On the Schauder fixed point property II

Abstract

The Schauder fixed point property (F) was introduced and studied by Lau and Zhang as a semigroup formulation in the general setting of convex spaces of the well-known Schauder fixed point theorem in Banach spaces. What amenability property should possess a semigroup or a topological group to satisfy the Schauder fixed point property. Recently, the author provided a partial answer to that question and as a sequel, it is the purpose of this paper to study in more deep this problem. Our main result establishes that for a compact semitopological semigroup S we have: LUC(S) is left amenable if, and only if, S has the fixed point property (F). Furthermore, we also prove that totally bounded topological groups, semitopological groups S with the property that LUC(S) $\subset$${\textrm{aa}}$(S), and strongly left amenable semitopological semigroups, possess all the Schauder fixed point property.

1 Introduction

The Schauder fixed point theorem is a theorem in topological fixed point theory that guarantees the existence of a fixed point for any continuous mapping of a non-empty compact convex set into itself in a normed space. This result is an extension of the celebrated Brouwer fixed point theorem to normed spaces, later generalized by Tychonoff to Hausdorff locally convex spaces. A semitopological semigroup, S,  will be said to have the Schauder fixed point property (F) introduced by Lau and Zhang [6], if for each compact convex set K in a separated locally convex space (E, Q),  and for each homomorphic jointly continuous representation of S as a semigroup of mappings of K into itself, there is in K a common fixed point of the representation. In terms of semigroups, the Schauder theorem is equivalent to saying that cyclic semigroups possess the Schauder fixed point property:

$({{\textbf {F}}}):$:

Whenever $S\times K\rightarrow K$ is a jointly continuous action of S on a non-empty compact convex subset K of a separated locally convex space (E, Q) there exists a point ${{\textbf {a}}}$ of K such that $s({{\textbf {a}}})={{\textbf {a}}},$ for all $s\in S.$

It is important to note that if a semigroup S satisfies the property $({{\textbf {F}}})$ then it is must be left amenable, i.e., there exists a left invariant on LUC(S). However, there is a commutative a fortiori amenable semigroup S which does not have property $({{\textbf {F}}}).$ Consider, see [3], two continuous functions $f,g:[0,1]\rightarrow [0,1]$ such that $f\circ g=g\circ f$ with no fixed point in common. Set $S:=\lbrace f^m\circ g^n;\ m,n\in {\mathbb {Z}}_+\rbrace$ with the discrete topology, then the action $(s,x)\mapsto s(x)$ from $S\times [0,1]$ into [0, 1] satisfies all conditions of property $({{\textbf {F}}})$ but has no common fixed point. Therefore, mere left amenability for a semigroup is not in general a sufficient condition for it to satisfy the Schauder fixed point property. However, see [6], there are amenable semigroups which possess the Schauder fixed point property, this is the case for: the additive group of the rationals, commutative semigroups of idempotents, the additive semigroup $({\mathbb {R}},+),$ the additive locally compact group of p-adic numbers $({\mathbb {Q}}_p,+).$ In this paper, we are interested in the following problems, see [6, Question 5], related to the fixed point property $({{\textbf {F}}})$:

• Problem 1 For a semigroup what amenability property of it may be characterized by the Schauder fixed point property?

• Problem 2 What amenability property of a topological group may be characterized by the Schauder fixed point property?

In [8], we have established that the class $\Sigma$-ELA of all n-extremely left amenable semitopological groups $(n\in {\mathbb {N}})$ possess the fixed point property $({{\textbf {F}}}).$ However, n-extreme amenability is not a necessary condition for a semigroup to satisfy the Schauder fixed point property; in fact, the bicyclic semigroup $S_1=\langle e,a,b\ :\ ab=e\rangle =\lbrace b^ma^n;\ m,n\in {\mathbb {N}},\ \text {with}\ ab=e\rbrace$ possesses the Schauder fixed point property, because b has a fixed point by the Tychonoff fixed point theorem, which is also a fixed point for a (since $ab=e$) and then for S. However, $S_1$ fails to be n-extremely left amenable for some n because it is not even left amenable as it is not left reversible (e.g. $aS_1\cap bS_1=\emptyset$).

In this paper, see Corollaries 3.4 and 3.6, Theorems 3.8, and 3.11, we are able to establish sufficient conditions for a semitopological semigroup S to satisfy the Schauder fixed point property $({{\textbf {F}}}),$ namely for the following cases:

• (i) S is a compact semitopological semigroup;

• (ii) S is a totally bounded topological group.

• (iii) S is a semitopological group for which LUC$(S)\subset$ ${\textrm{aa}}$(S);

• (iv) S is a strongly left amenable semitopological semigroup.

2 Preliminaries

Let S be a semitopological semigroup (i.e. a semigroup with a separated topology such that the operation is singly continuous in the sense that for each $a\in S,$ the mappings $s\mapsto as$ and $s\mapsto sa$ from S into itself are continuous). Let $C_b(S)=\ell ^{\infty }(S)\cap C(S)$ the Banach algebra of bounded continuous real-valued functions on S with the sup norm topology. For each $a\in S,$ let $\ell _a$ (resp. $r_a$) be the endomorphism of $C_b(S)$ defined by $\ell _af(s)=f(as)\ (\text {resp.}\ r_af(s)=f(sa)), (a,s\in S,\ f\in C_b(S)),$ called, respectively, the left translation and right translation operator by a. Let $\Phi \subset C_b(S)$ be a translation invariant (i.e. $\ell _a\Phi \subset \Phi$ and $r_a\Phi \subset \Phi ,$ for each $a\in S$) closed subspace containing the constant functions, and $\Phi ^*$ be its topological conjugate. A mean on $\Phi$ is an element m in $\Phi ^*$ such that $m(1)=1=\Vert m\Vert ;$ $\Phi$ is said to have a left invariant mean (or equivalently, is said to be left amenable) if there is a mean m on $\Phi$ such that $m(\ell _af)=m(f)$ for all $a\in S, f\in \Phi .$ Let (E, Q) be a locally convex space, and $K\subset E$ be a non-empty subset. An action of a semitopological semigroup S on K is a mapping $\pi : S\times K\rightarrow K$ such that $\pi (ss',x)=\pi (s,\pi (s',x)),\ (s,x)\in S\times K);$ the action is said to be jointly continuous if $\pi$ is continuous when $S\times K$ is given the product topology. A subset $L\subset K$ is said to be S-invariant, if $\pi (s,L)\subset L,$ for all $s\in S;$ an S-invariant one-point set is said to contain a common fixed point of S.

