Intersection properties of balls in Banach spaces

In this paper, 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 in [15]. In particular, we prove that if Y 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 injective tensor product spaces and polyhedral direct sums of Banach spaces are also derived.


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 space , we denote by   (, ) the closed ball in  of radius  > 0 around  ∈  and by   the closed unit ball of .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  as a subspace of  * * .Also, if a Banach space  is isometric to a subspace of the Banach space , then, without loss of generality, we will consider  as a subspace of .Our notations are otherwise standard.Any unexplained terminology can be found in [3].
Definition 3 (see [1,Definition 2.8]).One shall denote by (GC) the class of all Banach spaces  such that for every positive integer  and every  1 , . . .,   ∈ , 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]).Let  be a Banach space.One says that a subspace  ⊆  is a central subspace of  if every finite family of closed balls with centers in  that intersects in  also intersects in .
Clearly  ∈ (GC) if and only if  is a central subspace of  * * .It follows from [2, Proposition 2.2(a)] that  is a central subspace of a Banach space  if and only if, for any finite set {  }  =1 ⊂  and  ∈ , there exists a  ∈  such that ‖−  ‖ ≤ ‖ −   ‖ for 1 ≤  ≤ .
An infinite version of central subspace called almost constrained subspace was investigated in [4,5].Definition 5. A subspace  of a Banach space  is said to be an almost constrained (AC) subspace of  if any family of closed balls centered at points of  that intersects in  also intersects in .
We recall that a subspace  of a Banach space  is called 1-complemented in  if there exists a projection of norm one on  with range .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 subspace  of a Banach space  is said to be an ideal in  if  ⊥ is the kernel of a norm one projection on  * .Clearly 1-complemented subspaces are ideals.Also, every Banach space is an ideal in its bidual.For, if  is a Banach space, then the projection  :  * * * →  * * * defined by (Λ) = Λ|  is a projection of norm one with kernel  ⊥ .It is well known that  0 is an ideal in ℓ ∞ but it is not the range of a projection of norm one in ℓ ∞ .
An important concept in the -structure theory which is closely related to ball intersection properties is the well known concept called -ideal (see [7] for details).Definition 7. A projection  on a Banach space  is called an -projection (-projection) if ‖‖ = max{‖‖, ‖ − ‖} (‖‖ = ‖‖ + ‖ − ‖) for all  ∈ .A subspace  of  is called an -summand (-summand) if it is the range of an -projection (-projection).A subspace  of  is called an -ideal if  ⊥ is an -summand in  * .For two Banach spaces  and , one denotes by ⨁ 1  and ⨁ ∞  the direct sum of  and , equipped with the ℓ 1 -norm and supremum norm, respectively.
In Section 2, 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  is in the class (GC) if and only if it is a central subspace of  * * .We also derive several sufficient conditions for a semi -ideal to be an -ideal in terms of these intersection properties of balls.
In Section 3, 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 , , and  with  ⊆  ⊆ , if  is almost central or ideal in , then / is almost central or ideal in /, respectively, and we also prove the converse when  is an  1 -predual (that is  * =  1 (), for some positive measure ) and  is an -ideal in .
In the case of injective tensor product spaces, we show that if  is an  1 -predual space, then, for any almost central subspace  of a Banach space , the injective tensor product 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]).

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].
Hence  is an almost central subspace of .
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 every -ideal is an ideal, by Lemma 10, -ideals are almost central.We now give an example to show that a semi -ideal may not be an almost central subspace.But (3) shows that both of (1) and (2) cannot be positive.But the symmetric inequalities (4) and (5) rule out other possibilities.Thus  is not a central subspace of ℓ 3 1 .Then, by a compactness argument, we can see that  is not an almost central subspace of ℓ 3 1 .
In [1, Example 5.6], Veselý gave an example of a threedimensional Banach space  such that ([0, 1], ) 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. Let 𝑌 be an almost central subspace of a Banach space 𝑋 such that 𝑌 ∈ (GC).
Then  is a central subspace of .
Our next result gives a sufficient condition for an almost central subspace to be an AC-subspace.Proposition 14.Let  be an almost central subspace of a Banach space  such that  is isometric to the range of a projection of norm one in some dual space.Then  is an ACsubspace of .
We now give a class of Banach spaces where almost central subspaces are central.We recall that a Banach space  whose dual  * is isometric to  1 () for some positive measure  is called an  1 -predual.

