# Intersection Properties of Balls in Banach Spaces

Authors: C. R. Jayanarayanan
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce a weaker notion of central subspace called almost central subspace, and we study Banach spaces that belong to the class (GC), introduced by Veselý (1997). In particular, we prove that ifY is an almost central subspace of a Banach space X such that Y is in the class (GC), then Y is a central subspace of X**. We also prove that if Y is a semi M-ideal in a Banach space X such that Y⊥⊥ is an almost central subspace of X**, then Y is an M-ideal in X. Certain stability results for quotient spaces, injective tensor product spaces, and polyhedral direct sums of Banach spaces are also derived.


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## Body

## 1. Introduction

In [1], Veselý studied a new class of Banach spaces, namely, the class (GC), which were defined in terms of the existence of weighted Chebyshev centers (see below for definition and details). In the same paper, he characterized such spaces using intersection properties of balls. In [2], Bandyopadhyay and Rao considered some general results about the class (GC) by introducing a new class of subspaces called “central subspaces” of Banach spaces. Using this concept, they characterized the class (GC) and produced several examples of Banach spaces which belong to the class (GC) (see [2] for details). In this paper, we introduce and study a weaker notion of central subspace called almost central subspace (see Section 2 for definition and details). Using this concept, we obtain some new results about the class (GC) and also about some of the other types of intersection properties of balls studied in the literature.For a Banach spaceX, we denote by BX(x,r) the closed ball in X of radius r>0 around x∈X and by BX the closed unit ball of X. In this paper, we restrict ourselves to real scalars and all subspaces we consider are assumed to be closed. Under the canonical embedding, we will consider X as a subspace of X**. Also, if a Banach space Y is isometric to a subspace of the Banach space X, then, without loss of generality, we will consider Y as a subspace of X. Our notations are otherwise standard. Any unexplained terminology can be found in [3].We now recall the definition of the class (GC) from [1].Definition 1.
LetX be a Banach space. Let a1,…,an∈X and η1,…,ηn>0. Minimizers of the function ϕ:X→ℝ defined by ϕ(x)=max1≤i≤nηi∥x-ai∥ are called weighted Chebyshev centers with the weight η=(η1,…,ηn). Classical Chebyshev centers are the weighted Chebyshev centers with the weight η=(1,…,1).Theorem 2 (see [1, Theorem 2.7]).
For a Banach spaceX and a1,…,an∈X, the following assertions are equivalent.(i)
Ifr1,…,rn>0 and ⋂i=1n‍BX**(ai,ri)≠∅, then ⋂i=1n‍BX(ai,ri)≠∅.(ii)
a1,…,an admits weighted Chebyshev centers for all weights η=(η1,…,ηn), where ηi>0 for all i.Definition 3 (see [1, Definition 2.8]).
One shall denote by (GC) the class of all Banach spacesX such that  for every positive integer n and every a1,…,an∈X, one of the equivalent conditions (i) or (ii) of Theorem 2 is satisfied.Next we recall the definition of a central subspace which generalizes the notion of (GC).Definition 4 (see [2, Definition 2.1]).
LetX be a Banach space. One says that a subspace Y⊆X is a central subspace of X if every finite family of closed balls with centers in Y that intersects in X also intersects in Y.ClearlyX∈(GC) if and only if X is a central subspace of X**. It follows from [2, Proposition 2.2(a)] that Y is a central subspace of a Banach space X if and only if, for any finite set {yi}i=1n⊂Y and x∈X, there exists a y∈Y such that ∥y-yi∥≤∥x-yi∥ for 1≤i≤n.An infinite version of central subspace called almost constrained subspace was investigated in [4, 5].Definition 5.
A subspaceY of a Banach space X is said to be an almost constrained (AC) subspace of X if any family of closed balls centered at points of Y that intersects in X also intersects in Y.We recall that a subspaceY of a Banach space X is called 1-complemented in X if there exists a projection of norm one on X with range Y. One can easily observe that 1-complemented subspaces are AC-subspaces, and hence they are also central subspaces. The notion of an “ideal,” which is weaker than being a 1-complemented subspace, was introduced by Godefroy et al. in [6].Definition 6.
A subspaceY of a Banach space X is said to be an ideal in X if Y⊥ is the kernel of a norm one projection on X*.Clearly1-complemented subspaces are ideals. Also, every Banach space is an ideal in its bidual.  For, if X is a Banach space, then the projection P:X***→X*** defined by P(Λ)=Λ|X is a projection of norm one with kernel X⊥. It is well known that  c0 is an ideal in ℓ∞ but it is not the range of a projection of norm one in ℓ∞.An important concept in theM-structure theory which is closely related to ball intersection properties is the well known concept called M-ideal (see [7] for details).Definition 7.