3 Main results

In this section, we establish our main results. We recall that given a topological group G, it is said to be totally bounded if for each neighbourhood V of the identity there exists a finite subset F of G such that $G=\bigcup \lbrace g.V;\ g\in F\rbrace .$

Theorem 3.1

Let S be a semigroup satisfying either one of the following conditions : 

• 1. S is a compact semi-topological semigroup.

• 2. S is a totally bounded topological group.

Then, LUC(S) has a left invariant mean if,  and only if,  S possesses the Schauder fixed point property : 

(F) :  Whenever $S\times K\rightarrow K$ is a jointly continuous action on a non-empty compact convex set K in a Hausdorff locally convex space (E, Q) there exists a point x of K such that $s.{{\textbf {x}}}={{\textbf {x}}},$ for every $s\in S.$

Proof

It is known, see [1, Proposition 4.8], that in both cases we have the relation AP(S) = LUC(S) has a left invariant mean. Therefore, there is a compact right topological group G sitting inside the compact topological semigroup $S^{\text {AP}(S)},$ the spectrum of AP(S),  such that $\delta _g\odot G=G$ for all $g\in G.$ So by [1, Corollary 4.5], it follows that G is a compact topological group. Now, we extend the action to G as follows: given g in G and $x\in K,$ we let g.x denote the unique element in K that satisfies the property $f(g.x) = g(f_x)$ for all $f\in C(K)$ with $f_x(s)=f(s.x)\; (s\in S).$ Note that such an element does exist because of two facts: on one hand, the action being jointly continuous with compact Hausdorff K,  each map $f_x, f\in C(K)$ and $x\in K,$ lies in LUC(S) by [9, Lemma 2.2]; and on the other hand, for each pair $(g,x)\in G\times K\, g$ being a multiplicative mean on LUC(S) then the map ${\mathscr {L}}_{g,x}: C(K)\rightarrow {\mathbb {R}},\ f\mapsto g(f_x)$ defines a nonzero multiplicative linear functional on C(K),  therefore ${\mathscr {L}}_{g,x}$ is the evaluation map at some point in K which we denote by g.x. Moreover, the action is well-defined because given $g,g'\in G$ and $x\in K$, we have $g=\text {wk}^{*}\text {-}\lim \delta _{s_i}$ and $g'.x=\text {wk}^{*}\text {-}\lim _j\delta _{s_j}$ for some nets $(s_i)_i$ and $(s_j)_j,$ and for a fixed $\varphi \in C(K)$, we have:

$$\begin{aligned} \varphi \big ((g\odot g').x\big )= & {} (g\odot g')(\varphi _x)\\= & {} \lim _i\delta _{s_i}\odot g'(\varphi _x)= \lim _i g'(\ell _{s_i}\varphi _x)\\= & {} \lim _i g'\big ((\varphi \circ s_i)_x\big )=\lim _i(\varphi \circ s_i)(g'.x)\\= & {} \lim _i\delta _{s_i}(\varphi _{g'.x})=g\big (\varphi _{g'.x}\big )=\varphi \big (g.(g'.x)\big ). \end{aligned}$$

Since C(K) separates points and $\varphi$ is arbitrary, it follows that $(g\odot g').x=g.(g'.x).$ By a standard Zorn’s lemma argument let ${{{\textbf {K}}}}\subset K$ be a non-empty compact convex and G-invariant subset of K that is minimal (for the inclusion) with respect to having these properties; and by a second application of Zorn’s lemma, let ${{\textbf {F}}}\subset {{\textbf {K}}}$ be minimal with respect to being non-void Q-compact and G-invariant. Now, let us fix a point $\kappa \in {{\textbf {F}}}.$ Then, given $\varphi \in E^*,$ we define a seminorm $Q_{\varphi }$ on the linear space ${\mathscr {E}}_{{{\textbf {K}}}},$ spanned by ${{\textbf {K}}},$ by the formula

$$\begin{aligned} Q_{\varphi }\left( \sum _{i=1}^nt_i.x_i\right) = \sup _{g\in S}\left| \left\langle {{\textbf {m}}},\sum _{i=1}^nt_i.\varphi (g.x_i).\omega _{x_i}\right\rangle \right| \end{aligned}$$

where each $\omega _{x_i}\in \text {L}^{\infty }(G), (i=1,\ldots ,n)$ is given by

$$\begin{aligned} \omega _{x_i}(g) := {\left\{ \begin{array}{ll} 1 &{} \text {if } g.x_i\ne \kappa \\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

with ${{\textbf {m}}}$ being a right invariant mean on $\text {L}^{\infty }(G)$ (derived from the left invariant mean $m(f)=\int _{G}f\ d\lambda _{G},\ (f\in \text {L}^{\infty }(G))$ induced by normalized (left) Haar measure $\lambda _G$ on G) given by ${{\textbf {m}}}(f)=m(\check{f})$ with $\check{f}(g)=f(g^{-1}).$ Set ${\mathfrak {W}} := \lbrace Q_{\varphi }\ ;\ \varphi \in E^*\rbrace .$ Then $\big ({\mathscr {E}}_{{{\textbf {K}}}},{\mathfrak {W}}\big )$ becomes a locally convex space whose topology is Hausdorff on ${{\textbf {F}}}.$ In fact, given two distinct points x, y in ${{\textbf {F}}},$ as the continuous dual of E separates points let $\varphi \in E^*$ be such that $\varphi (x-y)>0.$ then we claim that the corresponding seminorm $Q_{\varphi }\in {\mathfrak {W}}$ does not vanish at $x-y.$ In fact, since ${{\textbf {F}}}=G.x$, we have $y=g.x$ for some $g\in G,$ and, therefore, we have