Proposition 15.
Let  be an  1 -predual and let  be an almost central subspace of .Then  is an  1 -predual.In particular,  is a central subspace of .
Proof.Let {  (  ,   )}  =1 be any family of  balls in  such that any two of them intersect in .Since  is an  1 -predual, by [10, Theorem 6.1], there exists an  ∈  such that ‖ −   ‖ ≤   for all .Also, since  is an almost central subspace of , ⋂  =1   (  ,   + ) ̸ = 0 for all  > 0.Then, by [10, Lemma 4.2 and Theorem 6.1], it follows that  is an  1 -predual.Now let {  (  ,   )}  =1 be a family of  balls in  that has nonempty intersection in .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 {  (  ,   )}  =1 is a pairwise intersecting family in .Since  is an  1 -predual, by [ It is well known that a semi -ideal need not be an ideal (see [7, Chapter I, Remarks 2.3(a)] for example).Our next theorem gives a sufficient condition for a semi -ideal to be an -ideal.
Theorem 19.Let  be a semi -ideal in a Banach space  such that  ⊥⊥ is an almost central subspace of  * * .Then  is an -ideal in .
We now give a sufficient condition for a semi -ideal to be an -summand.
Theorem 20.Let  be an AC-subspace of a Banach space .Then  is a semi -ideal in  if and only if  is an summand in .
Our next theorem gives another sufficient condition for a semi -ideal to be an -ideal.In fact, this result improves Proposition 23 of [11].
Theorem 21.Let  be a subspace of a Banach space  such that  is an ideal in span{, } for all  ∈ .Then  is a semi -ideal in  if and only if  is an -ideal in .
Proof.Suppose  is a semi -ideal in  and is an ideal in span{, } for all  ∈ .Then, by [11,Proposition 23],  is an -ideal in span{, } for all  ∈ .Hence, by [7, Chapter I, Theorem 2.2], it follows that  is an -ideal in .
We now recall the following theorem of Bandyopadhyay and Dutta that characterizes an AC-subspace of finite codimension in the space () of all continuous real-valued functions on a compact Hausdorff space , endowed with the supremum norm.
Theorem 22 (see [5,Theorem 1.1]).Let  be a compact Hausdorff space and  be a subspace of codimension  of ().Then the following are equivalent.
(iii) There exist measures  1 , . . .,   on  and distinct isolated points  1 , . . .,   of  such that In our next proposition, we observe a simple proof for the implication (iii) ⇒ (ii) of Theorem 22 when  is an extremely disconnected space.
We recall that a compact Hausdorff space  is extremely disconnected if the closure of each open set in  is again open in  (see [13,Section 7] for details).
For any infinite discrete set Γ, ℓ ∞ (Γ) denotes the space of all bounded real-valued functions on Γ, endowed with the supremum norm, and  0 (Γ) denotes its subspace consisting of all functions  ∈ ℓ ∞ (Γ) such that the set { ∈ Γ : |()| ≥ } is finite for all  > 0. Also, for any infinite discrete set Γ, ℓ 1 (Γ) denotes the space of all functions  : Γ → R such that 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.
Let  be a compact Hausdorff space and  be a closed subset of .Also, let B() be the class of Borel subsets of .Now, for  ∈ () * , we define μ ∈ () * as Lemma 25.Let  be a compact Hausdorff space and let  be a closed subset of  such that there exists a continuous map  :  →  which is identity on  and let Proof.Let  : () → () be a projection of norm one with range ⋂  =1 ker(μ  ).Now define   : () → () by Since In an  1 -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.

Stability Results
Coming to quotient spaces, one can easily observe that if  is 1-complemented in a Banach space , then, for any subspace  of , / is 1-complemented in /.Motivated by this, we consider the following problem.Let  be a subspace of a Banach space  having some property () in .If  is a subspace of , then when can we say that / has the property () in /?We study this problem when the property () under consideration is almost constrained, almost central, central, and ideal.
For a subspace  of a Banach space  and  ∈ , we denote by [] the equivalence class in / containing .
Our next result solves the above problem for AC-subspaces.
Proposition 27.Let  be an AC-subspace of  and let  be a subspace of .Then / is an AC-subspace of /.We recall that a subspace  of a Banach space  is said to be a factor reflexive subspace if the quotient space / is reflexive.Since any reflexive spaces are in the class (GC), the following corollary is easy to see.
Corollary 32.Let  be a subspace of a Banach space  such that  ⊥⊥ is an almost central subspace of  * * .Then, for any factor reflexive subspace  of , / is a central subspace of /.
We now prove the converse of Proposition 29 under some additional assumptions.
Proposition 33.Let  be an  1 -predual, let  be an -ideal in , and let  be a subspace of  such that  ⊆  ⊆ .If / is almost central in /, then  is a central subspace of .
The following corollary is the converse of Proposition 28 under some additional assumptions.
Corollary 34.Let  be an  1 -predual, let  be an -ideal in , and let  be a subspace of  such that  ⊆  ⊆ .If / is an ideal in /, then  is an ideal in .
Proof.Since / is an ideal in /, by Lemma 10, / is almost central in /.Thus, by Proposition 33,  is a central subspace of .Then, by Proposition 15,  is an  1 -predual.Hence, by [15, Proposition 1],  is an ideal in .
We now prove the stability of almost central subspaces in vector-valued continuous function spaces.For a compact Hausdorff space  and a Banach space , we denote by (, ) the space of all -valued continuous functions defined on , endowed with the supremum norm.
Let  be a compact Hausdorff space and let  be a Banach space.Then, for  ∈ () and  ∈ , an element  ⊗  ∈ (, ) is defined as ( ⊗ )() = () for  ∈ .For a central subspace  of a Banach space  and for a compact Hausdorff space , it is not known whether (, ) is a central subspace of (, ).But if (, ) ∈ (GC) and  is almost central in , then, by Proposition 36 and Theorem 13, (, ) is a central subspace of (, ).Now for a Banach space , Theorem 3.6 of [8] gives a sufficient condition for (, ) to be in the class (GC).Precisely, if  is a polyhedral Banach space such that  ∈ (GC) and { ∈   * : () = 1} ⋂ ext(  * ) is finite for each  ∈  with ‖‖ = 1, then (, ) ∈ (GC) (by ext(  * ), we denote the set of all extreme points of   * 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  is a finite dimensional polyhedral space, then (, ) ∈ (GC).This information together with Proposition 36 give the following corollary.
Corollary 37. Let  be an almost central subspace of a Banach space  and let  be a compact Hausdorff space.If  is a polyhedral Banach space such that  ∈ (GC) and { ∈   * : () = 1} ⋂ ext(  * ) is finite for each  ∈  with ‖‖ = 1, then (, ) is a central subspace of (, ).In particular, if  is a finite dimensional polyhedral central subspace of , then (, ) is a central subspace of (, ).
We now discuss the stability problem in injective tensor product spaces.We now answer a question raised in [2] and also improve Theorem 6 of [11].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.
Our next theorem proves that the property of being a central subspace is stable under polyhedral direct sums.
For  ∈ N and 1 ≤  ≤ , we denote by   the th canonical unit vector of R  .
Hence  is a central subspace of .
An argument similar to the one used to prove Theorem 44 gives the following.
Theorem 45.Let  be a polyhedral direct sum of Banach spaces   (1 ≤  ≤ ) and let   be a subspace of   (1 ≤  ≤ ).Let  be the corresponding polyhedral norm and suppose (  ) ̸ = 0 for all .Then the polyhedral sum  of   (1 ≤  ≤ ) is an AC-subspace of  if and only if   is an AC-subspace of   for all .
Definition 8.A subspace  of a Banach space  is called an almost central subspace if, for every finite set { 1 , . . .,   } ⊆ ,  ∈ , and  > 0, there exists a   ∈  such that ‖  −   ‖ ≤ ‖ −   ‖ +  for 1 ≤  ≤ .Clearly central subspaces of Banach spaces are almost central.As in the case of central subspace, it is easy to observe that  is an almost central subspace of a Banach space  if and only if, for each family {  (  ,   )}  =1 of closed balls in  having nonempty intersection in , the family {  (  ,   + )}  =1 of closed balls in  has nonempty intersection in  for all  > 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  is an almost central subspace of a Banach space  and  is an almost central subspace of a Banach space , then  is an almost central subspace of .Define  = span{ 1 , . . .,   , }.Since  is an ideal in , by [9, Theorem 1], there exists an operator   :  →  such that 10, Theorem 6.1], it follows that {  (  ,   )}  =1 intersects in .Hence  is a central subspace of .Remark 16.Following the same line of argument as in the proof of [2, Theorem 3.3], we can observe that a Banach space  is an  1 -predual if and only if  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].Let  be an almost central subspace of a Banach space .Then  is a central subspace of  * * if and only if  ∈ (GC). .If  is a central subspace of  * * , then, by [2, Proposition 2.2(d)],  ∈ (GC).Conversely suppose  ∈ (GC).Since  is an almost central subspace of  and  is an almost central subspace of  * * , by Remark 9,  is an almost central subspace of  * * .Hence, by Theorem 13, it follows that  is a central subspace of  * * .By a similar transitivity argument used in the proof of the Proposition 17, we can easily observe the following corollaries.Let  be a subspace of  such that  ⊥⊥ is an almost central subspace of  * * .Then  is an almost central subspace of  * * .In addition, if  ∈ (GC), then  is a central subspace of  * * .