A projectionP on a Banach space X is called an M-projection (L-projection) if ∥x∥=max{∥Px∥,∥x-Px∥} (∥x∥=∥Px∥+∥x-Px∥) for all x∈X. A subspace Y of X is called an M-summand (L-summand) if it is the range of an M-projection (L-projection). A subspace Y of X is called an M-ideal if Y⊥ is an L-summand in X*. For two Banach spaces X and Z, one denotes by X⨁1Z and X⨁∞Z the direct sum of X and Z, equipped with the ℓ1-norm and supremum norm, respectively.Forn∈ℕ, we recall that a subspace Y of a Banach space X is said to have the (strong) n-ball property if, given n closed balls {BX(ai,ri)}i=1n in X such that ⋂i=1n‍BX(ai,ri)≠∅ and Y⋂‍BX(ai,ri)≠∅ for all i, then Y⋂‍(⋂i=1n‍BX(ai,ri+ɛ))≠∅ for every (ɛ≥0)ɛ>0. It is well known that M-ideals are precisely the subspaces having the n-ball property for all n∈ℕ (see [7, Chapter I, Theorem 2.2] for details). There is a weaker notion of M-ideal called semi M-ideal, which is precisely the subspace having the 2-ball property (see [7, Page 43] for details).In Section2, we define an almost central subspace of a Banach space by a relative intersection property of balls. We will use this to give some sufficient conditions for subspaces to be central. We also consider some general results about the class (GC). In particular, we prove that an almost central subspace of a Banach space X is in the class (GC) if and only if it is a central subspace of X**. We also derive several sufficient conditions for a semi M-ideal to be an M-ideal in terms of these intersection properties of balls.In Section3, we prove the stability of some of the ball intersection properties in quotient spaces, direct sums, vector-valued continuous function spaces, and injective tensor product spaces (see Chapter VIII of [3] for the theory of injective tensor product spaces). In quotient spaces, we prove that for Banach spaces X,Y, and Z with Z⊆Y⊆X, if Y is almost central or ideal in X, then Y/Z is almost central or ideal in X/Z, respectively, and we also prove the converse when X is an L1-predual (that is X*=L1(μ), for some positive measure μ) and Z is an M-ideal in X. In the case of injective tensor product spaces, we show that if Z is an L1-predual space, then, for any almost central subspace Y of a Banach space X, the injective tensor product Z⨂∨Y is an almost central subspace of Z⨂∨X. We also prove that properties of being a central subspace and an AC-subspace are stable under a recently introduced concept called polyhedral direct sums of Banach spaces (see [8, Definition 2.1]).


## 2. Almost Central Subspaces

We begin this section with the definition of an “almost central subspace” of a Banach space which is the generalization of the concept central subspace, defined in [2].Definition 8.
A subspaceY of a Banach space X is called an almost central subspace if, for every finite set {y1,…,yn}⊆Y, x∈X, and ɛ>0, there exists a yɛ∈Y such that ∥yɛ-yi∥≤∥x-yi∥+ɛ for 1≤i≤n.Remark 9.
Clearly central subspaces of Banach spaces are almost central. As in the case of central subspace, it is easy to observe thatY is an almost central subspace of a Banach space X if and only if, for each family {BY(yi,ri)}i=1n of closed balls in Y having nonempty intersection in X, the family {BY(yi,ri+ɛ)}i=1n of closed balls in Y has nonempty intersection in Y for all ɛ>0. On the other hand, by a weak*-compactness argument, it is easy to see that weak*-closed almost central subspace of a dual space is a central subspace. Moreover, if Z is an almost central subspace of a Banach space Y and Y is an almost central subspace of a Banach space X, then Z is an almost central subspace of X.Lemma 10.
LetX be a Banach space and let Y be an ideal in X. Then Y is an almost central subspace of X.Proof.
Let{y1,…,yn}⊆Y, x∈X, and ɛ>0. Choose an η>0 such that η∥x-yi∥≤ɛ for all i∈{1,…,n}. Define F=span{y1,…,yn,x}. Since Y is an ideal in X, by [9, Theorem 1], there exists an operator Tη:F→Y such that
(1)Tη(y)=yfory∈F∩Y,∥Tη∥≤1+η.
Now define yη=Tη(x). Then yη∈Y and for 1≤i≤n,
(2)∥yη-yi∥=∥Tη(x)-Tη(yi)∥≤(1+η)∥x-yi∥≤∥x-yi∥+ɛ.
Hence Y is an almost central subspace of X.Since every Banach space is an ideal in its bidual, we have the following result.Corollary 11.
Every Banach space is almost central in its bidual.Since everyM-ideal is an ideal, by Lemma 10, M-ideals are almost central. We now give an example to show that a semi M-ideal may not be an almost central subspace.Example 12.
Letℓ13 denote the three-dimensional space ℝ3, endowed with the norm ∥x∥=|x(1)|+|x(2)|+|x(3)| for x=(x(1),x(2),x(3))∈ℝ3. Now consider the subspace G of ℓ13 defined as G={(x(1),x(2),-x(1)-x(2)):x(1),x(2)∈ℝ}⊆ℓ13. Then, by [7, Chapter I, Remark 2.3(a)], G is a semi M-ideal in ℓ13. But G is not a central subspace of ℓ13. For, let g1=(-1,-1,2),g2=(-1,2,-1),g3=(2,-1,-1), and let x=(-1,-1,-1). Then g1,g2,g3∈G and x∈ℓ13. Clearly ∥gi-x∥=3 for all i. Suppose there is an α∈G such that ∥α-gi∥≤3 for all i. Then
(3)|α(1)+1|+|α(2)+1|+|α(1)+α(2)+2|≤3,(4)|α(1)+1|+|α(2)-2|+|α(1)+α(2)-1|≤3,(5)|α(1)-2|+|α(2)+1|+|α(1)+α(2)-1|≤3.
But (3) shows that both of α(1) and α(2) cannot be positive. But the symmetric inequalities (4) and (5) rule out other possibilities. Thus G is not a central subspace of ℓ13. Then, by a compactness argument, we can see that G is not an almost central subspace of ℓ13.In [1, Example 5.6], Veselý gave an example of a three-dimensional Banach space X such that C([0,1],X) is not a central subspace of its bidual. Since every Banach space is an ideal in its bidual, the same example shows that an ideal (in particular, an almost central subspace) need not be a central subspace. We now give a sufficient condition for an almost central subspace to be a central subspace.Theorem 13.
