Vector valued invariant means, complementability and almost constrained subspaces

Abstract

In this paper we will study a connection between some generalization of AC-subspaces, vector valued invariant $\lambda$-means and $\lambda$-complementability.

1 Introduction

Invariant means on amenable groups are an important tool in many parts of Mathematics, especially in Harmonic analysis (see [10, 11]). For the basic properties of invariant means, we refer the reader to [10]. 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, 6, 7, 20]).

The space of all bounded functions from a set S into a Banach space X is denoted by $\ell _\infty (S,X)$. Let us recall the definition of an amenable semigroup (see [5]).

Definition 1.1

A semigroup $(S,+)$ is called left [resp. right] amenable if and only if there exists a linear map $L:\ell _\infty (S,{\mathbb {R}}) \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \inf f(S)&\le L(f)\le \sup f(S),\ f\in \ell _\infty (S,{\mathbb {R}})\\ L(_a f)&= L(f),\ a\in S,\ f\in \ell _\infty (S,{\mathbb {R}}),\\ [L(f_a)&= L(f),\ a\in S,\ f\in \ell _\infty (S,{\mathbb {R}})], \end{aligned}$$

where

$$\begin{aligned} a f(x)&= f(a+x),\ a,x\in S,\ f\in \ell _\infty (S,{\mathbb {R}}),\\ [f_a(x)&= f(x+a),\ a,x\in S,\ f\in \ell _\infty (S,{\mathbb {R}})]. \end{aligned}$$

If both left and right invariant means exist, then there exists a two-sided invariant mean and S is called amenable.

Remark 1.2

In the above definition the first condition

$$\begin{aligned} \inf f(S)\le L(f)\le \sup f(S) \end{aligned}$$

is equivalent to conditions $L(1_S)=1$ and $|L(f)|\le \Vert f \Vert :=\sup |f(S)|$.

It is known that every commutative semigroup is amenable (an easy consequence of the Markov–Kakutani fixed point theorem, see [18, Theorem 5.23]).

Certain generalizations of invariant means were investigated for vector-valued functions in [6] and the existence thereof appears to be related to properties such as reflexivity.

Some generalized definition of an invariant mean has been used by many mathematicians as a folklore (e.g. by Pełczyński [17]). The explicit form of this definition can be found e.g. in the work of Ger [7].

Definition 1.3

Let $(S,+)$ be a left [right] amenable semigroup and X be a Banach space. A linear map ${M:\ell _{\infty } (S,X) \rightarrow X}$ is called a left [right] X-valued invariant mean if

$$\begin{aligned} \Vert M \Vert&\le 1,\\ M(c 1_S)&=c,\ c\in X,\\ M(_a f)&=M(f),\ a\in S,f\in \ell _\infty (S,X),\\ [M(f_a)&=M(f),\ a\in S,f\in \ell _\infty (S,X),] \end{aligned}$$

where

$$\begin{aligned} {}_a f(x)&= f(a+x),\ a,x\in S,\ f\in \ell _\infty (S,X),\\ [f_a(x)&= f(x+a),\ a,x\in S,\ f\in \ell _\infty (S,X).] \end{aligned}$$

If M is a left and right invariant mean, then M is called an X-valued invariant mean.

If in the above definition the norm of map M is equal to at most $\lambda \ge 1$, then M is called an X-valued invariant $\lambda$-mean.

The existence of such invariant means for a fixed Banach space and for all amenable semigroups has been studied by Domecq [3, Theorem 1 and 2], Kania [12], Goucher and Kania [9], and by the author in [15, 16]. The most general result obtained is as follows (recall that for $\lambda \ge 1$ a subspace X of a Banach space Y is called $\lambda -$complemented if there is a projection of norm $\lambda$ from Y on X).

Corollary 1.4

(see [16, Corollary 3.3]) Let X be an infinite-dimensional Banach space, $\lambda \ge 1$. The following assertions are equivalent:

• (1) X is complemented in $X^{**}$ by a projection of norm at most $\lambda$;

• (2) for every amenable semigroup S there exists an X-valued invariant $\lambda$-mean on S;

• (3) for any commutative cancellative semigroup S of torsion-free rank $\delta$, ${\text {dens}}{X^{**}}=\max (\delta , \omega )$, there exists an X-valued invariant $\lambda$-mean on S.