$$\begin{aligned} Q_{\varphi }(x-y)= & {} \sup _{g\in G}\vert \langle {{\textbf {m}}},\varphi (g.x). \omega _x-\varphi (g.y).\omega _y\rangle \vert \\= & {} \sup _{g\in G}\vert \langle {{\textbf {m}}},\varphi (g.x-g.y). \omega _x-\varphi (g.y)\lbrace \omega _x-\omega _y\rbrace \rangle \vert \\\ge & {} \vert \langle {{\textbf {m}}},\varphi (x-y).\omega _x-\varphi (y)\lbrace \omega _x-\omega _y\rbrace \rangle \vert \\= & {} \vert \langle {{\textbf {m}}},\varphi (x-y).\omega _x-\varphi (y)\lbrace \omega _x-\omega _{g.x}\rbrace \rangle \vert \\= & {} \vert \langle {{\textbf {m}}},\varphi (x-y).\omega _x-\varphi (y)\lbrace \omega _x-r_g\omega _x\rbrace \rangle \vert \\= & {} \vert \langle {{\textbf {m}}},\varphi (x-y).\omega _x\rangle \vert = \vert \varphi (x-y)\vert .\langle {{\textbf {m}}},\omega _x\rangle >0 \end{aligned}$$

the last inequality holds because $\omega _x$ being the characteristic function of the non-empty open set $\lbrace g\in G\ :\ g.x\ne \kappa \rbrace$ then ${{\textbf {m}}}(\omega _x)=\lambda _G(\lbrace g\in G\ :\ g.x\ne \kappa \rbrace \big )>0.$ Therefore, we have $Q_{\varphi }(x-y)>0$ which proves that the ${\mathfrak {W}}$-topology on ${{\textbf {F}}}$ is Hausdorff. Next, we show that ${{\textbf {F}}}$ is ${\mathfrak {W}}$-compact. Given an arbitrary net $(x_{\alpha })_{\alpha \in J}$ in it, by compactness with respect to the Q-topology, one can extract a Q-convergent subnet which we shall relabel by the same symbol. Set $x=\text {Q-}\lim _{\alpha }x_{\alpha }.$ Fix a seminorm $Q_{\varphi }\in {\mathfrak {W}}.$ By contradiction let us assume that $Q_{\varphi }(x_{\alpha }-x)\nrightarrow 0.$ Then, there exists a subnet of $(x_{\alpha _t})_{t\in T}$ and a positive $\epsilon$ such that $Q_{\varphi }(x_{\alpha _t}-x)>\epsilon ,$ for all $t\in T.$ Since $G.x={{\textbf {F}}}$ by minimality, then it follows that

$$\begin{aligned} x_{\alpha _t}= h_{\alpha _t}.x,\quad \text {for some element}\ h_{\alpha _t}\in G. \end{aligned}$$

(3.1)

So together with the fact that ${{\textbf {m}}}$ is invariant, it follows that

$$\begin{aligned} Q_{\varphi }(x_{\alpha _t}-x)= & {} \sup _{g\in G}\vert \langle {{\textbf {m}}},\varphi (g.x_{\alpha _t}).\omega _{x_{\alpha _t}}- \varphi (g.x).\omega _x\rangle \vert \\= & {} \sup _{g\in G}\vert \langle {{\textbf {m}}},\varphi (gh_{\alpha _t}.x). \omega _{h_{\alpha _t}.x}-\varphi (g.x).\omega _x\rangle \vert \\= & {} \sup _{g\in G}\vert \langle {{\textbf {m}}},\varphi (gh_{\alpha _t}.x). r_{h_{\alpha _t}}\omega _x-\varphi (g.x).\omega _x\rangle \vert \\= & {} \sup _{g\in G}\vert \langle {{\textbf {m}}},\lbrace \varphi (g_{\alpha _t} h_{\alpha _t}.x)-\varphi (g.x)\rbrace .\omega _x\rangle \vert >\epsilon . \end{aligned}$$

For each t let $g_{\alpha _t}\in G$ such that $\vert \langle {{\textbf {m}}},\lbrace \varphi (g_{\alpha _t}h_{\alpha _t}.x)-\varphi (g_{\alpha _t}.x)\rbrace .\omega _x\rangle \vert >\epsilon .$ Due to the compactness of G, we may assume (by passing to a subnet if necessary) without loss of generality that $g_{\alpha _t}\rightarrow g$ and $h_{\alpha _t}\rightarrow h$ for some $g,h\in G.$ Then $g_{\alpha _t}h_{\alpha _t}\rightarrow gh$ (the operation of G being jointly continuous). Therefore, together with (3.1), it follows that

$$\begin{aligned} \vert \langle {{\textbf {m}}},\lbrace \varphi (g_{\alpha _t}h_{\alpha _t}.x)- \varphi (g_{\alpha _t}.x)\rbrace .\omega _x\rangle \vert\le & {} \vert \varphi (g_{\alpha _t}h_{\alpha _t}.x)- \varphi (g_{\alpha _t}.x)\vert .\Vert \omega _x\Vert \\&\xrightarrow [t]{}&\vert \varphi (gh.x-g.x)\vert =0 \end{aligned}$$

leading to $0<\epsilon \le 0$ which is absurd. Consequently, $Q_{\varphi }(x_{\alpha }-x)\rightarrow 0$ which establishes our assertion. The final step consists of showing that ${{\textbf {F}}}$ is a point set. By contradiction let us assume that it contains at least two points let us say x and y. since the ${\mathfrak {W}}$-topology is separated on ${{\textbf {F}}},$ there is a seminorm $Q_{\varphi }$ with respect to which ${{\textbf {F}}}$ has a positive diameter. So being ${\mathfrak {W}}$-compact there exists, see [4], some u in the convex hull of ${{\textbf {F}}}$ such that

$$\begin{aligned} \sup _{x\in {{\textbf {F}}}}Q_{\varphi }(x-u)<\sup _{x,y\in {{\textbf {F}}}}Q_{\varphi }(x-y). \end{aligned}$$

(3.2)

Set

$$\begin{aligned} {{\textbf {K}}}_* :=\bigcap \big \lbrace \overline{B_{Q_{\varphi }}'(x,{{\textbf {r}}})} ; x\in {{\textbf {F}}}\big \rbrace , \end{aligned}$$

where ${{\textbf {r}}}:=\sup _{x\in {{\textbf {F}}}}Q_{\varphi }(x-u)$ and $B_{Q_{\varphi }}'(x,{{\textbf {r}}})$ stands for the $Q_{\varphi }$-closed ball in ${{\textbf {K}}}$ centered at x with radius ${{\textbf {r}}}$ given by $\lbrace y\in {{\textbf {K}}} : Q_{\varphi }(x-y)\le {{\textbf {r}}}\rbrace .$ Then, ${{\textbf {K}}}_*$ possesses the following properties:

• 1. ${{\textbf {K}}}_*$ is non-empty, Q-compact and convex;

• 2. ${{\textbf {K}}}_*$ is G-invariant and properly contained in K.