LetY be an almost central subspace of a Banach space X such that Y∈(GC). Then Y is a central subspace of X.Proof.
Let{y1,…,yn}⊆Y and x∈X. Since Y∈(GC), by [2, Proposition 2.9], it is enough to show that ⋂i=1n‍BY(yi,∥x-yi∥+ɛ)≠∅ for all ɛ>0.
Now letɛ>0. Choose η>0 such that η∥x-yi∥≤ɛ for all i∈{1,…,n}. Since Y is an almost central subspace of X, there exists a y∈Y such that ∥y-yi∥≤(1+η)∥x-yi∥ for all i∈{1,…,n}. Hence ∥y-yi∥≤∥x-yi∥+ɛ for all i∈{1,…,n}, and the result follows.Our next result gives a sufficient condition for an almost central subspace to be an AC-subspace.Proposition 14.
LetY be an almost central subspace of a Banach space X such that Y is isometric to the range of a projection of norm one in some dual space. Then Y is an AC-subspace of X.Proof.
LetZ be a Banach space and P:Z*→Z* be a projection of norm one such that Y is isometric to range(P). Let ϕ:Y→range(P) be the corresponding onto isometry. Now let {BY(yα,rα)}α∈I be any family of closed balls in Y and x∈X be such that ∥x-yα∥≤rα for all α∈I. Consider the family {BY(yα,rα+ɛ)}α∈I,ɛ>0. Since Y is an almost central subspace of X, any finite collection of balls from this family has nonempty intersection in Y. Hence any finite collection of balls from the family {BZ*(ϕ(yα),rα+ɛ)}α∈I,ɛ>0 has nonempty intersection in range(P). Now, by weak* compactness, there exists a Λ∈Z* such that ∥Λ-ϕ(yα)∥≤rα+ɛ for all α∈I and for all ɛ>0. Hence ∥Λ-ϕ(yα)∥≤rα for all α∈I. Now define y=ϕ-1(P(Λ)). Then, for all α∈I, we have
(6)∥y-yα∥=∥ϕ-1(P(Λ))-ϕ-1(ϕ(yα))∥=∥P(Λ)-ϕ(yα)∥=∥P(Λ-ϕ(yα))∥≤rα.
Hence Y is an AC-subspace of X.We now give a class of Banach spaces where almost central subspaces are central. We recall that a Banach spaceX whose dual X* is isometric to L1(μ) for some positive measure μ is called an L1-predual.Proposition 15.
LetX be an L1-predual and let Y be an almost central subspace of X. Then Y is an L1-predual. In particular, Y is a central subspace of X.Proof.
Let{BY(yi,ri)}i=1n be any family of n balls in Y such that any two of them intersect in Y. Since X is an L1-predual, by [10, Theorem 6.1], there exists an x∈X such that ∥x-yi∥≤ri for all i. Also, since Y is an almost central subspace of X, ⋂i=1n‍BY(yi,ri+ɛ)≠∅ for all ɛ>0. Then, by [10, Lemma 4.2 and Theorem 6.1], it follows that Y is an L1-predual. Now let {BY(yi,ri)}i=1n be a family of n balls in Y that has nonempty intersection in X. It is well known that two balls intersect if and only if the distance between the centers is less than or equal to the sum of the radii. Thus {BY(yi,ri)}i=1n is a pairwise intersecting family in Y. Since Y is an L1-predual, by [10, Theorem 6.1], it follows that {BY(yi,ri)}i=1n intersects in Y. Hence Y is a central subspace of X.Remark 16.
Following the same line of argument as in the proof of [2, Theorem 3.3], we can observe that a Banach space X is an L1-predual if and only if X is an almost central subspace of every Banach space that contains it.Since every ideal is almost central, our next result generalizes Proposition 14 of [11].Proposition 17.
LetY be an almost central subspace of a Banach space X. Then Y is a central subspace of X** if and only if Y∈(GC).Proof.
IfY is a central subspace of X**, then, by [2, Proposition 2.2(d)], Y∈(GC).
Conversely supposeY∈(GC). Since Y is an almost central subspace of X and X is an almost central subspace of X**, by Remark 9, Y is an almost central subspace of X**. Hence, by Theorem 13, it follows that Y is a central subspace of X**.By a similar transitivity argument used in the proof of the Proposition17, we can easily observe the following corollaries.Corollary 18.
LetY be a subspace of X such that Y⊥⊥ is an almost central subspace of X**. Then Y is an almost central subspace of X**. In addition, if  Y∈(GC), then Y is a central subspace of X**.It is well known that a semiM-ideal need not be an M-ideal (see [7, Chapter I, Remarks 2.3(a)] for example). Our next theorem gives a sufficient condition for a semi M-ideal to be an M-ideal.Theorem 19.
LetY be a semi M-ideal in a Banach space X such that Y⊥⊥ is an almost central subspace of X**. Then Y is an M-ideal in X.Proof.
