An ultraproduct method via left reversible semigroups to study Bruck's generalized conjecture

TL;DR

This paper introduces an ultraproduct-inspired method to analyze fixed points of left reversible semigroups acting on Banach spaces, leading to a new fixed point theorem for nearly uniformly convex spaces.

Contribution

It develops a novel ultraproduct-like approach for fixed point analysis and proves a Bruck-type theorem for nearly uniformly convex Banach spaces.

Findings

Nearly uniformly convex Banach spaces have the weak fixed point property for left reversible semigroups.

The method extends fixed point results to broader classes of Banach spaces.

Provides a new tool for studying fixed points in semigroup actions.

Abstract

We use a method similar to ultraproducts to study the common fixed point of a left reversible semitopological semigroup acting on a Banach space. As an application, we prove a Bruck-type theorem for nearly uniformly convex Banach spaces to the effect that such spaces have weak fixed point property for left reversible semigroups.

Equations

$$r(K,(x_{\alpha }))=\inf \{\lim \underset{\alpha }{\sup }\parallel x_{\alpha }-k\parallel :k\in K\}$$

$$AC(K,(x_{\alpha }))=\{k\in K:\lim \underset{\alpha }{\sup }\parallel x_{\alpha }-k\parallel =r(K,(x_{\alpha }))\}$$

$$r(\dot{K},[(x_{\alpha })])=\inf \{\parallel [(x_{\alpha })]-\dot{k}\parallel k\in K\},$$

$$AC(\dot{K},[(x_{\alpha })])=\{\dot{k}\in \dot{K}:\parallel [(x_{\alpha })]-\dot{k}\parallel =r(\dot{K},[(x_{\alpha })])\}$$

Full text

1. Introduction

Let $K$ be a subset of a Banach space $E$. A self mapping $T$ on $K$ is said to be non-expansive if $\|T(x)-T(y)\|\leq\|x-y\|$ for all $x,y\in K$. We say that $E$ has the weak fixed point property (weak fpp) if for every weakly compact convex non-empty subset $K$ of $E$, any non-expansive self mapping on $K$ has a fixed point.

Let $S$ be a semi-topological semigroup, i.e., $S$ is a semigroup with a Hausdorff topology such that for each $a\in S$, the mappings $s\mapsto sa$ and $s\mapsto as$ from $S$ into $S$ are continuous. $S$ is called left reversible if any two closed right ideals of $S$ have non-void intersection.

An action of $S$ on a subset $K$ of a topological space $E$ is a mapping $(s,x)\mapsto s(x)$ from $S\times K$ into $K$ such that $(st)(x)=s(t(x))$ for $s,t\in S,x\in K$. The action is separately continuous if it is continuous in each variable when the other is kept fixed. We say that $S$ has a common fixed point in $K$ if there exists a point $x$ in $K$ such that $sx=x$ for all $s\in S$. When $E$ is a normed space, the action of $S$ on $K$ is non-expansive if $\|s(x)-s(y)\|\leq\|x-y\|$ for all $s\in S$ and $x,y\in K$. There are also other types of action for a semi-topological semigroup (see [1] and [5]).

We say that a Banach space $E$ has the weak fpp for left reversible semigroups if for every weakly compact convex non-empty subset $K$ of $E$, any non-expansively separately continuous action of a semi-topological semigroup $S$ on $K$ has a fixed point.

One of the celebrated results in fixed point theory is due to Bruck [2]. He has shown that if a Banach space $E$ has weak fpp, then it has weak fpp for abelian semigroups. Now, we call the following statement Bruck’s Generalized Conjecture (BGC):

(BGC) If a Banach space $E$ has weak fpp, then it has weak fpp for any left reversible semi-topological semigroup $S$.

In [6], it has been shown that BGC is true for the preduals of von Neumann algebras. The main results of the current work is to show BGC is also true for nearly uniformly convex Banach spaces.

2. Weak fixed point property of Bruck type

A Banach space $E$ is called nearly uniformly convex (NUC), if for each $\varepsilon>0$ there exists a $\delta>0$ such that for every sequence $(x_{n})$ in the closed unit ball of $E$ with $sep(x_{n}):=\inf\{\rVert\ {x_{n}}-{x_{m}}\rVert:n\neq m\}>\epsilon$ the distance $dist(0,co\{x_{n}\})$ is strictly less than $1-\delta$ where $co\{x_{n}\}$ denotes the convex hull of the sequence.

When $S$ is a left reversible semigroup, we make it to a directed set by declaring: $\alpha\geq\beta$ if and only if $\alpha S\subseteq\overline{\beta S}$. Thus, we can use $S$ as an index set for nets and speak about limit and limit-supremum with respect to this net. Also, note that when $S$ acts on a weakly compact subset $K$ of Banach space $E$, each $\alpha$ acts non-expansively on $K$ and sometimes we use the notation $T_{\alpha}$ instead of $\alpha$; even, we may use $\alpha.\beta$ to denote the composition $T_{\alpha}\circ T_{\beta}$.

