Invariant vector means and complementability of Banach spaces in their second duals

Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of X∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{**}$$\end{document}. We then show that X is complemented in X∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{**}$$\end{document} if and only if there exists an invariant mean M:ℓ∞(S,X)→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:\ell _\infty (S,X)\rightarrow X$$\end{document}. This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).


Introduction
Invariant means on amenable groups are an important tool in many parts of Mathematics, especially in Harmonic analysis (see [8,9]). For the basic properties of invariant means, we refer the reader to [8]. Invariant means and their generalizations for vector-valued functions also play an important role in the stability of functional equations and selections of set-valued functions (see [1,5,6,16]).
The space of all bounded functions from a set S into a Banach space X is denoted by ∞ (S, X). Let us recall the definition of an amenable semigroup (see [3]). Theorem 1.4. Let X be a Banach space and λ ≥ 1. Then the following assertions are equivalent.
1. X is complemented in X * * by a projection of norm at most λ; 2. for every amenable semigroup S there exists an X-valued invariant λmean on S; 3. for every commutative semigroup S there exists an X-valued invariant λ-mean on S; 4. for every free commutative group G of rank |X * * | there exists an X-valued invariant λ-mean on G; 5. there exists an X-valued invariant λ-mean on the additive group of X * * .
It is also demonstrated ( [7, Remark 1.1]) that there exists a commutative noncancellative semigroup S (that could be chosen as large as one wishes) such that there exists an X-valued invariant mean on S.
In this paper we will prove that if X is a Banach space and there exists an invariant X-valued mean on any arbitrary commutative cancellative semigroup S of torsion-free rank dens X * * , then X is λ-complemented in X * * .

Preliminaries
First we recall the definition of torsion-free rank (see [2]).
Let further A 0 be the family of all independent sets L in S consisting only of elements whose order is infinite and such that L is maximal with respect to these properties. The cardinal number of any set in A 0 is called a torsion-free rank of S and is denoted by r 0 (S) (all the sets in A 0 have the same cardinal number).

R. Lukasik AEM
The density character of a Banach space X, denoted dens X, is the smallest cardinal κ for which X has a dense subset of cardinality κ. B of X such that |B| = dens X, span B = X, span (B∩Y ) = Y . Moreover, we can assume that the norm of each x ∈ B is equal to 1.
Proof. 1. We observe that 2. Let Y be a closed subspace of X, D be a dense subset of X such that |D| = dens X and K be a dense subset of Y such that |K| ≤ dens X. Let further Let further B 1 be a maximal linearly independent subset of D 1 and B be a maximal linearly independent subset of We also note that the norm of each x ∈ B is equal 1.
We will also require the version of the principle of local reflexivity due to Lindenstrauss and Rosenthal [11]. We denote by κ : X → X * * the canonical embedding from a Banach space X into the second dual.
It is a standard fact that subgroups and quotients of amenable groups are amenable. Using exactly the same ideas one can prove that if a Banach space admits an invariant mean with respect to a group, then it also does so with respect to subgroups and quotients of the group (see [ (subsemigroups of an amenable group need not be amenable) but first we must say something about normal semigroups and quotients of semigroups (see also [17]). Let (S, +) be a semigroup, G be a subsemigroup of S. Then G is called a normal subsemigroup if x + G = G + x for every x ∈ S. Of course in a commutative semigroup each subsemigroup is normal. Let further S be a semigroup and G be a normal subsemigroup of S. We define the quotient semigroup S/G : It is easy to notice that for any g ∈ G the set [g]G ∼ is a neutral element of S/G. Moreover, if G is a group, then G is a neutral element of S/G.