SinceY is a semi M-ideal, by [12, Theorem 6.14], Y⊥⊥ is a semi M-ideal in X**. Also, since Y⊥⊥ is a weak*-closed almost central subspace of X**, Y⊥⊥ is an AC-subspace of X**. Hence, for any Λ∉Y⊥⊥, by [4, Proposition 2.2], Y⊥⊥ is 1-complemented in span{Y⊥⊥,Λ} and hence is an ideal in span{Y⊥⊥,Λ}. Now, for any Λ∉Y⊥⊥, since Y⊥⊥ is a semi M-ideal in span{Y⊥⊥,Λ}, by [11, Proposition 23], it follows that Y⊥⊥ is an M-ideal in span{Y⊥⊥,Λ}. Hence, by [7, Chapter I, Theorem 2.2], Y⊥⊥ is an M-ideal in X**. Since Y⊥⊥ is a weak*-closed M-ideal in X**, by [7, Chapter II, Corollary 3.6], Y⊥⊥ is an M-summand in X**. Hence, by [7, Chapter I, Theorem 1.9], there exists an L-summand V in X* such that X**=Y⊥⊥⨁∞V⊥. Then, by the duality between L- and M-projections, we get X*=Y⊥⨁1V and hence Y is an M-ideal in X.We now give a sufficient condition for a semiM-ideal to be an M-summand.Theorem 20.
LetY be an AC-subspace of a Banach space X. Then Y is a semi M-ideal in X if and only if Y is an M-summand in X.Proof.
SupposeY is a semi M-ideal in X and is an AC-subspace of X. Since Y is an AC-subspace of X, by [4, Proposition 2.2], Y is 1-complemented in span{Y,x} for all x∈X. Also, since Y is a semi M-ideal in X, Y is a semi M-ideal in span{Y,x} for all x∈X. Thus, by [11, Proposition 23], Y is an M-ideal in span{Y,x} for all x∈X. Then, by [7, Chapter I, Corollary 1.3], Y is an M-summand in span{Y,x} for all x∈X. Hence, by [7, Chapter II, Proposition 3.2], Y is an M-summand in X.Our next theorem gives another sufficient condition for a semiM-ideal to be an M-ideal. In fact, this result improves Proposition 23 of [11].Theorem 21.
LetY be a subspace of a Banach space X such that Y is an ideal in span{Y,x} for all x∈X. Then Y is a semi M-ideal in X if and only if Y is an M-ideal in X.Proof.
SupposeY is a semi M-ideal in X and is an ideal in span{Y,x} for all x∈X. Then, by [11, Proposition 23], Y is an M-ideal in span{Y,x} for all x∈X. Hence, by [7, Chapter I, Theorem 2.2], it follows that Y is an M-ideal in X.We now recall the following theorem of Bandyopadhyay and Dutta that characterizes an AC-subspace of finite codimension in the spaceC(K) of all continuous real-valued functions on a compact Hausdorff space K, endowed with the supremum norm.Theorem 22 (see [5, Theorem 1.1]).
LetK be a compact Hausdorff space and Y be a subspace of codimension n of C(K). Then the following are equivalent.(i)
Y is an AC-subspace of C(K).(ii)
Y is 1-complemented in C(K).(iii)
There exist measuresμ1,…,μn on K and distinct isolated points k1,…,kn of K such that
(a)
Y=⋂i=1n‍ker(μi),(b)
∥μi∥≤2|μi({ki})|,i=1,…,n.In our next proposition, we observe a simple proof for the implication(iii)⇒(ii) of Theorem 22 when K is an extremely disconnected space.We recall that a compact Hausdorff spaceK is extremely disconnected if the closure of each open set in K is again open in K (see [13, Section 7] for details).For any infinite discrete setΓ, ℓ∞(Γ) denotes the space of all bounded real-valued functions on Γ, endowed with the supremum norm, and c0(Γ) denotes its subspace consisting of all functions f∈ℓ∞(Γ) such that the set {γ∈Γ:|f(γ)|≥ɛ} is finite for all ɛ>0. Also, for any infinite discrete set Γ, ℓ1(Γ) denotes the space of all functions f:Γ→ℝ such that ∑γ∈Γ‍|f(γ)|<∞ with the norm ∥f∥1=∑γ∈Γ‍|f(γ)|.The following lemma is the uncountable version of the main theorem of [14] for the space ℓ∞(Γ). As the proof is similar to that of the theorem of [14], we omit the proof here.Lemma 23.
LetΓ be any infinite discrete set and Y be a subspace of codimension n in ℓ∞(Γ). Then Y is 1-complemented in ℓ∞(Γ) if and only if there exist n distinct elements t1,…,tn in  Γ and n linearly independent functionals f1,…,fn in (ℓ∞(Γ))* such that (a)
fi=hi+gi with hi∈l1(Γ), gi∈c0(Γ)⊥,i=1,…,n,(b)
Y=⋂i=1n‍fi-1(0),(c)
∥gi∥≤2|hi(ti)|-∥hi∥1,i=1,…,n.Our next result shows that the spaceℓ∞(Γ) cannot have a finite codimensional 1-complemented subspace containing c0(Γ).Corollary 24.
LetX be a Banach space such that c0(Γ)⊂X⊂ℓ∞(Γ) for some infinite discrete space Γ. If X is a finite codimensional subspace of ℓ∞(Γ), then X cannot be 1-complemented in ℓ∞(Γ).Proof.
SupposeX is 1-complemented in ℓ∞(Γ). Then, by Lemma 23, there exist n distinct elements t1,…,tn in  Γ and n linearly independent functionals f1,…,fn in (ℓ∞(Γ))* such that(a)
fi=hi+gi with hi∈l1(Γ), gi∈c0(Γ)⊥,i=1,…,n,(b)
X=⋂i=1n‍fi-1(0),(c)
∥gi∥≤2|hi(ti)|-∥hi∥1,i=1,…,n. Since X⊥⊆c0(Γ)⊥, fi∈c0(Γ)⊥ for all i∈{1,…,n}. Hence hi=fi-gi∈c0(Γ)⊥ for all i∈{1,…,n}. Since ℓ∞(Γ)*=ℓ1(Γ)⨁1c0(Γ)⊥, we get hi=0 for all i∈{1,…,n}. Then, by (c), gi=0 and hence fi=0 for all i∈{1,…,n}. This contradiction proves that X cannot be 1-complemented in ℓ∞(Γ).LetK be a compact Hausdorff space and E be a closed subset of K. Also, let ℬ(K) be the class of Borel subsets of K. Now, for μ∈C(E)*, we define μ~∈C(K)* as
(7)μ~(B)={μ(B),ifB∈ℬ(K),B⊂E,0,ifB∈ℬ(K),B⋂E=∅.Lemma 25.