For (1) is straightforward. For (2), we first show that ${{\textbf {K}}}_*$ is G-invariant. Fix $g\in G$ and $x\in {{\textbf {K}}}_*.$ Given $y\in {{\textbf {F}}}=g.{{\textbf {F}}},$ we have $y=g.x'$ for some $x'\in {{\textbf {F}}}.$ Then, using the fact that $x\in B_{Q_{\varphi }}'(x',{{\textbf {r}}})$ with ${{\textbf {m}}}$ being right invariant yield

$$\begin{aligned} Q_{\varphi }(g.x-y)= & {} Q_{\varphi }(g.x-g.x')\\= & {} \sup _{h\in G}\vert \langle {{\textbf {m}}},\varphi (hg.x). \omega _{g.x}-\varphi (hg.x').\omega _{g.x'}\rangle \vert \\= & {} \sup _{h\in G}\vert \langle {{\textbf {m}}},\varphi (h.x).r_g \omega _x-\varphi (h.x').r_g\omega _{x'}\rangle \vert \\= & {} \sup _{h\in G}\vert \langle {{\textbf {m}}},\varphi (h.x). \omega _x-\varphi (h.x').\omega _{x'}\rangle \vert \\= & {} Q_{\varphi }(x-x')\le {{\textbf {r}}} \end{aligned}$$

which implies that $g.x\in B_{Q_{\varphi }}'(y,{{\textbf {r}}})$ for all $y\in {{\textbf {F}}}.$ Therefore, $g.x\in {{\textbf {K}}}_*.$ Hence, we have established that ${{\textbf {K}}}_*$ shares the same properties as ${{\textbf {K}}}$ and consequently, ${{\textbf {K}}}_*={{\textbf {K}}}$ by minimality. Now to arrive to a contradiction, we only need to prove that ${{\textbf {K}}}_*$ is a proper subset of ${{\textbf {K}}}.$ The previous equality implies that ${{\textbf {F}}}={{\textbf {K}}}_*\cap {{\textbf {F}}}=\bigcap _{x\in {{\textbf {F}}}} \overline{B_{Q_{\varphi }}'(x,{{\textbf {r}}})}\cap {{\textbf {F}}}= \bigcap _{x\in {{\textbf {F}}}}B_{Q_{\varphi }}'(x,{{\textbf {r}}})\cap {{\textbf {F}}}$ because the Q and ${\mathfrak {W}}$ induce the same topology on ${{\textbf {F}}}.$ Now let us fix $a,b\in {{\textbf {F}}}$ such that $Q_{\varphi }(a-b)=\sup _{x,y\in {{\textbf {F}}}}Q_{\varphi }(x-y)$ (note that such elements do exist because ${{\textbf {F}}}$ is ${\mathfrak {W}}$-compact). Then, by (3.1), we have ${{\textbf {r}}}<Q_{\varphi }(a-b)$ which implies that $a\notin \bigcap _{x\in {{\textbf {F}}}}B_{Q_{\varphi }}'(x,{{\textbf {r}}})\supset {{\textbf {K}}}_*,$ which shows that ${{\textbf {K}}}_*\subsetneq {{\textbf {K}}}$ contradicting the fact that both sets are equal. Consequently, F must contain a single point which is a common fixed point for G. Set ${{\textbf {F}}}=\lbrace {{\textbf {x}}}\rbrace .$ Given $s\in S$ and $\varphi \in E^{*},$ since $\delta _s\odot G=G$ and $g.{{\textbf {x}}}={{\textbf {x}}}$ for all $g\in G,$ we have $(\delta _s\odot g).{{\textbf {x}}}={{\textbf {x}}}.$ Therefore, it follows that

$$\begin{aligned} \varphi ({{\textbf {x}}})= & {} \varphi \big ((\delta _s\odot g).{{\textbf {x}}}\big )=\delta _s\odot g(\varphi _{{{\textbf {x}}}})\\= & {} g(\ell _s\varphi _x)=g\big ((\varphi \circ s)_x\big )\\= & {} (\varphi \circ s)(g.{{\textbf {x}}})=\varphi \big (s.(g.{{\textbf {x}}})\big )=\varphi \big (s.{{\textbf {x}}}). \end{aligned}$$

Thus, $\varphi$ being free and the fact that $E^*$ separates points yield $s.{{\textbf {x}}}={{\textbf {x}}}.$ $\square$

Remark 3.2

It is natural at this point to ask whether the converse of Theorem 3.1 is true. Unfortunately, the answer is no. In fact, any infinite cyclic (discrete) group has the fixed point property (F), but such a group cannot be strongly left amenable, because the only strongly left amenable locally compact groups are the compact ones.

Proposition 3.3

Let S be a semitopological semigroup satisfying the fixed point property (F). Then, LUC(S) possesses a left invariant mean. In particular,  if S is a topological group then LUC(S) possesses an invariant mean.

Proof

Consider the action of G on the non-empty compact convex subset K of all means on LUC(G) in the weak$^{*}$ topology of LUC(G)$^{*},$ given by $(g,m)\mapsto \delta _g\odot m;$ with $\delta _g\odot m(f)=m(\ell _gf)$ for $f\in$ LUC(G) and $g\in G.$ It is readily checked that the action is jointly continuous and, therefore, there exists an $m_1\in K$ such that $\delta _g\odot m_1 = m_1$ for all $g\in G;$ which precisely means that $m_1$ is a left invariant mean on LUC(G). We now assume that G is a topological group. Then, we derive a right invariant mean on LUC(G) by setting $m_2(f) :=m_1(\check{f})$ with $\check{f}(g)=f(g^{-1}).$ It is easy to checked that if $f\in$ LUC(G) then $\check{f}\in$ LUC(G) too and also $r_gf =\ell _{g^{-1}}\check{f}$ So $m_2$ is a well-defined right invariant mean on LUC(S). We define then an invariant mean ${{\textbf {m}}}$ on LUC(S) by just letting ${{\textbf {m}}} :=m_1\odot m_2,$ i.e. ${{\textbf {m}}}(f)=m_1(T_f)$ with $T_f(g)=m_2(\ell _gf)\ (g\in S),$ LUC(S) being left introverted. $\square$

Corollary 3.4

Let S be a compact semitopological semigroup,  then the following properties are equivalent : 

• 1. S is left amenable; 

• 2. S has the fixed point property (F).