Let $l_{\infty}(E)=\{x=(x_{\alpha}):x_{\alpha}\in E;\quad\rVert x\rVert=\sup\rVert x_{\alpha}\rVert<\infty\}$, and $\mathcal{N}=\{x=(x_{\alpha}):x_{\alpha}\in E;\quad\lim_{\alpha}\rVert x_{\alpha}\rVert=0\}$. Put $\tilde{E}=l_{\infty}(E)/\mathcal{N}$ and endow it with the quotient norm $\rVert[(x_{\alpha})]\rVert=\lim\sup_{\alpha}\rVert x_{\alpha}\rVert$. One can embed $E$ and its subsets into $\tilde{E}$ by using constant classes. For example, for $x\in K$ let $\dot{x}=[(x)]$ denotes the equivalence class containing the constant net $(...,x,x,x,...)$. So, $\dot{K}=\{\dot{x}:x\in K\}$ is a subset of $\tilde{E}$. Also, we define $\tilde{K}=\{[(k_{\alpha})]\in\tilde{E}:k_{\alpha}\in K\quad\text{for each }\alpha\}$; and note that, $\dot{K}\subseteq\tilde{K}$. The process of embedding preserves the properties of being closed and convex for subsets of $E$. If each $T_{\alpha}$ is non-expansive on $K$, then the mega mapping $\tilde{T}:\tilde{K}\longrightarrow\tilde{K}$ defined by $\tilde{T}[(x_{\alpha})]=[(T_{\alpha}x_{\alpha})]$ is also non-expansive. Another piece of notation, when $(x_{k})$ is a sequence in $E$, then $f_{k}=[(\alpha.x_{k})_{\alpha}]$ is a sequence in $\tilde{E}$.

Definition 2.1**.**

(a) Let $K$ be a non-empty subset of a Banach space $E$ and $(x_{\alpha})_{\alpha\in S}$ be a bounded net in $E$. For each $k\in K,\alpha\in S$ define

[TABLE]

[TABLE]

The set $AC(K,(x_{\alpha}))$ (the number $r(K,(x_{\alpha}))$ ) will be called the asymptotic center (asymptotic radius) of $(x_{\alpha})_{\alpha\in S}$ in $K$. These are the generalizations of Chebyshev center and radius and are due to Edelstein[4]. The asymptotic center is always non-empty for weakly compact set $K$.

(b) When viewed in $\tilde{E}$, these notions are seen as:

[TABLE]

[TABLE]

.

The proof of the following theorem uses some ideas from [3] and [7].

Theorem 2.2**.**

Let $K$ be a bounded closed convex non-empty subset of a Banach space $E$, and let $S$ be a left reversible semigroup acting non-expansively and separately continuous on $K$. Suppose there exists a $\lambda\in[0,1)$ such that for each net $(x_{\alpha})$ in $K$ and each net $(y_{\alpha})$ in $AC(K,(x_{\alpha}))$ we have $r(K,(y_{\alpha}))\leq\lambda r(K,(x_{\alpha}))$. Then, $S$ has a common fixed point in $K$.

Proof. Let $x_{0}\in K$. Put $r_{0}=r(K,(\alpha.x_{0}))$ and $A_{1}=AC(K,(\alpha.x_{0}))$. Let $x_{1}\in A_{1}$. Since the action is non-expansive we see that $\alpha.x_{1}\in A_{1}$ and $[(\alpha.x_{1})]\in\tilde{A_{1}}$. Put $r_{1}=r(K,(\alpha.x_{1}))$ and $A_{2}=AC(K,(\alpha.x_{1}))$. By induction, we get a sequence $x_{k}$ such that $x_{k}\in A_{k}$, $[(\alpha.x_{k})]\in\tilde{A_{k}}$, $r_{k}=r(K,(\alpha.x_{k}))$, $A_{k+1}=AC(K,(\alpha.x_{k}))$ and $r_{k}\leq{\lambda}^{k}r_{0}$. As in the proof of [3, Theorem 3.17] and [7, Theorem 5.3], there exists a $u\in K$ such that the sequence $f_{k}=[(\alpha.x_{k})]$ converges to $\dot{u}$ in $\tilde{K}$. Note that the use of $\alpha\in S$ instead of $n\in\mathbb{N}$ in those theorems for the part we need, cause no problem since the convergence occurs in $k$. So, the limit of $[(\alpha.x_{k})]$ is $[(u)]$ regardless of the indexing $\alpha$s. Let $\beta\in S$ be arbitrary and consider the elements $[(\beta.\alpha.x_{k})]$ and the constant element $[(\beta.u)]$ which are indexed by $\alpha$s. Since the action is non-expansive we get the $\lVert[(\beta.\alpha.x_{k})]-[(\beta.u)]\rVert\leq\lVert[(\alpha.x_{k})]-[(u)]\rVert$. Hence the sequence $[(\beta.\alpha.x_{k})]$ converges to $[(\beta.u)]$ in $\tilde{K}$. But, $\beta.\alpha\in S$, so by the first part of the current proof, $[(\beta.\alpha.x_{k})]$ is a another sequence like $[(\alpha.x_{k})]$ with the same limit. That is, by the same discussion, $[(\beta.\alpha.x_{k})]$ must have the limit $[(\beta.u)]=[(u)]$. Hence, $(\beta.u-u)_{\alpha}\in\mathcal{N}$. Therefore, $\beta.u=u$, which is the desired result.$\blacksquare$

Corollary 2.3**.**

Let $K$ be a weakly compact convex non-empty subset of a nearly uniformly convex Banach space $E$, and let $S$ be a left reversible semigroup acting non-expansively and separately continuous on $K$. Then, $S$ has a common fixed point in $K$.

Proof. The set $K$ is closed and bounded. Also, nearly uniformly convex Banach spaces satisfy the inequality in Theorem 2.2. So, the result follows.$\blacksquare$