It is also demonstrated ([9, 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.

Definition 1.5

A Banach space X is said to have the finite-infinite intersection property ($IP_{f,\infty }$) if every family of closed balls in X with empty intersection contains a finite subfamily with empty intersection.

It is known that dual spaces and their constrained subspaces have $IP_{f,\infty }$. It can be shown (see [8]) that a Banach space X has $IP_{f,\infty }$ if and only if any family of closed balls centered at points of X that intersects in $X^{**}$ also intersects in X. With this in mind Bandyopadhyay and Dutta ([2]) defined

Definition 1.6

A subspace X of a Banach space Y is said to be an almost constrained (AC) subspace of Y if any family of closed balls centered at points of X that intersects in Y also intersects in X.

Clearly, any constrained subspace of a Banach space is an AC-subspace. In the case of $IP_{f,\infty }$, whether the converse is also true remains an open question. However, they give an example (from [13]) that an AC-subspace need not, in general, be constrained.

In this paper we will study a connection between some generalization of AC-subspaces, vector valued invariant $\lambda$-means and $\lambda$-complementability.

2 Preliminaries

First we recall the definition of torsion-free rank (see [4]).

Definition 2.1

Let S be a commutative cancellative semigroup. A set $A\subset S$ is independent if $\sum _{i=1}^n k_i a_i=\sum _{i=1}^n m_i a_i$ for any $n\in {\mathbb {N}}$ and $a_i\in A$, $k_i,m_i \in {\mathbb {N}}_0$, $i\in \{ 1,\ldots ,n \}$ implies $k_i=m_i$ for $i\in \{ 1,\ldots ,n \}$.

Let further ${\mathcal {A}}_0$ 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 ${\mathcal {A}}_0$ is called the torsion-free rank of S and is denoted by $r_0 (S)$ (all the sets in ${\mathcal {A}}_0$ have the same cardinal number).

The density character of a Banach space X, denoted ${\text {dens}}X$, is the smallest cardinal $\kappa$ for which X has a dense subset of cardinality $\kappa$.

We will also require the version of the Principle of Local Reflexivity due to Lindenstrauss and Rosenthal [14]. We will identify any element of a Banach space X with its canonical image in $X^{**}$.

Theorem 2.2

Let X be a Banach space. Then for every finite-dimensional subspace $F\subset X^{**}$ and each $\varepsilon \in (0, 1]$ there exists a linear map $P_F^\varepsilon :F\rightarrow X$ such that

• (1) $(1-\varepsilon )\Vert x \Vert \le \Vert P_F^\varepsilon (x) \Vert \le (1+\varepsilon )\Vert x \Vert$, $x\in F$;

• (2) $P_F^\varepsilon (x) =x$ for $x\in F\cap X$.

We also recall some result from [16].

Theorem 2.3

(see [16, Theorems 3.1 and 3.2]) Let X be an infinite-dimensional Banach space, $\lambda \ge 1$, S be a commutative cancellative semigroup of torsion-free rank $\delta$, $\gamma = \max (\delta ,\omega )$. If there exists an X-valued invariant $\lambda$-mean $M:\ell _{\infty } (S,X)\rightarrow X$, then for every subspace E of $X^{**}$ such that ${\text {dens}}E=\gamma$ there exists a linear map $P:E\rightarrow X$ such that $\Vert P \Vert \le \lambda$ and $P(x)=x$ for $x\in X\cap E$.

Now, we introduce generalizations of the finite-infinite intersection property and almost constrained subspaces.

Definition 2.4

Let $\gamma \ge \omega$ be a cardinal number, $\lambda \ge 1$. We say that a Banach space X has the $\lambda -$finite-$\gamma$ intersection property ($\lambda -IP_{f,\gamma }$) if for every collection ${\mathcal {F}} = \{ B_X (x_\alpha ,r_\alpha ):\alpha < \gamma \}$ of closed balls in X such that $\bigcap \nolimits _{\alpha <\gamma } B_X(x_\alpha ,\lambda r_\alpha ) = \emptyset$, there exists a finite set $J\subset \gamma$ such that $\bigcap \nolimits _{\alpha \in J } B_X(x_\alpha ,r_\alpha ) = \emptyset$.