Remark 3.5

Corollary 3.4 answers partially Problem 1, see Sect. 1, about what amenability property for a semigroup may be characterized by the Schauder fixed point property, for compact semitopological semigroups; and also [8, Question 5] for this special class of strongly amenable topological semigroups. Concerning the question, see [6, Question 5], about what amenability property of a topological group may be characterized by the fixed point property (F), we have the following result.

Corollary 3.6

Totally bounded topological groups have the Schauder fixed point property.

Remark 3.7

Corollary 3.6 answers positively Problem 2, see Sect. 1, for the class of totally bounded groups; and partially [8, Question 5].

We recall that for a semitopological group S,  a bounded continuous function $f:S\rightarrow {\mathbb {R}}$ is said to be almost automorphic, see [1], if $(s_t)_t$ is a net in S such that $g(s)=\lim _t f(ss_t)$ and $h(s)=\lim _t g(ss^{-1}_t)$ for all $s\in S$ for some functions $h,g: S\rightarrow {\mathbb {R}}$ exist, then $h=f.$ The set of all such functions on S,  denoted by ${\textrm{aa}}$(S), is a translation invariant sub-algebra of $C_b(S).$ Let $\tau$ be the Bohr topology of S (i.e. the initial topology on S for ${\textrm{aa}}$(S)). When S is a totally bounded topological group then ${\textrm{aa}}$$(S)=C_b(S)\supset$ LUC(S),  see [1, Corollary 7.9]. It is also always true that AP$(S)\subset$${\textrm{aa}}$(S),  where AP(S) is the closed translation invariant sub-algebra of $C_b(S)$ consisting of those functions f for which their left orbits are relatively norm compact.

Theorem 3.8

Let S be a semitopological group such that LUC$(S)\subset$${\textrm{aa}}$(S). Then, S possesses the Schauder fixed point property $(\mathbf{{F}}).$

Proof

We first note that when S is given the Bohr topology $\tau$ then the action is separately continuous. In fact, left continuity is trivial, and for the right one let us fix a convergent net $s_{\alpha }\rightarrow s,\ (\alpha \in J)$ in $(S,\tau )$ and $x\in K.$ The original action being jointly continuous with K compact, it follows that $\varphi _x\in$ LUC(S) for all $\varphi \in E^*$ by [9, Lemma 2.2] or [7, page 633]. So from the inclusion LUC$(S)\subset$ ${\textrm{aa}}$(S) it follows that $\varphi (s_{\alpha }.x)=\varphi _x(s_{\alpha })\rightarrow \varphi _x(s)=\varphi (s.x).$ Therefore, $s_{\alpha }.x\rightarrow s.x$ weakly; and since the original topology and the weak topology agree on K we conclude that $s_{\alpha }.x\rightarrow s.x$ strongly. On the other hand, since $(S,\tau )$ is a totally bounded topological group then AP$(S,\tau )$ = LUC$(S,\tau ),$ see [1, Proposition 4.8 and Theorem 7.5]. Therefore, AP$(S,\tau )$ being amenable there is a compact right topological group G,  therefore, a compact topological group, sitting inside $S^{\text {AP}(S,\tau )}.$ Let ${{\textbf {m}}}$ be an invariant mean on $\text {L}^{\infty }(G).$ Using the condition LUC$(S)\subset$ ${\textrm{aa}}$(S), we extend the action of S on K to a right continuous action of $S^{\text {LUC}(S,\tau )}$ by letting g.x denote the unique element of K with the property that $\varphi (g.x)=g(\varphi _x), (\varphi \in E^*)$ (see proof of Theorem 3.1). By a Zorn’ lemma argument, let ${{\textbf {M}}}\subset K$ be minimal with respect to the properties of K and ${{\textbf {F}}}\subset {{\textbf {M}}}$ with same properties but the convexity. Fix a point $\kappa \in {{\textbf {F}}}$ and consider the family of seminorms $Q_{\varphi }, (\varphi \in E^*)$ on the linear manifold spanned by K as in the proof of Theorem 3.1. Then, a same argument allows us to conclude that ${\mathbb {F}}$ consists of a single point ${{\textbf {x}}}$ which for sure a common fixed point S. $\square$

Remark 3.9

Theorem 3.8 provides a positive answer to Problem 2 for semitopological groups S for which LUC$(S)\subset$${\textrm{aa}}$(S).

Corollary 3.10

Let S be a semitopological group such that LUC$(S)\subset$${{\textrm{aa}}}$(S),  then the following properties hold : 

• 1. LUC(S) possesses a left invariant mean; 

• 2. AP(S) = WAP(S) $\subset$ LUC$(S)\subset$${{\textrm{aa}}}$(S).

Proof

(1) follows from Proposition 3.3, while (2) follows from [1, Corollary 4.11 and Corollary 7.13] with the inclusion AP$(S)\subset$ LUC(S) being always true. $\square$

Theorem 3.11

Let S be a semitopological semigroup,  and suppose that LUC(S) is strongly left amenable. Then, S possesses the Schauder fixed point property (F).

Proof

Using the construction in the proof of Theorem 3.1, we extend the action $S\times K\rightarrow K$ to an action of $S^{\text {LUC}(S)},$ the spectrum of LUC(S), on K. On the other hand, S being strongly left amenable there is a compact right topological group $\Delta \subset S^{\text {LUC}(S)}$ such that $\delta _s\odot \Delta =\Delta$ for all $s\in S.$ Let ${{\textbf {m}}}$ be a right invariant mean on ${\textbf {L}}^{\infty }(\Delta )$ given by ${{\textbf {m}}}(f)=\int _{\Delta }\check{f}\ d\lambda ,$ with $\check{f}(s)=f(s^{-1}), \lambda$ standing for normalized (left) Haar measure on $\Delta .$ Let ${\mathfrak {S}}:=\lbrace \sigma \subset S: \sigma \ \text {is non-void and finite}\rbrace$ ordered by $\sigma \le \sigma '$ if, and only if, $\sigma \subset \sigma '.$ Using Zorn’s lemma, let F$\subset$ K$\subset K$ be non-empty compact S-invariant sets, with K being convex and both being minimal with respect to having all these properties. Then, LUC(S) being left amenable with the action being jointly continuous, [10, Lemma 2.12] implies that $s.{{\textbf {F}}}={{\textbf {F}}},$ for all $s\in S.$ Given $\varphi \in E^*$ and $\sigma \in {\mathfrak {S}}$, we define a seminorm $Q_{\varphi ,\sigma }$ on the linear space ${\mathscr {E}}_K$ (generated by K) as follows: given $x=\sum _{i=1}^p\lambda _i.x_i\in {\mathscr {E}}_K$ and $\varphi \in E^*,$ set

$$\begin{aligned} Q_{\varphi ,\sigma }(x):=\sup _{s\in \sigma }\left| \sum _{i=1}^p\lambda _i. \varphi (s.x_i)\langle \omega _{x_i},{{\textbf {m}}}\rangle \right| . \end{aligned}$$

Then, it is clear that $Q_{\varphi ,\sigma }\le Q_{\varphi ,\sigma '}$ for all $\varphi \in E^*$ and $\sigma ,\sigma '\in {\mathfrak {S}}$ with $\sigma \le \sigma '.$ On the other hand, due to the fact that $E^*$ separates points of K, it follows that ${\mathscr {E}}_K$ equipped with the topology induced by ${\mathscr {Q}} :=\lbrace Q_{\varphi ,\sigma }\ ; \varphi \in E^*,\sigma \in {\mathfrak {S}}\rbrace$ is a locally convex space that is separated on ${{\textbf {F}}}.$ In fact, ${\mathscr {Q}}$ generates a topology finer than the convex topology of ${\mathscr {E}}_K$ induced by the $Q_{\varphi ,\sigma }$’s $(\varphi \in E^*,\sigma \in {\mathfrak {S}})$ which is Hausdorff on ${{\textbf {F}}}.$ We now wish to show that ${{\textbf {F}}}$ is ${\mathscr {Q}}$-compact. To do it, we fix an arbitrary net $(x_{\alpha })_{\alpha \in J}$ in F and show that one can extract a convergent subnet. First of all, ${{\textbf {F}}}$ being Q-compact there is a Q-convergent subnet $(x_{\alpha _t})_t.$ Set $x= Q\text {-}\lim x_{\alpha _t}$ and fix a seminorm $Q_{\varphi ,\sigma }\in {\mathscr {Q}}.$ By contradiction assume that $Q_{\varphi ,\sigma }(x_{\alpha _t}-x)\nrightarrow 0.$ Then, there is $\epsilon >0$ and a subnet of $(x_{\alpha _t})_t$ which we still denote by $(x_{\alpha _t})_t$ such that $Q_{\varphi ,\sigma }(x_{\alpha _t}-x)>\epsilon$ for all t. For each t fix $g_{\alpha _t}\in \sigma$ such that $\vert \langle \varphi (g_{\alpha _t}.x_{\alpha _t}).\omega _{x_{\alpha _t}}-\varphi (g_{\alpha _t}.x).\omega _x,{{\textbf {m}}}\rangle \vert \ge \epsilon .$ Then $\sigma$ being finite, we may assume without loss of generality, by passing to a suitable subnet if necessary, that $g_{\alpha _t}=g_{\alpha _{t_o}}$ for all t. On the other hand, since $\Delta .x={{{\textbf {F}}}}$ by minimality, then $x_{\alpha _t}=\mu _{\alpha _t}.x,$ for some $\mu _{\alpha _t}\in \Delta .$ Moreover, by passing to a subnet if necessary, we may also assume that $\mu _{\alpha _t}\rightarrow \mu$ weak$^{*}$ due to the compactness of $\Delta .$ Therefore, we have:

$$\begin{aligned} \epsilon\le & {} \vert \langle \varphi (g_{\alpha _{t_o}}.x_{\alpha _t}). \omega _{x_{\alpha _t}}-\varphi (g_{\alpha _{t_o}}.x).\omega _x,{{\textbf {m}}}\rangle \vert \\= & {} \big \vert \langle \varphi (g_{\alpha _{t_o}}.x_{\alpha _t}-g_{\alpha _{t_o}}.x). \omega _{x_{\alpha _t}}-\varphi (g_{\alpha _{t_o}}.x).\lbrace \omega _{x_{\alpha _t}}- \omega _x\rbrace ,{{\textbf {m}}}\rangle \big \vert \\= & {} \big \vert \langle \varphi (g_{\alpha _{t_o}}.x_{\alpha _t} -g_{\alpha _{t_o}}.x).\omega _{x_{\alpha _t}}-\varphi (g_{\alpha _{t_o}}.x). \lbrace \omega _{\mu _{\alpha _t}.x}-\omega _x\rbrace ,{{\textbf {m}}}\rangle \big \vert \\= & {} \big \vert \langle \varphi (g_{\alpha _{t_o}}.x_{\alpha _t}-g_{\alpha _{t_o}}.x). \omega _{x_{\alpha _t}}-\varphi (g_{\alpha _{t_o}}.x).\lbrace r_{\mu _{\alpha _t}}\omega _x-\omega _x\rbrace ,{{\textbf {m}}}\rangle \big \vert \\\le & {} \vert \varphi (g_{\alpha _{t_o}}.x_{\alpha _t}-g_{\alpha _{t_o}}.x)\vert + \vert \varphi (g_{\alpha _{t_o}}.x)\vert .\underbrace{\vert \lbrace r_{\mu _{\alpha _t}}\omega _x-\omega _x\rbrace ,{{\textbf {m}}}\rangle \big \vert }_{=0}\\= & {} \vert \varphi (g_{\alpha _{t_o}}.x_{\alpha _t}-g_{\alpha _{t_o}}.x)\vert \xrightarrow [t]{}0 \end{aligned}$$

the last equation following by continuity of the action. Therefore, we have shown that $\epsilon \le 0$ which is absurd; therefore $\lim _tQ_{\varphi ,\sigma }(x_{\alpha _t}-x)=0$ which proves that ${{\textbf {F}}}$ is ${\mathscr {Q}}$-compactness. Now, we wish to show that ${{\textbf {F}}}$ is a one-point set. By contradiction let us assume that this is not the case. Then, pick $x\ne y\in {{\textbf {F}}}$ and $\varphi \in E^*$ such that $\sup _{x,y\in {{\textbf {F}}}}\vert \varphi (x-y)\vert >0.$ Since ${{\textbf {F}}}$ has positive diameter with respect to the seminorms $Q_{\varphi ,\sigma },$ then an application of [4] guarantees the existence of a sequence $(u_{\sigma })_{\sigma }$ of elements of the convex envelope, $\text {co}({{\textbf {F}}}),$ of ${{\textbf {F}}}$ such that

$$\begin{aligned} \sup _{x\in {{\textbf {F}}}}Q_{\varphi ,\sigma }(x-u_{\sigma })< \sup _{x,y\in {{\textbf {F}}}}Q_{\varphi ,\sigma }(x-y),\quad \text {for all }\sigma \in {\mathfrak {S}}. \end{aligned}$$