Lemma 2.4. Let S be an amenable semigroup and G be a normal subsemigroup of S. If there exists an
Proof. We define a map M 1 : ∞ (S/G, X) → X by the formula

Main results
Throughout this section we fix an infinite-dimensional Banach space X, λ ≥ 1. Let γ be a cardinal number. We denote by S γ the commutative semigroup comprising all finite subsets of γ endowed with the operation of taking the union of sets. It is easy to observe that |S γ | = γ. Theorem 3.1. Let γ be an infinite cardinal number. If there exists an X-valued invariant λ-mean M : ∞ (S γ , X) → X, then for every subspace E of X * * such that dens E = γ there exists a linear map P : E → X such that P ≤ λ and Proof. Let K be a scalar field of X. In view of Lemma 2.2 there exists a linearly independent subset B of E such that span B = E, span (B ∩κ(X)) = κ(X)∩E, |B| = dens E = γ. Let T : γ → B be a bijection and M : ∞ (S γ , X) → X be an X-valued invariant λ-mean .
For A ∈ S γ we define ε A := 1 |A|+1 and P εA spanT (A) is a fixed linear operator satisfying the conditions of Theorem 2.3.

R. Lukasik AEM
We define the map P : E → X in the following way (on the dense subspace spanB, the map is simply continuously extended to the closure): for x ∈ spanB we put P ( For x, y ∈ spanB and α ∈ K we notice that φ αx+y = αφ x + φ y . Thus so P is linear on spanB. Since A 0 is arbitrary, we get P (x) ≤ λ x . Moreover, if x ∈ κ(X), then from the properties of B we get x 1 , . . . , x n ∈ κ(X) and Hence Proof. First we observe that we can assume that S contains only elements of infinite order. Indeed the set G of all elements of finite order is a group and a torsion-free rank of S/G is equal to γ. In view of Lemma 2.4 there exists an X-valued invariant λ-mean on S/G. Let A ⊂ S be a maximal linearly independent set. Hence |A| = δ.
• First assume that |A| = γ and let A = {x α : α < γ}. For each x ∈ S we define a set First, we show that the above set is well-defined. If there exist k, m ∈ N, As A is linearly independent, we have I \ J = J \ I = ∅, which means that I = J. Thus we get that km i = mk i for i ∈ {1, . . . , n}, so D x is well-defined. We define a map ϕ : ∞ (S γ , X) → ∞ (S, X) by the formula It is easy to observe that ϕ is linear, ϕ(f ) = f for f ∈ ∞ (S γ , X) and ϕ(c1 Sγ ) = c1 S for c ∈ X.
Let M S : ∞ (S, X) → X be an X-valued invariant λ-mean. We define M : ∞ (S γ , X) → X by the formula From the properties of ϕ we obtain that M is linear, M (c1 Sγ ) = c for c ∈ X, and M ≤ M S ≤ λ. Now we show that M is invariant. Let f ∈ ∞ (S γ , X) and A ∈ S γ . Since A = {α 1 , . . . , α n }, from the invariance on each singleton {α i } we obtain Hence we need to prove the invariance on each singleton, so we can assume that A = {β} for some β < γ.
Since S is cancellative, from (3.1) we obtain that Let g ∈ ∞ (S, X) be such that g(x) = 0 for x ∈ S \ Z. From (3.2) we get Hence • Now assume that |A| < γ.
Since S can be embedded in a group, for each x ∈ S there exist k(x) ∈ N, It is easy to observe that ϕ is linear, ϕ(f ) ≤ f for f ∈ ∞ (S ω , X) and ϕ(c1 Sω ) = c1 S for c ∈ X. Let M S : ∞ (S, X) → X be an X-valued invariant λ-mean. We define M : ∞ (S ω , X) → X by the formula M (f ) := M S (ϕ(f )), f ∈ ∞ (S ω , X).
From the properties of ϕ we obtain that M is linear, M (c1 Sγ ) = c for c ∈ X, and M ≤ λ. Now we show that M is invariant. Let f ∈ ∞ (S ω , X) and A ∈ S ω . Similarly as in the previous case we need only to prove the invariance on each singleton, so we can assume that A = {β} for some β ∈ ω. Let Z := x ∈ S : |k 1 (x)| < βk(x) .