If X has $\lambda -IP_{f,\gamma }$ for every cardinal number $\gamma$, then X is said to haves the $\lambda$ -finite-infinite intersection property ($\lambda -IP_{f,\infty }$).

Definition 2.5

Let $\gamma \ge \omega$ be a cardinal number, $\lambda \ge 1$. A subspace X of a Banach space Y is said to be a $(\lambda ,\gamma )$ almost constrained ($(\gamma ,\lambda )-AC$) subspace of X if for every family ${\mathcal {F}} = \{ B_X (x_\alpha ,r_\alpha ):\alpha < \gamma \}$ of closed balls in X, if ${\mathcal {F}}$ intersects in Y, then $\{ B_X (x_\alpha ,\lambda r_\alpha ):\alpha < \gamma \}$ intersects in X.

If X is a $( \lambda ,\gamma )-AC$ subspace of Y for every cardinal number $\gamma$, then X is said to be a $\lambda$ -almost constrained ($\lambda -AC$) subspace of Y.

Lemma 2.6

Let X, Y be Banach spaces, $X\subset Y$. Then X is a $(2+\varepsilon )-AC$ subspace of Y for every $\varepsilon >0$.

Proof

Let ${\mathcal {F}} = \{ B_X (x_\alpha ,r_\alpha ): \alpha \in I \}$ intersects in Y. Let further $r:=\inf \{ r_\alpha : \alpha \in I \}$ and $\varepsilon >0$. If $r=0$, then there exists a sequence $(x_{\alpha _n})_{n\in {\mathbb {N}}}$ such that $\lim \nolimits _{n\rightarrow \infty } r_{\alpha _n} =0$. Hence $(x_{\alpha _n})_{n\in {\mathbb {N}}}$ is convergent to some $x\in X$. We have

$$\begin{aligned}&\Vert x_\alpha -x \Vert \le \Vert x_\alpha - x_{\alpha _n} \Vert +\Vert x_{\alpha _n} -x \Vert \le r_\alpha +r_{\alpha _n} +\Vert x_{\alpha _n} -x \Vert ,\ n\in {\mathbb {N}}, \end{aligned}$$

so

$$\begin{aligned}&\Vert x_\alpha -x \Vert \le r_\alpha +\lim \limits _{n\rightarrow \infty } (r_{\alpha _n} +\Vert x_{\alpha _n} -x \Vert ) = r_\alpha , \end{aligned}$$

which means that $x\in \bigcap {\mathcal {F}}$.

Assume that $r>0$. Let $x_{\alpha _0}$ be such that $r_{\alpha _0} <r(1+\varepsilon )$ and $y\in Y$ be a point from the intersection of ${\mathcal {F}}$ in Y. Then

$$\begin{aligned} \Vert x_\alpha -x_{\alpha _0} \Vert&\le \Vert x_\alpha -y \Vert + \Vert y -x_{\alpha _0} \Vert \le r_\alpha + r_{\alpha _0}\\&\le r_\alpha + (1+\varepsilon )r\le r_\alpha + (1+\varepsilon )r_\alpha = (2+\varepsilon )r_\alpha . \end{aligned}$$

$\square$

Lemma 2.7

Let X, Y be Banach spaces, $\lambda \ge 1$. If X is $\lambda$-complemented in Y, then X is a $\lambda -AC$ subspace of Y.

Proof

Let $P:Y\rightarrow X$ be a norm $\lambda$ projection, $\gamma$ be a cardinal number, ${\mathcal {F}} = \{ B_X (x_\alpha ,r_\alpha ): \alpha < \gamma \}$ be a family of closed balls in X such that ${\mathcal {F}}$ intersects in Y. Let $z\in Y$ be any point from this intersection. Then

$$\begin{aligned}&\Vert P(z) -x_\alpha \Vert = \Vert P(z-x_\alpha ) \Vert \le \lambda \Vert z-x_\alpha \Vert \le \lambda r_\alpha ,\ \alpha < \gamma , \end{aligned}$$

so $P(z) \in \bigcap \nolimits _{\alpha < \gamma } B_X (x_\alpha ,\lambda r_\alpha )$. $\square$

Lemma 2.8

Let X be a Banach space, $\lambda \ge 1$, $\gamma \ge \omega$ be a cardinal number. Then the following are equivalent:

• (a) X has $\lambda -IP_{f,\gamma }$.