(3.3)

Set:

• 1. $r_{\varphi ,\sigma } := \sup _{x\in {{\textbf {F}}}}Q_{\varphi ,\sigma }(x-u_{\sigma });$

• 2. ${{\textbf {r}}}_{\varphi ,\sigma } := \sup _{j\ge \sigma }r_{\varphi ,\sigma }\in [0,\infty );$

• 3. ${{\textbf {F}}}_{\varphi ,\sigma } :=\bigcap _{x\in {{\textbf {F}}}}B^{\prime }_{\varphi ,\sigma }(x,{{\textbf {r}}}_{\sigma }),$

with $B^{\prime }_{\varphi ,\sigma }(x,{{\textbf {r}}}_{\varphi ,\sigma }) :=\lbrace y\in {{\textbf {K}}} : Q_{\varphi ,\sigma }(x-y)\le {{\textbf {r}}}_{\varphi ,\sigma }\rbrace .$ We need to justify that ${{\textbf {r}}}_{\varphi ,\sigma }$ is a finite number for $\sigma$ and $\varphi$ fixed.

$$\begin{aligned} {{\textbf {r}}}_{\varphi ,\sigma }= & {} \sup _{\tau \ge \sigma }\left\{ \sup _{x\in {{\textbf {F}}}} \sup _{s\in \tau }\vert \varphi (s.x)\omega _x -\varphi (s.u_{\tau }).\omega _{u_{\tau }},{{\textbf {m}}}\rangle \vert \right\} \\= & {} \sup _{\tau \ge \sigma }\left\{ \sup _{x\in {{\textbf {F}}}} \sup _{s\in \tau }\vert \varphi \big (s.x-s.u_{\tau }).\omega _x- \varphi (s.u_{\tau }).\lbrace \omega _x-\omega _{u_{\tau }}\rbrace , {{\textbf {m}}}\rangle \vert \right\} \\\le & {} \sup _{\tau \ge \sigma }\left\{ \sup _{x\in {{\textbf {F}}}} \sup _{s\in \tau }\vert \varphi \big (s.x-s.u_{\tau })\vert \langle \omega _x, {{\textbf {m}}}\rangle +\vert \varphi (s.u_{\tau })\vert .\langle \vert \omega _x- \omega _{u_{\tau }}\vert ,{{\textbf {m}}}\rangle \right\} \\\le & {} \sup _{\tau \ge \sigma }\left\{ \sup _{x\in {{\textbf {F}}}} \sup _{s\in \tau }\vert \varphi \big (s.x-s.u_{\tau })\vert +2\vert \varphi (s.u_{\tau })\vert \right\} \\\le & {} \sup _{x\in K-K}\vert \varphi (x)\vert +2\sup _{x\in K}\vert \varphi (x)\vert . \end{aligned}$$

It follows that $\sup _{\sigma }{{\textbf {r}}}_{\varphi ,\sigma }\le \sup _{x\in K-K}\vert \varphi (x)\vert +2\sup _{x\in K}\vert \varphi (x)\vert <\infty$, because $\varphi$ is continuous on the compact sets $K-K$ and K. We have the following properties:

• 1. $u_{\sigma }\in {{\textbf {F}}}_{\varphi ,\sigma }\ne \emptyset ,$ for all $\sigma \in {\mathfrak {S}};$

• 2. $\Sigma :=\bigcap _{\sigma \in {\mathfrak {S}}} \overline{{{\textbf {F}}}}_{\varphi ,\sigma }\ne \emptyset ;$

• 3. $\Sigma$ is S-invariant, convex, and Q-compact.

(1) is trivial and for (2), due to the Q-compactness of ${{\textbf {K}}}$ it is enough to show that $\lbrace \overline{{{\textbf {F}}}}_{\varphi ,\sigma };\ \sigma \in {\mathfrak {S}}\rbrace$ has the finite intersection property. Given $\sigma$ and $x\in {{\textbf {F}}},$ whenever $y\in {{\textbf {F}}}_{\varphi ,\tau }$ with $\sigma \subset \tau$ we have $Q_{\varphi ,\sigma }(x-y)\le Q_{\varphi ,\tau }(x-y)\le {{\textbf {r}}}_{\varphi ,\tau } \le {{\textbf {r}}}_{\tau }\le {{\textbf {r}}}_{\sigma },$ which implies that $y\in {{\textbf {F}}}_{\sigma }.$ Consequently, $\emptyset \ne {{\textbf {F}}}_{\varphi ,\tau }\subset {{\textbf {F}}}_{\sigma } \Rightarrow \emptyset \ne \overline{{{\textbf {F}}}}_{\varphi ,\tau }\subset \overline{{{\textbf {F}}}}_{\varphi ,\sigma },$ for all $\sigma .$ Finally, for (3), convexity and Q-compactness are trivial; so it remains to show that $\Sigma$ is S-invariant. Let $s\in S$ fixed. We have $s.\Sigma \subset \bigcap _{\sigma \in \mathfrak {S}} \overline{s.{{{\textbf {F}}}}_{\varphi ,\sigma }}.$ Therefore, it is enough to prove that $\overline{s.{{\textbf {F}}}_{\varphi ,\sigma }}\subset \overline{{{\textbf {F}}}}_{\varphi ,\sigma }$ for each $\sigma .$ Fix $\sigma \in {\mathfrak {S}}, y\in {{\textbf {F}}}_{\varphi ,\sigma }.$ Since ${{\textbf {F}}}=s.{{\textbf {F}}}$ then given $x\in \textbf{F}$ we have $x=s.x'$ for some $x'\in {\mathbb {F}};$ so with the fact that ${{\textbf {m}}}$ is right invariant, we obtain

$$\begin{aligned} Q_{\varphi ,\sigma }(s.y-x)= & {} Q_{\varphi ,\sigma }(s.y-s.x')\\= & {} \sup _{z\in \sigma }\vert \varphi (zs.y)\omega _{s.y}-\varphi (zs.x'). \omega _{s.x'},{{\textbf {m}}}\rangle \vert \\= & {} \sup _{z\in \sigma .\lbrace s\rbrace }\vert \varphi (z.y)\omega _{s.y}- \varphi (z.x').\omega _{s.x'},{{\textbf {m}}}\rangle \vert \\= & {} \sup _{z\in \sigma .\lbrace s\rbrace }\vert \varphi (z.y)\omega _y- \varphi (z.x').\omega _{x'},{{\textbf {m}}}\rangle \vert \\\le & {} \sup _{z\in \sigma .\lbrace s\rbrace \cup \sigma }\vert \varphi (z.y)\omega _y-\varphi (z.x').\omega _{x'},{{\textbf {m}}} \rangle \vert \\= & {} Q_{\varphi ,\sigma .\lbrace s\rbrace \cup \sigma }(x-y)\le {{\textbf {r}}}_{\varphi ,\sigma .\lbrace s\rbrace \cup \sigma }\le {{\textbf {r}}}_{\varphi ,\sigma }. \end{aligned}$$

It follows that $s.y\in B^{\prime }_{\varphi ,\sigma }(x,{{\textbf {r}}}_{\varphi ,\sigma })$ for all $\sigma \in {\mathfrak {S}}.$ Therefore, $s.y\in {{\textbf {F}}}_{\varphi ,\sigma }.$ Consequently, $s.{{\textbf {F}}}_{\varphi ,\sigma }\subset {{\textbf {F}}}_{\varphi ,\sigma }$ and the conclusion follows by continuity. Thus, $\Sigma$ possessing all the properties of ${{\textbf {K}}}$ we must have by minimality ${{\textbf {K}}}=\Sigma .$ Therefore, ${{\textbf {F}}}=\Sigma \cap {{\textbf {F}}}=\bigcap _{\sigma \in {\mathfrak {S}}} \overline{{{\textbf {F}}}}_{\varphi ,\sigma }\cap {{\textbf {F}}}= \bigcap _{\sigma \in {\mathfrak {S}}}{{\textbf {F}}}_{\varphi ,\sigma }\cap {{\textbf {F}}};$ because the topologies induced by Q and ${\mathscr {Q}}$ agree on ${{\textbf {F}}}.$ On the other hand, let $\sigma \in {\mathfrak {S}}$ be fixed. ${{\textbf {F}}}$ being ${\mathscr {Q}}$-compact, let $a,b\in {{\textbf {F}}}$ be such that $Q_{\varphi ,\sigma }(a-b)=\sup _{x,y\in {{\textbf {F}}}}Q_{\varphi ,\sigma }(x-y).$ Then, $a\notin B^{\prime }_{\varphi ,\sigma }(x,{{\textbf {r}}}_{\varphi ,\sigma })\supset {{\textbf {F}}}_{\varphi ,\sigma }$ which is absurd. Hence, necessarily ${{\textbf {F}}}=\lbrace {{\textbf {x}}}\rbrace .$ $\square$

Remark 3.12

Since $\sigma$-extreme left amenability $\Rightarrow$ strong left amenability, then Theorem 3.11 extends [8, Theorem 3.1] and answers [8, Question 5].

Corollary 3.13

Let S be a semitopological semigroup such that the minimal right ideals of $S^{\text {LUC}(S)}$ are closed. Then, LUC(S) is left amenable if,  and only if,  S satisfies the fixed point property (F).

Proof

In fact, for such a semigroup S, LUC(S) is left amenable if and only if LUC(S) is strongly left amenable, see [2, Theorem 3.5.1]. $\square$

Corollary 3.14

Let S be a compact semitopological semigroup. Then, LUC(S) is left amenable if,  and only if,  S satisfies the fixed point property (F).

Proof

In fact, $S^{\text {LUC}(S)}\simeq S$ and sS is closed for all $s\in S.$ $\square$

4 Remarks and open problem

Remark 4.1

Theorem 3.11 answers Problem 1 (see Sect. 1) for strongly left amenable semitopological semigroups, and affirmatively [8, Question 5].

Remark 4.2

It is worth to point out that for a semitopological semigroup S,  strong left amenability property is not equivalent to possessing the fixed point property (F), because there are semigroups, for example the bicyclic semigroup $S_1$ (see Sect. 1) and the additive topological group ${\mathbb {Q}}$ of the rationals, that satisfy the Schauder fixed point property without being strongly left amenable.

Remark 4.3

Let SLA denote the class of all strongly left amenable semitopological semigroups, LA that of all left amenable semitopological semigroups. We have the following relationships: $\Sigma$-LA $\subsetneq$ SLA $\subsetneq$ LA. In fact, inclusions are trivial. The commutative semigroup S generated by two continuous commuting self-mappings of the unit interval $I=[0,1]$ constructed by Boyce [3] sits in LA$\setminus$ SLA, because it fails to satisfy the fixed point property $({{\textbf {F}}}),$ see Theorem 3.11; on the other hand, any infinite compact topological group G,  e.g. the unit sphere ${\mathbb {S}}^1,$ sits in SLA$\setminus \Sigma$-LA. In fact, such a group is clearly strongly left amenable and it cannot be in $\Sigma$-LA because $S^{\text {LUC}(G)}\simeq G$ and the only left ideal group in G is G itself. So to answer the question on what amenability property should possess a semigroup to satisfy the Schauder fixed point property, we need to find a certain amenability property $({{\textbf {P}}})$ such that SLA $\subsetneq ({{\textbf {P}}})\subsetneq$ LA.

The bicyclic semigroup $S_1=\langle e,a,b: ab=e\rangle$ satisfies the fixed point property $({{\textbf {F}}}),$ it is amenable but it is not strongly left amenable. In fact, if it were strongly left amenable then, see [3, Theorem 3.5.5], as a left amenable discrete semigroup it would be n-extremely left amenable (for some n) and, therefore, the factor semigroup $S_1/(r)$ would be a group of order n,  see [5, Theorem 5.2] and [11, Theorem 3.3.6]. However this is not possible because for each $s\in S_1,$ if we let $\bar{s}$ denote its equivalence class then we have $\bar{s}=\lbrace s\rbrace$ and $S_1$ is infinite, which is a contradiction. Hence, $S_1\in$ LA$\setminus \Sigma$-LA.

Question: Is there an amenability property (P) of $S_1$ with SLA $\subsetneq (P)\subsetneq$ LA?