LetK be a compact Hausdorff space and let E be a closed subset of K such that there exists a continuous map φ:K→E which is identity on E and let μi∈C(E)*  (1≤i≤n). If ⋂i=1n‍ker(μi~) is 1-complemented in C(K), then ⋂i=1n‍ker(μi) is 1-complemented in C(E).Proof.
LetP:C(K)→C(K) be a projection of norm one with range ⋂i=1n‍ker(μi~).
Now defineP′:C(E)→C(E) by
(8)P′(f)=P(f∘φ)|Eforf∈C(E).
Since
(9)∫EP′(f)dμi=∫EP(f∘φ)|Edμi=∫KP(f∘φ)dμi~=0∀f∈C(E),
we get P′(f)∈⋂i=1n‍ker(μi) and hence P′ is well defined. Clearly P′ is a linear map.
Now letf∈⋂i=1n‍ker(μi). Since φ is identity on E, we have
(10)∫KP(f∘φ)dμi~=∫Efdμi=0∀i.
Thus, f∘φ∈⋂i=1n‍ker(μi~) and P(f∘φ)=f∘φ. Therefore P′(f)=f∘φ|E=f and hence P′ is a projection on C(E) with range ⋂i=1n‍ker(μi). Now, since ∥P′(f)∥=∥P(f∘φ)|E∥≤∥P(f∘φ)∥≤∥f∘φ∥=∥f∥, ∥P′∥=1. Hence P′ is the required projection.Proposition 26.
LetK be an extremely disconnected space. If there exist measures μ1,…,μn on K and distinct isolated points k1,…,kn of K such that ∥μi∥≤2|μi({ki})| for 1≤i≤n, then ⋂i=1n‍ker(μi) is 1-complemented in C(K).Proof.
LetΓ be a dense subset of K. Since each ki’s are isolated points of K, ki∈Γ for all i. Now consider Γ with the discrete topology and its Stone-Čech compactification β(Γ). Then, by [13, Section 7, Lemma 3 and Theorem 3], K is homeomorphically embedded into β(Γ), and also there exists a continuous map φ:β(Γ)→K such that φ is identity on K. Now consider measures μi~ on β(Γ) such that μi~(D)=0 for any Borel set D disjoint from K and μi~(D)=μi(D) for any Borel set D⊂K. Since ki∈Γ, 2|μi({ki})|≥∥μi∥=∥μ~∥. Since C(β(Γ)) is isometric to ℓ∞(Γ), by Lemma 23, ⋂i=1n‍ker(μi~) is 1-complemented in ℓ∞(Γ). Then, by Lemma 25, ⋂i=1n‍ker(μi) is 1-complemented in C(K).In anL1-predual space, we do not know whether every AC-subspace of finite codimension is the range of a norm one projection and/or is the intersection of AC-subspaces of codimension one.


## 3. Stability Results

Coming to quotient spaces, one can easily observe that ifY is 1-complemented in a Banach space X, then, for any subspace Z of Y, Y/Z is 1-complemented in X/Z. Motivated by this, we consider the following problem. Let Y be a subspace of a Banach space X having some property (P) in X. If Z is a subspace of Y, then when can we say that Y/Z has the property (P) in X/Z? We study this problem when the property (P) under consideration is almost constrained, almost central, central, and ideal.For a subspaceY of a Banach space X and x∈X, we denote by [x] the equivalence class in X/Y containing x.Our next result solves the above problem for AC-subspaces.Proposition 27.
LetY be an AC-subspace of X and let Z be a subspace of Y. Then Y/Z is an AC-subspace of X/Z.Proof.
Let{BY/Z([yi],ri)}i∈I be a family of balls in Y/Z and also let x∈X be such that [x]∈⋂i∈I‍BY/Z([yi],ri). Then, for each ɛ>0 and i∈I, there exists a zɛ,i∈Z such that
(11)∥x-yi+zɛ,i∥≤∥[x]-[yi]∥+ɛ≤ri+ɛ∀i∈I,ɛ>0.
We now consider the family {BY(yi-zɛ,i,ri+ɛ)}i∈I,ɛ>0 of closed balls in Y. Clearly x∈⋂i∈I,ɛ>0‍BY(yi-zɛ,i,ri+ɛ). Since Y is an AC-subspace of X, there exists a y∈Y such that y∈⋂i∈I,ɛ>0‍BY(yi-zɛ,i,ri+ɛ). Then, for i∈I, we have
(12)∥[y]-[yi]∥≤∥y-yi+zε,i∥≤ri+ɛ∀ɛ>0.
Therefore ∥[y]-[yi]∥≤ri for all i∈I, and hence Y/Z is an AC-subspace of X/Z.We now prove the stability of ideals in quotient spaces.Proposition 28.
LetY be an ideal in X and let Z be a subspace of Y. Then Y/Z is an ideal in X/Z.Proof.
SinceY is an ideal in X, by [9, Theorem 1], Y⊥⊥ is 1-complemented in X**. Then Y⊥⊥/Z⊥⊥ is 1-complemented in X**/Z⊥⊥. But X**/Z⊥⊥ is isometric to (X/Z)**, and this isometry takes Y⊥⊥/Z⊥⊥ onto (Y/Z)⊥⊥. Hence (Y/Z)⊥⊥ is 1-complemented in (X/Z)**. Then, again by [9, Theorem 1], Y/Z is an ideal in X/Z.Our next result proves the stability of almost central subspaces in quotient spaces.Proposition 29.
LetY be an almost central subspace of X and let Z be a subspace of Y. Then Y/Z is an almost central subspace of X/Z.Proof.
Let[x]∈X/Z, {[y1],…,[yn]}⊆Y/Z and ɛ>0. Then, for 1≤i≤n, there exists an element zɛ,i∈Z such that (13)∥x-yi+zε,i∥≤∥[x]-[yi]∥+ε2.
Since Y is an almost central subspace of X, there exists an element yɛ∈Y such that
(14)∥yɛ-yi+zɛ,i∥≤∥x-yi+zɛ,i∥+ε2for1≤i≤n.
Now, for 1≤i≤n, we have
(15)∥[yε]-[yi]∥≤∥yε-yi+zε,i∥≤∥x-yi+zε,i∥+ε2≤∥[x]-[yi]∥+ε.
Hence Y/Z is an almost central subspace of X/Z.Now, for Banach spacesX, Y, Z with Z⊆Y⊆X, our next set of results give some sufficient conditions for Y/Z to be a central subspace of X/Z.Combining Proposition29 and Theorem 13, we get the following.Corollary 30.
LetY be an almost central subspace of X and let Z be a subspace of Y. If Y/Z∈(GC), then Y/Z is a central subspace of X/Z.As a consequence of the above corollary, we have the following result.Corollary 31.
LetY be a subspace of X and let Z be a subspace of Y such that Y/Z∈(GC). If Y⊥⊥ is an almost central subspace of X**, then Y/Z is a central subspace of X/Z.Proof.
By Corollary18, Y is an almost central subspace of X**. Hence Y is an almost central subspace of X. Since Y/Z∈(GC), by Corollary 30, Y/Z is a central subspace of X/Z.We recall that a subspaceY of a Banach space X is said to be a factor reflexive subspace if the quotient space X/Y is reflexive. Since any reflexive spaces are in the class (GC), the following corollary is easy to see.Corollary 32.
LetY be a subspace of a Banach space X such that Y⊥⊥ is an almost central subspace of X**. Then, for any factor reflexive subspace Z of Y, Y/Z is a central subspace of X/Z.We now prove the converse of Proposition29 under some additional assumptions.Proposition 33.
LetX be an L1-predual, let Z be an M-ideal in X, and let Y be a subspace of X such that Z⊆Y⊆X. If Y/Z is almost central in X/Z, then Y is a central subspace of X.Proof.
Letx∈X, y1…,yn∈Y, and ɛ>0. Then, by assumption, there exists a yɛ∈Y such that
(16)∥[yɛ]-[yi]∥≤∥[x]-[yi]∥+ε2≤∥x-yi∥+ε2.
Let zɛ,i∈Z be such that ∥yɛ-yi-zi∥≤∥x-yi∥+ɛ for all i∈{1,…,n}. Now consider the finite collection of balls {BX(yɛ-yi,∥x-yi∥+ɛ)}i=1n in X. Since this is a pairwise intersecting family of balls in X and X is an L1-predual, ⋂i=1n‍BX(yɛ-yi,∥x-yi∥+ɛ)≠∅. Also, since X is an L1-predual, by [12, Proposition 6.5], it follows that Z has the strong n-ball property in X. Hence there exists an element zɛ∈Z such that ∥zɛ-yɛ+yi∥≤∥x-yi∥+ɛ for all i∈{1,…,n}. Therefore Y is an almost central subspace of X and hence, by Proposition 15, Y is a central subspace of X.The following corollary is the converse of Proposition28 under some additional assumptions.Corollary 34.
LetX be an L1-predual, let Z be an M-ideal in X, and let Y be a subspace of X such that Z⊆Y⊆X. If Y/Z is an ideal in X/Z, then Y is an ideal in X.Proof.
SinceY/Z is an ideal in X/Z, by Lemma 10, Y/Z is almost central in X/Z. Thus, by Proposition 33, Y is a central subspace of X. Then, by Proposition 15, Y is an L1-predual. Hence, by [15, Proposition 1], Y is an ideal in X.We recall that, for any collection{Xα:α∈Γ} of Banach spaces, ℓ∞-sum of Xα (α∈Γ) and c0-sum of Xα (α∈Γ) are defined as ⨁∞Xα={x∈∏‍Xα:∥x∥=sup∥x(α)∥<∞} and ⨁c0Xα={x∈∏‍Xα:for  allɛ>0,{α∈Γ:∥x(α)∥>ɛ}is  finite}, respectively, and equip both spaces with supremum norm. For a finite family of Banach spaces {X1,…,Xk}, ℓ∞-sum of Xi (1≤i≤k) is denoted by (X1⨁⋯⨁Xk)ℓ∞k.Remark 35.