• (b) For every Banach space Y, $X\subset Y\subset X^{**}$, X is a $(\lambda ,\gamma ) -AC$ subspace of Y.

• (c) X is a $(\lambda ,\gamma ) -AC$ subspace of $X^{**}$.

Proof

(a)$\Rightarrow$(b) Assume that X has $\lambda -IP_{f,\gamma }$. Let Y be a Banach space such that $X\subset Y\subset X^{**}$. Let further ${\mathcal {F}} = \{ B_X (x_\alpha ,r_\alpha ): \alpha < \gamma \}$, $z\in Y\subset X^{**}$ be a point from the intersection of ${\mathcal {F}}$ in Y. Let J be a finite subset of $\gamma$ and $\varepsilon >0$. From the Principle of Local Reflexivity there exists a linear map $P:{\text {span}}\left\{ \{ x_\alpha :\alpha \in J \} \cup \{ z\} \right\} \rightarrow X$ such that $P(x_\alpha )=x_\alpha$, $\alpha \in J$ and $\Vert P \Vert \le 1+\varepsilon$.

$$\begin{aligned}&\Vert P(z)-x_\alpha \Vert =\Vert P(z-x_\alpha ) \Vert \le (1+\varepsilon ) \Vert z-x_\alpha \Vert \le (1+\varepsilon )r_\alpha . \end{aligned}$$

Hence $P(z)\in B_X (x_\alpha ,(1+\varepsilon )r_\alpha )$, $\alpha \in J$ so $\bigcap \nolimits _{\alpha \in J } B_X (x_\alpha ,(1+\varepsilon )r_\alpha )\ne \emptyset$. Thus, from $\lambda -IP_{f,\gamma }$ we have

$$\begin{aligned}&\bigcap \limits _{\alpha< \gamma } B_X (x_\alpha ,\lambda r_\alpha ) = \bigcap \limits _{\alpha < \gamma } \bigcap \limits _{n\in {\mathbb {N}}} B_X (x_\alpha ,\frac{n+1}{n}\lambda r_\alpha ) \ne \emptyset . \end{aligned}$$

(b)$\Rightarrow$(c) is trivial.

(c)$\Rightarrow$(a) Let ${\mathcal {F}} = \{ B_X (x_\alpha ,r_\alpha ): \alpha < \gamma \}$ be such that for every finite subset J of $\gamma$ we have $\bigcap \nolimits _{\alpha \in J } B_X (x_\alpha ,r_\alpha )\ne \emptyset$. Then for every finite subset J of $\gamma$ we have $\bigcap \nolimits _{\alpha \in J } B_{X^{**}}(x_\alpha ,r_\alpha )\ne \emptyset$ and since $X^{**}$ is a dual space, it has $IP_{f,\infty }$, then $\bigcap \nolimits _{\alpha < \gamma } B_{X^{**}}(x_\alpha ,r_\alpha )\ne \emptyset$. Since X is a $(\lambda ,\gamma )-AC$ subspace of $X^{**}$, we obtain that $\bigcap \nolimits _{\alpha < \gamma } B_X ( x_\alpha ,\lambda r_\alpha )\ne \emptyset$. $\square$

Lemma 2.9

Let X, Y be Banach spaces, $\lambda \ge 1$. If X is a $(\lambda ,{\text {dens}}(X) )-AC$ subspace of Y, then X is a $\lambda -AC$ subspace of Y.

