μ on X is left [right] invariant if μ(ℓ_{a}f) = μ(f) [μ(r_{a}f) = μ(f)] for each a ∈ S, f ∈ X.

We abbreviate left invariant submean to LISM and left invariant mean to LIM. Depending on time and circumstances, the value of a submean (or mean) μ at f ∈ X

will also be denoted by μ(f), ⟨μ, f⟩ or μ_{t}f(t).

A semitopological semigroup S is left reversible if any two closed right ideals of S have non-void intersection.

Lemma 3.1 Let S be a semitopological semigroup and X be a left translation invariant subspace of CB(S) containing constants and which separates closed subsets of S. If X has a LISM, then S is left reversible.

Proof. Let μ be a LISM of X, I_{1} and I_{2} be disjoint nonempty closed right ideals of S. By assumption, there exists f ∈ X such that f ≡ 1 on I_{1} and f ≡ 0 on I_{2}. Now if a_{1} ∈ I_{1}, then ℓ_{a_{1}}f = 1. So,

μ(f) = μ(ℓ_{a_{1}}f) = 1.

But if a_{2} ∈ I_{2} , then ℓ_{a_{2}}f ≡ 0 . So μ(f) = μ(ℓ_{a_{2}}f) = 0, which is impossible. ■

Corollary 3.2 If S is normal and CB(S) has a LISM, then S is left reversible.

Corollary 3.3 If S is normal and CB(S) has a LISM, then AP (S) has a LIM.

Proof. This follows from Corollary 3.2 and [?, Corollary 3.3]. ■

If S is a left reversible semitopological semigroup, then (S, ⪯) is a directed system when the binary relation “⪯” on S is defined by a ⪯ b if and only if \{a\}∪\overline{aS} ⊇ \{b\}∪\overline{bS}, a, b ∈ S.

Lemma 3.4 Let S be a semitopological semigroup, J be a non-empty subset of S and f ∈ LUC(S). If sup{f(t); t ⪰ u} ≥ β for each u ∈ J, then sup{f(t); t ⪰ p} ≥ β for each p ∈ \overline{J}.

Proof. Let p ∈ \overline{J} and sup\{f(t); t ⪰ p\} ≤ β − δ, δ > 0. Then

f(ps) ≤ β − δ for each s ∈ S ∪ {e},

where xe = ex = e. Let u_{α} ∈ J be a net such that u_{α} → p. Hence ∥ℓ_{u_{α}}f − ℓ_{p}f∥ → 0. Consequently there exists α_{0} such that

f(u_{α}s) ≤ β − \frac{δ}{2} for each s ∈ S ∪ \{e\}, α ≥ α_{0}.

Hence for α ≥ α_{0}, we have sup{f(t); t ⪰ u_{α}} ≤ β − δ/2, which contradicts the assumption. ■

A semitopological semigroup S is left amenable [resp. subamenable] if LUC(S) has a LIM [resp. LISM].

Proposition 3.5 Let S be a semitopological semigroup. If S is left reversible, then S is left subamenable.

Proof. For each f ∈ CB(S), define

μ(f) = inf_{s} sup_{t⪰s} f(t).

Then μ is a submean on CB(S). Indeed, if f, g ∈ CB(S), and ϵ > 0, choose a, b ∈ S such that

sup_{t⪰a} f(t) ≤ μ(f) + ϵ and sup_{t⪰b} g(t) ≤ μ(g) + ϵ.

Let c ∈ \overline{aS} ∩ \overline{bS} (which is non-empty by left reversibility). Then c ⪰ a, and c ⪰ b. Hence sup_{t⪰c} f(t) ≤ μ(f) + ϵ and sup_{t⪰c} g(t) ≤ μ(g) + ϵ. So

sup_{t⪰c}(f(t) + g(t)) ≤ sup_{t⪰c} f(t) + sup_{t⪰c} g(t)
≤ μ(f) + μ(g) + 2ϵ.

Consequently

μ(f + g) ≤ μ(f) + μ(g) + 2ϵ.

Since ϵ > 0 is arbitrary, condition (1) for submean holds. Proofs of conditions (2), (3) and (4) are routine.

To see that μ is left invariant, let f ∈ LUC(S) and a ∈ S. Then

μ(ℓ_{a}f) = inf_{s} sup_{t⪰s} f(at)

= inf_{s} \{sup\{f(at); t ∈ \overline{sS} ∪ \{s\}\}\}

= inf_{s} { sup{f(at); t ∈ sS ∪ {s}}}

(by continuity of f and multiplication in S)

= inf_{s} { sup{f(ast); t ∈ S ∪ {e}}}