It is easy to observe that, for any family{Xα:α∈Γ} of Banach spaces, if Yα is an almost central subspace of a Banach space Xα, then ⨁∞Yα is an almost central subspace of ⨁∞Xα.We now prove the stability of almost central subspaces in vector-valued continuous function spaces. For a compact Hausdorff spaceK and a Banach space X, we denote by C(K,X) the space of all X-valued continuous functions defined on K, endowed with the supremum norm.LetK be a compact Hausdorff space and let X be a Banach space. Then, for f∈C(K) and x∈X, an element f⊗x∈C(K,X) is defined as (f⊗x)(k)=f(k)x for k∈K.Proposition 36.
LetY be an almost central subspace of Banach space X and let K be a compact Hausdorff space. Then C(K,Y) is an almost central subspace of C(K,X).Proof.
Letf1,…,fn∈C(K,Y), f∈C(K,X), and ɛ>0. Then, by the proof of [10, Page 43, Corollary 2], for the finite family {f1,…,fn}, there exists a partition of unity {φj}j=1m and a closed subspace B of C(K,Y) spanned by the elements of the form ∑j=1m‍φj⊗yj with yj∈Y such that d(fi,B)<ɛ/4 for 1≤i≤n and B is isometric to (Y⨁⋯⨁Y)ℓ∞m. Similarly for f, there exists a partition of unity {φl′}l=1k and a closed subspace B′ of C(K,X) spanned by the elements of the form ∑l=1k‍φl′⊗xl with xl∈X such that d(f,B′)<ɛ/4, and B′ is isometric to (X⨁⋯⨁X)ℓ∞k. Now let f~∈B′ be such that ∥f-f~∥<ɛ/4 and let fi~∈B be such that ∥fi-fi~∥<ɛ/4 for 1≤i≤n.
Case
1 (m≤k).  Since B is isometric to (Y⨁⋯⨁Y)ℓ∞m, B is an M-summand in (Y⨁⋯⨁Y)ℓ∞k (up to an isometry). Since M-summands are central, by Remark 35 and Remark 9, B is an almost central subspace of B′. Then there exists a g∈B such that ∥g-fi~∥≤∥f~-fi~∥+ɛ/4 for 1≤i≤n. Hence we have
(17)∥g-fi∥≤∥g-fi~∥+∥fi~-fi∥≤∥f~-fi~∥+ɛ4+ɛ4≤∥f~-f∥+∥f-fi∥+∥fi-fi~∥+ɛ2≤∥f-fi∥+ɛ.
Case
2 (k<m).  In this case, we can isometrically embed B′ into (X⨁⋯⨁X)ℓ∞m. Since B is isometric to (Y⨁⋯⨁Y)ℓ∞m, by Remark 35, B is an almost central subspace of (X⨁⋯⨁X)ℓ∞m (up to an isometry). Then there exists an element g∈B such that ∥g-fi~∥≤∥f~-fi~∥+ɛ/4 for 1≤i≤n. Hence we have
(18)∥g-fi∥≤∥g-fi~∥+∥fi~-fi∥≤∥f~-f∥+∥f-fi∥+∥fi-fi~∥+ɛ2≤∥f-fi∥+ɛ.
Therefore in all cases there exists a g∈B⊆C(K,Y) such that ∥g-fi∥≤∥f-fi∥+ɛ for 1≤i≤n. Hence C(K,Y) is an almost central subspace of C(K,X).For a central subspaceY of a Banach space X and for a compact Hausdorff space K, it is not known whether C(K,Y) is a central subspace of C(K,X). But if C(K,Y)∈(GC) and Y is almost central in X, then, by Proposition 36 and Theorem 13, C(K,Y) is a central subspace of C(K,X). Now for a Banach space X, Theorem 3.6 of [8] gives a sufficient condition for C(K,X) to be in the class (GC). Precisely, if X is a polyhedral Banach space such that X∈(GC) and {f∈BX*:f(x)=1}⋂‍ext(BX*) is finite for each x∈X with ∥x∥=1, then C(K,X)∈(GC) (by ext(BX*), we denote the set of all extreme points of BX* and a Banach space is called polyhedral if the unit ball of each of its finite dimensional subspace is a polytope). In particular, by [8, Fact 1.3(e)], if X is a finite dimensional polyhedral space, then C(K,X)∈(GC). This information together with Proposition 36 give the following corollary.Corollary 37.
LetY be an almost central subspace of a Banach space X and let K be a compact Hausdorff space. If Y is a polyhedral Banach space such that Y∈(GC) and {g∈BY*:g(y)=1}⋂‍ext(BY*) is finite for each y∈Y with ∥y∥=1, then C(K,Y) is a central subspace of C(K,X). In particular, if  Y is a finite dimensional polyhedral central subspace of X, then C(K,Y) is a central subspace of C(K,X).We now discuss the stability problem in injective tensor product spaces.Proposition 38.
LetK be a compact Hausdorff space and let A be an almost central subspace of C(K). Then, for any Banach space X, the injective tensor product A⨂∨X is almost central in C(K,X).Proof.
SinceA is an almost central subspace of C(K), by Proposition 15, A is an L1-predual. Then, by [15, Proposition 1], A is an ideal in C(K). Hence, by [15, Lemma 2], A⨂∨X is an ideal in C(K)⨂∨X. Since C(K,X)=C(K)⨂∨X (up to an isometry), by Lemma 10, A⨂∨X is almost central in C(K,X).Theorem 39.
LetK be a compact Hausdorff space and let A be an almost central subspace of C(K). If Y is an almost central subspace of a Banach space X, then the injective tensor product A⨂∨Y is an almost central subspace of C(K)⨂∨X. In particular, A⨂∨Y is an almost central subspace of A⨂∨X.Proof.
By Proposition38, A⨂∨Y is almost central in C(K)⨂∨Y=C(K,Y). Then, by Proposition 36 and Remark 9, A⨂∨Y is an almost central subspace of C(K,X)=C(K)⨂∨X. Since A⨂∨Y⊆A⨂∨X⊆C(K,X), A⨂∨Y is an almost central subspace of A⨂∨X.Corollary 40.
LetZ be an L1-predual. Then, for any almost central subspace Y of a Banach space X, the injective tensor product Z⨂∨Y is an almost central subspace of Z⨂∨X.Proof.
SinceZ is an L1-predual, by [15, Proposition 1], Z is isometric to an ideal in C(K) for some compact Hausdorff space K. Then, by Lemma 10, Z is an almost central subspace of C(K) (up to an isometry). Therefore, by Theorem 39, Z⨂∨Y is an almost central subspace of Z⨂∨X.We now answer a question raised in [2] and also improve Theorem 6 of [11].Proposition 41.
LetΓ be an index set and let Xα(α∈Γ) be Banach spaces. Then ⨁c0Xα is a central subspace of ⨁∞Xα.Proof.
Letx∈⨁∞Xα and y1,…,yn∈⨁c0Xα. Let r=min1≤i≤n∥x-yi∥. Since y1,…,yn∈⨁c0Xα, there exists a finite set A such that ∥yi(α)∥≤r whenever α∉A.
Definez∈⨁c0Xα as
(19)z(α)={x(α),ifα∈A,0,ifα∉A.
Now for 1≤i≤n,ifα∈A, then ∥z(α)-yi(α)∥=∥x(α)-yi(α)∥≤∥x-yi∥ andifα∉A, then ∥z(α)-yi(α)∥=∥yi(α)∥≤r≤∥x-yi∥.Hence ∥z-yi∥≤∥x-yi∥ for all i.Corollary 42.
The class(GC) is stable under c0-direct sum of Banach spaces.Proof.
IfXα∈(GC) for all α∈Γ, then, by [2, Theorem 4.7] and Proposition 41, ⨁c0Xα is a central subspace of ⨁∞Xα**=(⨁c0Xα)**. Hence the result follows.In [8], Veselý defined a new direct sum called polyhedral direct sum. We now prove the stability of some ball intersection properties under polyhedral direct sums.Definition 43 (see [8]).
A normπ on ℝ+n is called polyhedral if it is of the form π(t)=max1≤j≤mgj(t), where g1,…,gm∈(ℝn)*. In this case, we say that the family {g1,…,gm} generates π. Now Lemma 1.5 of [8] shows that if {g1,…,gm} is a minimal family generating π, then gj(i)≥0 for all i=1,…,n and j=1,…,m.
We say that a Banach spaceX is the polyhedral direct sum of Banach spaces X1,…,Xn if X=X1⊕⋯⊕Xn and the norm on X is of the form ∥x∥X=π(∥x(1)∥,…,∥x(n)∥), where π is a polyhedral nondecreasing norm on ℝ+n. In this case, we write X=(X1⊕⋯⊕Xn)π.Our next theorem proves that the property of being a central subspace is stable under polyhedral direct sums.Forn∈ℕ and 1≤i≤n, we denote by ei the ith canonical unit vector of ℝn.Theorem 44.
LetX be a polyhedral direct sum of Banach spaces Xi  (1≤i≤n) and let Yi be a subspace of Xi  (1≤i≤n). Let π be the corresponding polyhedral norm and suppose π(ei)≠0 for all i. Then the polyhedral sum Y of Yi  (1≤i≤n) is a central subspace of X if and only if Yi is a central subspace of Xi for all i.Proof.
SupposeY is a central subspace of X. Fix an m∈{1,…,n}. Let xm∈Xm and ym,k∈Ym(1≤k≤p). Define x∈X and yk∈Y (1≤k≤p) as(20)x(i)={xm,ifm=i,0,otherwise,yk(i)={ym,k,ifm=i,0,otherwise.
Then there exists a y∈Y such that ∥y-yk∥X≤∥x-yk∥X for 1≤k≤p. Therefore, for 1≤k≤p, we have
(21)∥y(m)-yk(m)∥π(em)=π(∥y(m)-yk(m)∥em)≤π(∥y(1)-yk(1)∥,…,∥y(n)-yk(n)∥)≤π(∥x(1)-yk(1)∥,…,∥x(n)-yk(n)∥)=∥x(m)-yk(m)∥π(em).
Since π(ei)≠0 for all i, we get ∥y(m)-ym,k∥≤∥x(m)-ym,k∥ for 1≤k≤p. Hence Ym is a central subspace of Xm.
Conversely supposeYi is central in Xi for all i. Let x∈X and yk∈Y  (1≤k≤p). Then, for 1≤m≤n, there exists a ym∈Ym such that ∥ym-yk(m)∥≤∥x(m)-yk(m)∥ for 1≤k≤p. Define y∈Y as y(i)=yi  (1≤i≤n). Now, by the monotonicity of π,
(22)∥y-yk∥X≤π(∥(x-yk)(1)∥,…,∥(x-yk)(n)∥)=∥x-yk∥Xfor1≤k≤p.
Hence Y is a central subspace of X.An argument similar to the one used to prove Theorem44 gives the following.Theorem 45.
LetX be a polyhedral direct sum of Banach spaces Xi  (1≤i≤n) and let Yi be a subspace of Xi  (1≤i≤n). Let π be the corresponding polyhedral norm and suppose π(ei)≠0 for all i. Then the polyhedral sum Y of Yi  (1≤i≤n) is an AC-subspace of X if and only if Yi is an AC-subspace of Xi for all i.



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