Proof

Assume that X is a $(\lambda ,{\text {dens}}(X))-AC$ subspace of Y. Let $D\subset X$ be a dense set in X such that $\vert D\vert ={\text {dens}}(X)$. For every $x_{\alpha }\in X$ and $r_\alpha >0$ there exist $(x_{\alpha ,n})_{n\in {\mathbb {N}}} \subset D$, $(q_{\alpha ,n})_{n\in {\mathbb {N}}} \subset {\mathbb {Q}}_+$ such that $\lim \nolimits _{n\rightarrow \infty } x_{\alpha ,n} =x_\alpha$, $q_{\alpha ,n}\in [r_\alpha +\Vert x-x_{\alpha ,n} \Vert ,r_\alpha +\Vert x-x_{\alpha ,n} \Vert +\frac{1}{n}]$. Hence

$$\begin{aligned}&\bigcap \limits _{\alpha \in I} B_X (x_\alpha , \lambda r_\alpha ) = \bigcap \limits _{\alpha \in I} \bigcap \limits _{n\in {\mathbb {N}}} B_X (x_{\alpha ,n}, \lambda q_{\alpha ,n} )\ne \emptyset , \end{aligned}$$

because the intersection on the right side contains at most ${\text {dens}}(X)$ different balls. $\square$

3 Main results

We will use the following notation:

Let X be a subspace of a Banach space Y, $\lambda \ge 1$. For all $y\in Y$

$$\begin{aligned} \beta _\lambda (y):= \bigcap _{x\in X} B_X (x,\lambda \Vert y-x \Vert ). \end{aligned}$$

Clearly $\beta _\lambda (y) = \{ y \}$ for $y\in X$.

$$\begin{aligned}&O_\lambda (X,Y) := \{ y\in Y: \lambda \Vert y+x \Vert \ge \Vert x \Vert \text { for all } x\in X \}. \end{aligned}$$

$O_\lambda (X,X^{**})$ is denoted by $O_\lambda (X)$.

Clearly if $x\in O_\lambda (X,Y)$ and $\alpha \in {\mathbb {K}}$ (${\mathbb {K}}$ is a scalar field of X and Y), then $\alpha x\in O_\lambda (X,Y)$.

Theorem 3.1

For a subspace X of a Banach space Y and $\lambda \ge 1$ the following are equivalent:

• (a) X is a $\lambda -AC$ subspace of Y.

• (b) For every $y\in Y$ $\beta _\lambda (y)\ne \emptyset$.

• (c) For every $y\in Y$ there exist $x\in X$ and $u\in O_\lambda (X,Y)$ such that $y=x+u$.

• (d) For every subspace Z of Y, $X\subset Z$, ${\text {dim}} (Z/X)=1$, X is $\lambda -$complemented in Z.

• (e) There exists a map (not necessarily linear) $P:Y\rightarrow X$ such that

  • $P(\alpha y)=\alpha P(y)$, $\alpha \in {\mathbb {K}}$, $y\in Y$;

  • $P(x+y)=x+P(y)$, $x\in X$, $y\in Y$;

  • $\Vert P(y) \Vert \le \lambda \Vert x \Vert$, $y\in Y$.

Proof

(a)$\Rightarrow$(b) is trivial.

(b)$\Rightarrow$(c) Let $y\in Y$, $z\in \beta _\lambda (y)$. Then

$$\begin{aligned}&\Vert z-x \Vert \le \lambda \Vert y-x \Vert ,\ x\in X, \end{aligned}$$

so

$$\begin{aligned}&\Vert x \Vert \le \lambda \Vert y-z+x \Vert , x\in X. \end{aligned}$$

Hence $u:=y-z\in O_\lambda (X,Y)$.

(c)$\Rightarrow$(d) Let Z be a subspace of Y such that $X\subset Z$ and ${\text {dim}} (Z/X)=1$. We have $Z={\text {span}}{\{ X\cup \{y_0 \}\}}$ for some $y_0 \in Y$. From (c) there exist $x_0\in X$, $u_0\in O_\lambda (X,Y)$ such that $y_0 =x_0 +u_0$. Hence $Z=X\oplus {\mathbb {K}}u_0$. Let $P:Z\rightarrow X$ be a map given by $P(x+\alpha u_0)=x$, $x\in X$, $\alpha \in {\mathbb {K}}$. Since

$$\begin{aligned}&\Vert x+\alpha u_0 \Vert =\Vert x \Vert \le \lambda \Vert x+\alpha u_0 \Vert ,\ x\in X,\ \alpha \in {\mathbb {K}}, \end{aligned}$$

P is a norm $\lambda$ projection.

(d)$\Rightarrow$(b) Let $y\in Y{\setminus } X$, $Z={\text {span}}\{ X\cup \{ y \} \}$. From (d) there exists a norm $\lambda$ projection $P:Z\rightarrow X$. Hence

$$\begin{aligned}&\Vert P(y)-x \Vert =\Vert P(y-x) \Vert \le \lambda \Vert y-x \Vert ,\ x\in X, \end{aligned}$$

so $P(y)\in \beta _\lambda (y)$.

(c)$\Rightarrow$(e) Let $z\in O_\lambda (X,Y)$, $X_z:= X\oplus {\mathbb {K}}z$ and $P_z:X_z\rightarrow X$ be a norm $\lambda$ projection. From (c) we have $Y=\bigcup \nolimits _{z\in O_\lambda (X,Y)} X_z$. For $z_1,z_2\in O_\lambda (X,Y)$ we also have $X_{z_1} = X_{z_2}$ or $X_{z_1} \cap X_{z_2} = X$, so if $X_{z_1} = X_{z_2}$, then we can assume that $P_{z_1}=P_{z_2}$. Then we can define $P:Y\rightarrow X$ by the formula $P(y):=P_z (y)$ if $y\in X_z$. It is easy to observe that $\Vert P(y) \Vert \le \lambda \Vert y \Vert$, $y\in Y$ and $P(\alpha y)=\alpha P(y)$, $\alpha \in {\mathbb {K}}$, $y\in Y$. We also have

$$\begin{aligned} P(x+y)&=P_z (x+y)=P_z (x) +P_z (y)=x+P_z(y)\\&=x+P(y),\ x\in X,\ y\in X_z. \end{aligned}$$

(e)$\Rightarrow$(a) Let $x_\alpha \in X$, $r_\alpha >0$ for $\alpha \in I$, a family $\{ B_Y (x_\alpha ,r_\alpha ):\alpha \in I \}$ intersects in Y, and let $z\in \bigcap \nolimits _{\alpha \in I} B_Y (x_\alpha ,r_\alpha )$. Let $P:Y\rightarrow X$ be a map from (e). Then

$$\begin{aligned}&\Vert P(z)-x_\alpha \Vert =\Vert P(z-x_\alpha ) \Vert \le \lambda \Vert z-x_\alpha \Vert \le \lambda r_\alpha ,\ \alpha \in I, \end{aligned}$$

so $P(z)\in \bigcap \nolimits _{\alpha \in I } B_X (x_\alpha ,\lambda r_\alpha )$. $\square$

From Theorems 2.3, 3.1 (d) and Lemma 2.8 we get

Corollary 3.2

Let X be a Banach space, $\lambda \ge 1$, S be a commutative cancellative semigroup of torsion-free rank $\delta$, ${\text {dens}}(X)=\max \{ \delta ,\omega \}$. If there exists an X-valued invariant $\lambda$-mean $M:\ell _{\infty } (S,X)\rightarrow X$, then X has $\lambda -IP_{f,\infty }$.

Remark 3.3

It is known (Sobczyk [19]) that there is no projection of $\ell ^{\infty }$ on $c_0$. On the other hand for every separable subspace Y of $\ell ^{\infty }$, $c_0 \subset Y$, there exists a projection of norm not greater than 2 from Y onto $c_0$ (and 2 is a best constant for the norm of this projection). From Lemma 2.8, Theorems 3.1, 2.3 we obtain that $c_0$ has $2-IP_{f,\infty }$, so in general $\lambda -IP_{f,\infty }$ and $\lambda$-complementability aren’t equivalent. Moreover, there doesn’t exist any $c_0$-valued invariant $\lambda$-mean from any semigroup of countable non-zero torsion-free rank for any $\lambda \ge 1$ (so the converse of Theorem 2.3 doesn’t hold). Indeed, if $M:\ell _{\infty } (S_\omega ,c_0)\rightarrow c_0$ is a $c_0$-valued invariant $\lambda$-mean, then we define the map $P:\ell ^\infty \rightarrow c_0$ given by $P(y):=M({\widetilde{y}})$, where ${\widetilde{y}}(A)=\sum \nolimits _{n\in A} y_n e_n$, $y\in \ell ^\infty$, A is a finite subset of ${\mathbb {N}}$. It is easy to see that P is a norm $\lambda$ projection which is impossible.

Remark 3.4

Now we can see that we have the following implications and open problems: