Ball remotality in Banach spaces and related topics

(1)

Ball Remotality In Banach Spaces And Related Topics

Author: Tanmoy P

^AUL

Thesis submitted to the Indian Statistical Institute

in partial fulfillment of the requirements for the award of the degree of

Doctor of Philosophy

INDIAN STATISTICAL INSTITUTE - KOLKATA July, 2010

(2)

(3)

Ball Remotality In Banach Spaces And Related Topics

TANMOY PAUL

Indian Statistical Institute 203, B. T. Road, Kolkata, India.

(4)

(5)

Dedicated to all those people who love Banach spaces . . .

(6)

(7)

Acknowledgments

To produce a faithful acknowledgment I have to go back about ten years. It was an article published in a daily newspaper on the ever discussed Fermat’s last theorem. It was like a fairy tale to me : how a man reached his dream that was waiting for him for three hundred years. So let’s give a sincere thanks to the writer of that article.

Though it is not possible to make a single list of all those people who have much or less contribution in my research, I mention some of them.

First, my sincere thanks go to my supervisor Professor Pradipta Bandyopadhyay. With- out his help, this work was not possible.

Besides my supervisor, I met so many academicians throughout my research career in ISI, but the first name will go for Professor A. K. Roy. I came to know him not only as a person who has wide knowledge in analysis, but also as a person who knows how to get optimal output from a student. His encouragement and advice stimulated my research.

Prof. Roy taught me so many courses. Those class notes are still in my bookshelves as a memoir of my research life in ISI.

During my research I have visited Professor T.S.S.R.K. Rao, sometimes in ISI Bangalore, sometimes during his Kolkata visit. He helped me whenever he got time for that.

Next I have to mention Professor B. V. Rao. Such a knowledgeable and down to earth person is an asset for any academic institute. Always he was ready to discuss with any research scholar in their research area. I have learned so many things from him.

Lastly, I would like to thank some of the nonacademic stuff of the Stat-Math Unit as well as of ISI, Kolkata and some of my colleagues for such a good and friendly atmosphere here.

(8)

(9)

Introduction

Let us fix some notations and conventions first. Unless otherwise stated, most of our results hold for both real (R) or complex (C) scalars. We will denote the scalar field byF.

The closed unit ball and the unit sphere of a Banach spaceXwill be denoted byB_X and S_X respectively. By a subspace, we will mean a norm closed linear subspace. We denote by N A(X)the set of allx^∗ ∈X^∗which attain their norm onB_X. We will identifyx∈X with its canonical image inX^∗∗.

For a closed and bounded setC in a Banach spaceX, the farthest distance mapφ_C is defined as

φ_C(x) = sup{kz−xk:z∈C}, x∈X.

φ_C is a Lipschitz continuous convex function. Forx ∈X, we denote the set of points inC farthest fromxbyF_C(x),i.e.,

F_C(x) ={z∈C :kz−xk=φ_C(x)}

Note that this set may be empty. Let R(C, X) = {x ∈ X : F_C(x) 6= ∅}. We will write R(C)when there is no confusion about the ambient space. Call a closed and bounded setC remotal ifR(C, X) =Xand densely remotal ifR(C, X)is norm dense inX.

Clearly, a compact set is remotal. The study of densely remotal sets was initiated by Edelstein [19] who proved that any closed and bounded set in a uniformly convex space is densely remotal. Asplund [3] extended this to show that any closed and bounded set in a reflexive locally uniformly convex Banach space is densely remotal. In [39], Zizler generalized Asplund’s result by showing that ifX^∗ is an Asplund space with a LUR dual norm, then any closed and bounded set inXis densely remotal. Then Lau [30] showed that Theorem 1.0.1. [30, Theorem 2.3]Any weakly compact set in any Banach space is densely remotal with respect to any equivalent norm.

Deville and Zizler [17] proved a partial converse of this result :

Theorem 1.0.2. [17, Proposition 4] LetX be a Banach space andC is a closed bounded convex subset ofX. IfCis densely remotal for every equivalent renorming onX, thenCis weakly compact.

(12)

2 Chapter 1. Introduction

They also proved that

Theorem 1.0.3. [17, Proposition 3] If X has the Radon-Nikodým Property (RNP), every w*- compact set inX^∗is densely remotal with respect to any equivalent dual norm.

The survey article [14] contains many results on the existence of nearest and farthest points of sets and its relation with some geometric properties of Banach spaces.

Note thatB_X is always a remotal set in any Banach spaceX. So it is natural to ask what happens in case ofB_Y for a subspaceY? This issue was addressed in a recent paper [10].

Definition 1.0.4. Let us call a subspaceY of a Banach spaceX (a) ball remotal (BR), ifB_Y is remotal inX;

(b) densely ball remotal (DBR), ifB_Y is densely remotal inX.

The main object of this thesis is to study these properties more extensively. As noted in [10], it follows from the results noted above that :

(a) Any finite-dimensional subspace is BR.

(b) Any reflexive subspace is DBR.

(c) IfX^∗is an Asplund space with a LUR dual norm, then any subspace ofX^∗is DBR.

(d) IfXhas the RNP, then for any w*-closed subspace ofX^∗ is DBR.

The point of departure in [10] is the observation that the spacec₀ is DBR in`_∞. This was proved by first showing thatφ_B_c₀(x) =kxk+ 1for allx ∈`_∞and henceR(B_c₀, `_∞) = N A(`₁)and then appealing to the Bishop-Phelps Theorem.

They observed that the nice expression ofφ_B_c₀ is shared by a class of subspaces.

LetY be a subspace ofX. Clearly,φ_B_Y(x)≤φ_B_X(x) =kxk+ 1for allx∈X.

Definition 1.0.5. Let us call a subspaceY of a Banach spaceXa(∗)-subspace ofXif φ_B_Y(x) =kxk+ 1 for allx∈X.

We will encounter these subspaces rather frequently in this thesis. We have the complete description ofR(B_Y, X)in this case [10] (see Proposition2.3.13below).

In [10], the authors used a different definition and proved that their definition implies ours. They, however, used only the above expression forφ_B_Y in most applications. We show in Chapter2that the two definitions are actually equivalent.

A natural example of(∗)-subspace isXas a subspace ofX^∗∗. Sincec^∗∗₀ =`_∞, it follows thatc₀is a(∗)-subspace of`_∞. To prove thatR(B_c₀, `_∞) = N A(`₁), the authors in [10] also used the fact thatc₀is anM-embedded space,i.e., it is anM-ideal in its bidual.

(13)

3 Definition 1.0.6. [26] A subspace Y of a Banach space X is an M-ideal inX if there is a projectionP onX^∗withker(P) =Y^⊥and for allx^∗ ∈X^∗,kx^∗k=kP x^∗k+kx^∗−P x^∗k.

They showed that for any two Banach spaceXandY, the spaceK(X, Y)of all compact operators fromX toY, is(∗)-subspace of the spaceL(X, Y)of all bounded operators from XtoY. And they showed that :

(a) For a large class of Banach spaces that include all reflexive spaces,C(K)spaces and L¹(µ)spaces,K(X)is DBR inL(X).

(b) IfX is a rotund Banach space with the RNP, andµis the Lebesgue measure on[0,1], thenK(L¹(µ), X)is DBR inL(L¹(µ), X).

They also noted that for a compact Hausdorff space K, the space C(K, X) of all X- valued continuous functions onK is a(∗)-subspace of the spaceW C(K, X) of all continuous functions from K to X whenX is endowed with the weak topology. And if X is reflexive,C(K, X)is DBR inW C(K, X).

Coming to stability results, they observed that many natural summands are BR. For example,

(a) Any subspace of a Hilbert space is BR.

(b) For any measure space(Ω,Σ, µ),L¹(µ)is BR inL¹(µ)^∗∗. In particular,`₁is BR in`^∗∗₁ . Now, c₀ is an M-ideal as well as a DBR subspace of `_∞. The paper [10] has more examples of this phenomenon. Hence, it is natural to ask if anyM-ideal is DBR, or, more specifically, is any M-embedded space DBR in its bidual. Both the questions have been answered in the negative in [10] (see Remark7.2.13).

Observe that ifz ∈ B_Y is farthest fromx ∈ X, then it is nearest from any point on the line[x, z]extended beyondz. So a natural question is : Is there any relation between ball proximinality as introduced in [9] and ball remotality? Since anM-ideal is ball proximinal [29, Corollary 5.1.1, p 86], the above example shows that a ball proximinal subspace need not be DBR. We will show later a DBR subspace too need not be ball proximinal, or even proximinal (Remark4.2.34).

We now provide a chapter-wise summary of the principal results of this thesis.

In Chapter2, we obtain several characterizations of(∗)-subspaces, including the equiva- lence of our definition with that of [10]. In the process, we obtain a farthest distance formula, which is also of independent interest. We completely characterize(∗)- and DBR/BR subspaces of a Banach space. In this chapter, we also characterize 1-dimensional(∗)-subspaces.

It turns out that this depends on the existence of a strong unitary (Definition2.4.2), a notion related to that of geometric unitaries studied in [8].

(14)

4 Chapter 1. Introduction In Chapter3, we study ball remotality in the classical Banach spacesc₀,cand`_∞. The farthest distance formula of Chapter2takes a simpler form in these spaces. Using this, we characterize(∗)- and DBR subspaces ofc₀,cand`_∞. We observe that one can prove thatc₀ is DBR in`_∞without using the Bishop-Phelps Theorem. In the process, we prove that

(a) For a subspaceY ⊆c₀, the following are equivalent : (i) Y is(∗)- and DBR inc₀

(ii) Y is(∗)- and BR inc₀ (iii) Y is(∗)- and DBR inc (iv) Y is(∗)- and DBR in`_∞.

(b) In particular,c₀is a(∗)- and DBR subspace of bothcand`_∞.

(c) If a subspace ofcor`_∞contains the constant sequence1, then it is(∗)- and BR.

(d) candbc, the canonical image ofc, are(∗)- and BR in`_∞.

We also show thatc₀ has no finite dimensional (∗)-subspace. We characterize all hyperplanes inc₀ which are(∗)- and DBR in terms of the defining linear functionals.

Then we come to the space`₁and observe that`₁with its usual norm provides a simple non-reflexive example of a Banach space in which every subspace is DBR (Corollary3.3.2).

The existence of such spaces was observed in [10] with a more involved argument. We show that a hyperplane in`₁is BR if and only if it contains a strong unitary.

We note that most of the above results extend without difficulty to subspaces ofc₀(Γ),

`_∞(Γ)and`₁(Γ)for some arbitrary index setΓ. However, ifΓis uncountable, a hyperplane in`₁(Γ)is always(∗)- and BR.

From sequence spaces, we shift our attention to function spaces. In Chapter4, we study ball remotality of subspaces of the space C(K) of all scalar-valued continuous functions on K, where K is a compact Hausdorff space. We characterize(∗)- and (∗)- & DBR/BR subspaces ofC(K)in terms of the density of certain subsets ofK. In the process, we prove that any Banach space embeds isometrically as a (∗)- and DBR subspace of some C(K) space.

We also study boundaries of a general subspaceY ofC(K). In particular, we relate the Choquet boundary ofY with other boundaries, in the process recapturing some classical results. We show that ifY is a subspace of co-dimensionninC(K), then any closed boundary forY can miss at mostnpoints ofK. In particular, ifKhas no isolated points, then any finite co-dimensional subspace cannot have any proper closed boundary.

Applying these results to the question of DBR subspaces, we show that an infinite compact Hausdorff spaceK has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane in C(K) is DBR. We characterize (∗)- and DBR hy-

(15)

5 perplanes inC(K) in terms of the defining measures. We show that a Banach spaceX is reflexive if and only if X is a DBR subspace of any superspace. We also prove that any M-ideal or any closed self-adjoint subalgebra ofC(K)is DBR.

In Chapter 5, we study ball remotality ofM-ideals in some function spaces and function algebras. Isolating a common feature ofM-ideals in subspaces ofC(K), we define an Urysohn pair(A, D).

Definition 1.0.7. LetKbe a compact Hausdorff space,A ⊆C(K)a subspace andD⊆Ka closed set. We say that(A, D)is anUrysohn pairif

For anyt₀∈K\D, there existsf ∈Asuch thatkfk_∞= 1,f|_D ≡0andf(t₀) = 1.

In Theorem5.2.4, we show that for an Urysohn pair(A, D), the subspaceY ={f ∈A : f|_D ≡0}forms a DBR subspace ofA. As corollaries, we show that :

(a) AnyM-ideal inC(K)is DBR, recapturing our earlier result with a new proof.

(b) For a locally compact Hausdorff spaceL, anyM-ideal in the spaceC₀(L)of all scalar- valued continuous functions onL“vanishing at infinity”, is a DBR subspace.

(c) AnyM-ideal in the disc algebraAis DBR inA.

In Chapter5, we also consider the Banach spaceA_F(Q)of scalar-valued affine continuous functions, whereQis a compact convex set in some locally convex topological vector spaceE. We denote by∂_eQthe set of all extreme points ofQ. Our main result in this chapter is that ifQis a Choquet simplex and∂_eQ\∂_eQis at most finite, then anyM-ideal is a DBR subspace ofA_F(Q). Some variants of this result are also considered.

In Chapter6, we explore the stability of the properties(∗), BR and DBR. These properties are better behaved with respect to superspaces than subspaces. A p-summand is a (∗)- subspace if and only ifp= 1.

Coming to sequence spaces, we show that thec₀- or the`_p-sum(1< p ≤ ∞)ofY_α’s is a(∗)-/(∗)- and DBR/(∗)- and BR subspace in the corresponding sum ofX_α’s if and only if eachY_αis such a subspace inX_α. In the process, we answer [10, Question 2.17] in the affir- mative. On the other hand, if at least oneY_α is a(∗)-/(∗)- and DBR/(∗)- and BR subspace ofX_α, then the`₁-sum ofY_α’s is such a subspace of the corresponding sum ofX_α’s.

Coming to function spaces, we show thatY is a(∗)-/(∗)- and DBR/(∗)- and BR subspace ofXif and only ifC(K, Y)is such a subspace ofC(K, X). For BR, the(∗)- assumption may also be removed. IfY is a(∗)-/(∗)- and DBR subspace ofX and(Ω,Σ, µ) is a probability space, then the spaceL₁(µ, Y)ofY-valued Bochner integrable functions is such a subspace ofL₁(µ, X).

In Chapter7, we study ball remotality of a Banach spaceXin its bidual. In particular, we consider the following properties :

(16)

6 Chapter 1. Introduction Definition 1.0.8. We will say that a Banach spaceX

(a) is BR in its bidual (BRB) ifR(B_X, X^∗∗) =X^∗∗. (b) is DBR in its bidual (DBRB) ifR(B_X, X^∗∗) =X^∗∗.

(c) has remotally spanned bidual (RSB) ifspan(R(B_X, X^∗∗)) =X^∗∗. (d) is anti-remotal in its bidual (ARB) ifR(B_X, X^∗∗) =X.

It is clear that reflexivity⇒ BRB ⇒ DBRB ⇒ RSB. We show that none of the con- verse holds. We show that a Banach space having a strong unitary is BRB, producing a large class of non-reflexive examples. We show thatX is wALUR [7] if and only if X is rotund and ARB. We also obtain characterizations of reflexivity in terms of these phenom- ena. For example, we show that a separable Banach space is reflexive if and only if it is BRB/DBRB/RSB in every equivalent renorming. Some stability results are also obtained.

(17)

C

HAPTER

2

(∗)-subspaces and the farthest distance formula

2.1 Summary of results

As a generalization of(∗)-subspaces, we introduce(∗)-subsets and obtain several characterizations. In the process, we obtain a farthest distance formula for a closed bounded balanced subset of a Banach space, which is also of independent interest. We completely characterize (∗)- and DBR/BR subspaces of a Banach space. In this chapter, we also characterize 1-dimensional (∗)-subspaces. It turns out that this depends on the existence of a strong unitary (Definition2.4.2), a notion related to that of geometric unitaries studied in [8].

2.2 The farthest distance formula

Notation 1. LetT={z∈F:|z|= 1}. Definesgn:F→Tby sgn(z) =

( 1 if z= 0

|z|/z if z6= 0 That is, for anyz∈F,|sgn(z)|= 1andsgn(z)·z=|z|.

Definition 2.2.1. Forx∈X, letD(x) ={x^∗ ∈S_X^∗ :x^∗(x) =kxk}.

We say thatA⊆B_X^∗ is a norming set forXifkxk= sup{|x^∗(x)|:x^∗∈A}for allx∈X.

We say thatB ⊆ S_X^∗ is a boundary forX if for everyx ∈ X, there existsx^∗ ∈B such thatkxk=|x^∗(x)|.

Theorem 2.2.2. LetCbe a closed, bounded and balanced subset of a Banach spaceX. Forx^∗ ∈X^∗, letkx^∗k_C = sup_z∈C|x^∗(z)|.

(a) LetA⊆B_X^∗be a norming set forX. Then for anyx∈X,

φ_C(x) = sup{|x^∗(x)|+kx^∗k_C :x^∗ ∈A}. (2.1)

(18)

8 Chapter 2. (∗)-subspaces and the farthest distance formula (b) If A ⊆ S_X^∗ is a boundary forX, thenx ∈ R(C) if and only if there existsx^∗ ∈ A and

z∈Csuch that the supremum in (2.1) is attained atx^∗andkx^∗k_C =|x^∗(z)|.

Proof. (a). Letx∈X.

φ_C(x) = sup

z∈C

kx−zk= sup

z∈C

sup

x^∗∈A

|x^∗(x−z)|= sup

x^∗∈A

sup

z∈C

|x^∗(x−z)|

= sup

x^∗∈A

sup

z∈C

[|x^∗(x)|+|x^∗(z)|] (sinceCis balanced)

= sup

x^∗∈A[|x^∗(x)|+kx^∗k_C].

(b). Suppose x^∗ ∈ A and z ∈ C are such that the sup in (2.1) is attained at x^∗ and kx^∗k_C =|x^∗(z)|. Then for someα∈T,

φ_C(x) =|x^∗(x)|+kx^∗k_C =|x^∗(x)|+|x^∗(z)|=|x^∗(x−αz)| ≤ kx−αzk ≤φ_C(x).

SinceCis balanced,αz∈Cand hence,αz ∈F_C(x)andx∈R(C).

This argument does not needAto be a boundary.

Conversely, supposex₀ ∈R(C). Letz₀ ∈Cbe such thatkx₀−z₀k=φ_C(x₀). SinceAis a boundary, there existsx^∗ ∈Asuch thatkx₀−z₀k=|x^∗(x₀−z₀)|, then

|x^∗(x₀)|+kx^∗k_C ≥ |x^∗(x₀)|+|x^∗(z₀)| ≥ |x^∗(x₀−z₀)|=kx₀−z₀k=φ_C(x₀)≥ |x^∗(x₀)|+kx^∗k_C Hence, equality must hold everywhere. This completes the proof.

Corollary 2.2.3. LetXbe a Banach space andY ⊆Xa subspace.

(a) LetA⊆B_X^∗be a norming set forX. Then for anyx∈X,

φ_B_Y(x) = sup{|x^∗(x)|+kx^∗|_Yk:x^∗ ∈A}. (2.2) (b) IfA ⊆ S_X^∗ is a boundary forX, thenx ∈ R(B_Y)if and only if there existsx^∗ ∈ Aand

z∈B_Y such that thesupin (2.2) is attained atx^∗andkx^∗|_Yk=|x^∗(z)|.

2.3 Characterization of (∗)-subsets

Definition 2.3.1. LetC ⊆B_X be closed andsup_z∈Ckzk = 1. We callC a(∗)-subset ofX if for allx∈X,φ_C(x) =kxk+ 1.

Definition 2.3.2. For a closed and balanced subsetC⊆Xwithsup_x∈Ckxk= 1, define A_C ={x^∗∈S_X^∗ :kx^∗k_C = 1}

IfY is a subspace ofX, we will simply writeA_Y forA_B_Y.

(19)

2.3. Characterization of(∗)-subsets 9 Definition 2.3.3. LetK be a subset of a vector spaceE. A nonempty setM ⊆ K is said to be an extremal subset ofK ifx₁, x₂ ∈ K,0 < λ < 1, andλx₁+ (1−λ)x₂ ∈ M implies x₁, x₂ ∈M. A convex extremal set is called a face. A singleton face is an extreme point.

Proposition 2.3.4. For a closed and bounded setC ⊆ X, ifF_C(x) 6= ∅,F_C(x) is a norm closed extremal subset ofC, but need not be a face.

Proposition 2.3.5. IfC ⊆ X is closed and balanced withsup_x∈Ckxk = 1, then A_C is a norm closed extremal subset ofB_X^∗, but is not a face.

Proof. Clearly¯

¯kx^∗k_C−ky^∗k_C¯

¯≤ kx^∗−y^∗kand hence, the functionk·k_Cis norm continuous.

It follows thatA_C is a norm closed set.

Letx^∗₁, x^∗₂ ∈ B_X^∗ and0 < λ < 1be such thatλx^∗₁ + (1−λ)x^∗₂ ∈ A_C. Then there exists (y_n)⊆C⊆B_X such thatlim_n[(λx^∗₁+ (1−λ)x^∗₂)(y_n)] = 1. It follows that

limn x^∗₁(x_n) = 1 and lim

n x^∗₂(y_n) = 1.

And hence,x^∗₁, x^∗₂ ∈A_C.

SinceA_CisT-invariant,A_C cannot be convex.

We will also need the following lemma repeatedly.

Lemma 2.3.6. IfA ⊆S_X^∗ is such that{x∈X :D(x)∩A 6=∅}is norm dense inX, thenAis a norming set forX.

Proof. Letx∈Xandε > 0. Findz∈ {x ∈X :D(x)∩A6=∅}such thatkx−zk< ε/2. Let z^∗ ∈D(z)∩A. Then

|z^∗(x)|=|z^∗(z)−z^∗(z−x)| ≥ kzk − kz−xk>kxk −ε/2−ε/2 =kxk −ε.

Sinceεis arbitrary,Anormsx.

Definition 2.3.7. Forf :X →R, we define the subdifferential off at anx∈Xas

∂f(x) ={x^∗ ∈X^∗ :Re x^∗(z−x)≤f(z)−f(x), for allz∈X}

As a simple consequence of Hahn-Banach theorem, we have for each continuous convex functionfonX,∂f(x)is a nonempty w*-compact, convex set inX^∗.

We will also need the following result of Lau [30] (see also [16, Proposition II.2.7]). This needs the scalars to be real. For any bounded setK, anyx ∈X, and anyx^∗ ∈∂φ_K(x), we havekx^∗k ≤1, and hence,sup_z∈Kx^∗(x−z)≤φ_K(x). Moreover, the set

G(K) ={x∈X : sup

z∈Kx^∗(x−z) =φ_K(x)for allx^∗∈∂φ_K(x)}

(20)

10 Chapter 2. (∗)-subspaces and the farthest distance formula is a denseG_δinX.

Now we return to arbitrary scalars and to our main result of this chapter.

Theorem 2.3.8. LetC⊆B_X be closed and balanced withsup_x∈Ckxk= 1. Then the following are equivalent :

(a) A_C is a norming set forX.

(b) Cis a(∗)-subset ofX.

(c) There is a dense setG⊆Xsuch thatφ_C(x) =kxk+ 1for allx∈G.

(d) {x∈X :D(x)⊆A_C}is a denseG_δset inX.

(e) {x∈X:D(x)∩A_C 6=∅}contains a denseG_δset inX.

(f) {x∈X:D(x)∩A_C 6=∅}is dense inX.

(g) For every boundaryB forX,B∩A_Cis a norming set forX.

(h) For some boundaryBforX,B∩A_C is a norming set forX.

(i) Intersection of all balls containingCequalsB_X.

Proof. Clearly,(d)⇒ (e) ⇒(f),(g) ⇒(h) ⇒(a)and(b)⇔ (i). Sinceφ_C andk · kare both norm continuous,(b)⇔(c).

(f)⇒(a)follows from Lemma2.3.6withA=A_C. (a)⇒(b). By Theorem2.2.2(a)withA=A_C, we get

φ_C(x) = sup{|x^∗(x)|+kx^∗k_C :x^∗ ∈A_C}= sup{|x^∗(x)|+ 1 :x^∗ ∈A_C}=kxk+ 1.

(b)⇒(d). First let us assume that the scalars are real.

It is easy to see that iff(x) =kxk, then∂f(x) =D(x). Thus,(b)⇒∂φ_C(x) =D(x).

By the result of [30] quoted above,G(C) ={x∈X :for allx^∗∈D(x),sup_z∈Cx^∗(x−z) = kxk+ 1}is a denseG_δsubset ofX.

CLAIM: G(C) ={x∈X:D(x)⊆A_C}.

Letx ∈G(C)andx^∗ ∈D(x). Thensup_z∈Cx^∗(x−z) = kxk+ 1. SinceCis balanced, it follows thatkx^∗k_C = 1. Hencex^∗ ∈A_C.

Conversely, ifx ∈ X is such thatD(x) ⊆ A_C andx^∗ ∈ D(x), thensup_z∈Cx^∗(x−z) = x^∗(x) +kx^∗k_C =kxk+ 1. Thus,x∈G(C). This proves the claim.

If the scalars are complex, consider the real restrictionX_RofX. Recall thatx^∗ →Re x^∗ establishes a real linear isometry between(X^∗)_Rand(X_R)^∗.

If(b)holds, then∂φ_C(x) =D⁰(x) ={Re x^∗ ∈S_X^∗

R :Re x^∗(x) =kxk}. LetA⁰_C ={Re x^∗∈ S_X_R^∗ : kRe x^∗k_C = 1}. Again, by the above arguments, G⁰ = {x ∈ X_R : D⁰(x) ⊆ A⁰_C}

(21)

2.3. Characterization of(∗)-subsets 11 is a dense G_δ set in X_R. As x → x is an isometry from X_R to X, it follows that the set G={x∈X:D(x)⊆A_C}is a denseG_δsubset ofX.

(d) ⇒ (g). Since B is a boundary, for every x ∈ X, TB ∩D(x) 6= ∅. Since {x ∈ X : D(x) ⊆A_C} ⊆ {x ∈ X : D(x)∩[TB∩A_C]6= ∅}, by(d), the right hand set is dense inX.

By Lemma2.3.6,TB ∩A_C is norming forX. SinceA_C isT-invariant,B∩A_C is norming forX.

PuttingC =B_Y in Theorem2.3.8we obtain the following characterization theorem:

Theorem 2.3.9. For a subspaceY of a Banach spaceX, the following are equivalent : (a) A_Y is a norming set forX.

(b) Y is a(∗)-subspace ofX.

(c) There is a dense setG⊆Xsuch thatφ_B_Y(x) =kxk+ 1for allx∈G.

(d) {x∈X :D(x)⊆A_Y}is a denseG_δset inX.

(e) {x∈X:D(x)∩A_Y 6=∅}contains a denseG_δset inX.

(f) {x∈X:D(x)∩A_Y 6=∅}is dense inX.

(g) For every boundaryB forX,B∩A_Y is a norming set forX.

(h) For some boundaryBforX,B∩A_Y is a norming set forX.

(i) Intersection of all balls containingB_Y equalsB_X.

Remark 2.3.10. In [10], the authors used(a)as the definition of a(∗)-subspace and proved (a)⇒(b).

Here are some natural examples of(∗)-subspaces, as observed in [10].

Example 2.3.11. (a) Xis a(∗)-subspace ofX^∗∗, sinceS_X^∗ ⊆A_X (See Chapter7).

(b) IfY ⊆Z ⊆XandY is a(∗)-subspace ofX, thenZ is a(∗)-subspace ofXandY is a(∗)-subspace ofZ.

(c) For any two Banach spacesXandY,K(X, Y)is a(∗)-subspace ofL(X, Y). To see this, note thatA_K(X,Y₎⊇ {x⊗y^∗:x∈S_X, y^∗ ∈S_Y^∗}, which already normsL(X, Y).

(d) For a compact Hausdorff spaceK,C(K, X)is a(∗)-subspace ofW C(K, X).

Remark 2.3.12. Recall that a Banach spaceX has the Mazur Intersection Property (MIP) if every closed bounded convex set inXis the intersection of closed balls containing it. From (i)above, it follows that a space with the MIP cannot have a proper(∗)-subspace. On the other hand, it has been noted in [10, Proposition 2.8] that a wLUR Banach space also has no proper(∗)-subspace.

(22)

12 Chapter 2. (∗)-subspaces and the farthest distance formula It now follows from Corollary2.2.3that

Proposition 2.3.13. [10, Proposition 2.10]IfY is a(∗)-subspace ofX, thenx₀ ∈R(B_Y)if and only if there existsx^∗ ∈S_X^∗andy∈S_Y such that

x^∗(x₀) =kx₀k and x^∗(y) = 1.

As noted in [10, Remark 2.11], it follows from the result above that for(∗)-subspaces, R(B_Y)is closed under scalar multiplications. We will show later in Example5.2.11that this need not be true ifY is not a(∗)-subspace.

Since the sets involved areT-invariant, we actually get

Corollary 2.3.14. IfY is a(∗)-subspace ofX, thenx₀ ∈R(B_Y)if and only if there existsx^∗ ∈S_X^∗ andy ∈S_Y such that

|x^∗(x₀)|=kx₀k and |x^∗(y)|= 1.

Proposition 2.3.15. LetN_Y ={x^∗ ∈S_X^∗:x^∗(y) = 1for somey ∈S_Y}. Then

(a) Y is(∗)and DBR if and only if{x∈X:N_Y ∩D(x)6=∅}is dense inX. In particular, if Y is(∗)- and DBR, thenN_Y is norming forX.

(b) Y is a(∗)and BR subspace ofXif and only ifN_Y is a boundary forX.

Proof. Clearly,N_Y ⊆A_Y. If{x∈X:N_Y ∩D(x)6=∅}is dense inX; in particular, ifN_Y is a boundary forX, then by Theorem2.3.9,Y is a(∗)-subspace.

On the other hand, ifY is a(∗)-subspace ofXthen, by Proposition2.3.13,R(B_Y) ={x∈ X:N_Y ∩D(x)6=∅}. Hence the result.

The last part of(a)follows from the first part and Lemma2.3.6.

Corollary 2.3.16. IfY is a(∗)-subspace andA_Y ∩N A(X) =N_Y, thenY is DBR inX.

Proof. Clearly,N_Y ⊆A_Y ∩N A(X). Note thatN A(X)is aT-invariant boundary forX. So, if Y is a(∗)-subspace then as in the proof of Theorem2.3.8(d)⇒(g),{x∈X:A_Y ∩N A(X)∩ D(x)6=∅}is dense inX. SinceA_Y ∩N A(X) =N_Y andR(B_Y) ={x∈X :N_Y ∩D(x)6=∅}, Y is DBR.

Corollary 2.3.17. (a) IfY ⊆Z ⊆X andY is a(∗)- and BR subspace ofX, thenZ is a(∗)- and BR subspace ofX.

(b) IfY ⊆Z ⊆XandY is a(∗)- and DBR subspace ofX, thenZis a(∗)- and DBR subspace ofX.

(23)

2.4. Strong unitaries and 1-dimensional(∗)-subspaces 13 Remark 2.3.18. However, ifY ⊆ Z ⊆ XandY is a(∗)- and DBR subspace of X, then we do not know ifY must be a(∗)- and DBR subspace ofZ. Also ifY ⊆Z ⊆XandY is a DBR subspace ofX, thenZneed not be a DBR subspace ofX(Example4.2.50).

Proposition 2.3.19. (a) If Y is a (∗)-subspace of a strictly convex Banach space X, then R(B_Y) =Y.

(b) A strictly convex space cannot have a proper(∗)and DBR subspace.

(c) A reflexive strictly convex space has no proper(∗)subspace.

(d) Any Hilbert space, the spacesL_p([0,1])and`_p,1< p <∞has no proper(∗)subspace.

Proof. (a). LetXbe a strictly convex space andY be a(∗)-subspace ofX.

Letx∈R(B_Y). We may assumekxk = 1. Then there existsy ∈B_Y such thatkx+yk= φ_B_Y(x) = 2. SinceXis strictly convex,x=y.

Now,(a)⇒(b)⇒(c)by Theorem1.0.1and(c)⇒(d).

Remark 2.3.20. (d)also follows from [10, Proposition 2.8] since these spaces are LUR.

2.4 Strong unitaries and 1-dimensional (∗)-subspaces

It may seem that a(∗)-subspace must be somewhat large. This, however, is not the case. A Banach space may even have 1-dimensional(∗)-subspaces.

Theorem 2.4.1. LetXbe a Banach space andx₀ ∈S_X. The following are equivalent : (a) D(x₀)is a norming set forX.

(b) D(x₀)is a boundary forX.

(c) Fx₀is a(∗)-subspace ofX.

Proof. (a) ⇔ (c). Observe that ifY = Fx₀, then for anyx^∗ ∈ X^∗, kx^∗|_Yk = |x^∗(x₀)|. It follows thatA_Y ={x^∗ ∈S_X^∗ :|x^∗(x₀)|= 1}=TD(x₀).

Thus,Y is a(∗)-subspace ofX⇔D(x₀)is a norming set forX.

SinceD(x₀)is w*-compact,(a)⇔(b).

Definition 2.4.2. Let X be a Banach space. Let us call x₀ ∈ S_X a strong unitary if the equivalent conditions in Theorem2.4.1are satisfied.

For the origin of this terminology and related results, see [8] or the survey article [33].

Corollary 2.4.3. Ifx₀∈S_X is a strong unitary inXandY is a subspace withx₀ ∈Y, then (a) x₀ is a strong unitary inY.

(24)

14 Chapter 2. (∗)-subspaces and the farthest distance formula (b) Y is(∗)and BR inX.

Example 2.4.4. Here are some natural examples of strong unitaries in Banach spaces.

(a) The constant sequence1incor`_∞is a strong unitary.

(b) The canonical unit vectors in`₁ are strong unitaries.

(c) Any unimodular function inC(K)is a strong unitary inC(K).

(d) A commutativeC^∗ algebraAwith identity contains strong unitaries. To see this, note that the Gelfand transform induces an isometric(∗)-isomorphism fromAonto C(Σ), whereΣis the maximal ideal space ofA[15, Theorem VIII.2.1].

On the other hand, ifAis a commutativeC^∗ algebrawithoutidentity then it does not contain any strong unitaries. Indeed, we show later (Corollary4.3.8) that such a space has no finite-dimensional(∗)-subspaces.

There are also 2-dimensional(∗)-subspaces that do not contain a strong unitary.

Example 2.4.5. Consider the subspace Y ⊆ c spanned by x = (sin¹_n) and y = (cos_n¹).

Taking vectors of the formsin¹_k·x+ cos¹_k·y, one can see thatA_Y contains all the coordinate functionals. Hence,Y is a(∗)-subspace.

This example appears in a related context in [4, Example 2.34].

(25)

C

HAPTER

3

Ball remotality in some classical Banach spaces

3.1 Summary of results

In this chapter, we study ball remotality in the classical Banach spacesc₀(Γ),c(Γ),`_∞(Γ)for some arbitrary index setΓ. The farthest distance formula—Theorem2.2.2—takes a simpler form in these spaces. Using this, we first characterize(∗)- and densely remotal subsets (The- orem3.2.3) and thereby,(∗)- and DBR subspaces ofc₀(Γ),c(Γ)and`_∞(Γ)(Corollary3.2.4).

In the process, we prove that (Corollaries3.2.5and3.2.6) : (a) ForY ⊆c₀(Γ), the following are equivalent :

(1) Y is(∗)- and DBR inc₀(Γ) (2) Y is(∗)- and BR inc₀(Γ) (3) Y is(∗)- and DBR inc(Γ) (4) Y is(∗)- and DBR in`_∞(Γ).

(b) In particular,c₀(Γ)is a(∗)- and DBR subspace of bothc(Γ)and`_∞(Γ).

(c) If a subspace ofc(Γ)or`_∞(Γ)contains the constant vector1, then it is(∗)- and BR.

(d) c(Γ)andbc(Γ), the canonical image ofc(Γ), are(∗)- and BR in`_∞(Γ).

We also show thatc₀(Γ)has no finite dimensional(∗)-subspace (Theorem3.2.8). We characterize all hyperplanes in c₀(Γ) which are (∗)- and DBR in terms of the defining linear functionals (Theorem3.2.16).

Then we come to the space`₁(Γ)and observe that`₁(Γ)with its usual norm provides a simple non-reflexive example of a Banach space in which every subspace is DBR (Corol- lary3.3.2). We show that a hyperplane in`₁is BR if and only if it contains a strong unitary (Theorem3.3.6). However, ifΓis uncountable, a hyperplane in`₁(Γ)is always(∗)- and BR (Theorem3.3.8).

(26)

16 Chapter 3. Ball remotality in some classical Banach spaces

3.2 Ball remotality in c

₀

(Γ), c(Γ) and `

_∞

(Γ)

Notation 2. LetΓbe an arbitrary index set. Define

(a) c₀(Γ) = {x = (x_γ)_γ∈Γ :given ε > 0, there exists a finite subsetΓ₁ ⊆ Γsuch that

|x_γ|< εfor allγ /∈Γ₁}.

(b) `_∞(Γ) ={x= (x_γ)_γ∈Γ:kxk_∞= sup_γ∈Γ|x_γ|<∞}.

(c) `₁(Γ) = {x = (x_γ)_γ∈Γ :there existsM > 0such thatP

γ∈A|x_γ| ≤ M, for all finite subsetsAofΓ}.

(d) c(Γ) ={x = (x_γ)_γ∈Γ :there existsλ∈Csuch that for allε >0there exists a finite subsetΓ₁ ⊆Γsuch that|x_γ−λ|< εfor allγ /∈Γ₁}.

IfΓ =N, we get back the classical sequence spacesc₀,`_∞,`₁andc.

Let{e_γ}denote the canonical unit vectors in X = c₀(Γ),c(Γ)or `_∞(Γ)and{e^∗_γ} is the coordinate functionals in`₁(Γ) ⊆ X^∗. Note that{e^∗_γ : γ ∈ Γ} ⊆ `₁(Γ) is a boundary for X=c₀(Γ)and is a norming set forX=c(Γ)or`_∞(Γ).

Remark 3.2.1. (a) It is clear thatc(Γ) =C(Γ_∞), whereΓ_∞= Γ∪ {∞}is the one-point compactification ofΓendowed with discrete topology. Clearly,c₀(Γ)is identified with the subspace ofC(Γ_∞)that "vanish at"∞.

(b) Clearly,c(Γ)is a subspace of`_∞(Γ). Sincec(Γ)^∗∗ =`_∞(Γ), there is also a canonical embeddingbc(Γ)ofc(Γ)in`_∞(Γ). However, due to the nature of the action of`₁(Γ)on c(Γ),bc(Γ)6=c(Γ). For a fixedγ₀ ∈Γ, it can be shown thatbc(Γ) ={x∈c(Γ) :x_γ₀ =λ}

whereλ∈Ccorresponds toxas in the definition ofc(Γ).

Proposition 3.2.2. Let X be one of c₀(Γ), c(Γ) or`_∞(Γ). Let C ⊆ X be closed, bounded and balanced. Forγ ∈Γ, let

M_γ= sup

z∈C

|z_γ| (3.1)

Then

(a) for anyx₀ = (x_γ)∈X,

φ_C(x₀) = sup{|x_γ|+M_γ:γ ∈Γ}. (3.2) (b) If there exist γ ∈ Γ andz ∈ C such that φ_C(x₀) = |x_γ|+M_γ = |x_γ|+|z_γ|, then

x₀ ∈R(C). IfX=c₀(Γ), the converse is also true.

IfX=c(Γ)andC ⊆c(Γ)is closed, bounded and balanced. Then (c) for anyx₀∈c(Γ),

φ_C(x₀) = sup{|x_γ|+M_γ:γ ∈Γ_∞}, (3.3) whereM_∞= sup_z∈C|z_∞|.

(27)

3.2. Ball remotality inc₀(Γ), c(Γ)and`_∞(Γ) 17 (d) x₀ ∈ R(C, c(Γ)) if and only if there exist γ ∈ Γ_∞ and z ∈ C such that φ_C(x₀) =

|x_γ|+M_γ =|x_γ|+|z_γ|.

Proof. For(a)and(b), apply Theorem2.2.2withA={e^∗_γ:γ ∈Γ} ⊆`₁(Γ)⊆X^∗. For(c)and(d), takeA={e^∗_γ :γ ∈Γ_∞} ⊆c(Γ)^∗.

Theorem 3.2.3. LetXbe one of the spaces c₀(Γ),c(Γ)or`_∞(Γ)andC ⊆ X be closed, balanced andsup_x∈Ckxk= 1.

(a) Cis a(∗)-subset ofXif and only if for allγ ∈Γ,M_γ= sup_z∈C|z_γ|= 1.

(b) C is a(∗)-subset and densely remotal if and only if for allγ ∈ Γ, there existsy ∈C such that|y_γ|= 1.

(c) A(∗)-subsetC ⊆c₀(Γ)is densely remotal if and only if it is remotal.

(d) IfX=c(Γ)or`_∞(Γ)andCcontains the constant vector1, thenCis remotal.

Proof. (a). Necessity follows from Proposition3.2.2.

If for someα ∈Γ,M_α <1, let0 < δ < 1−M_α. Then for ally ∈C,|y_α| ≤M_α <1−δ.

Therefore, for ally ∈ C, ke_α−yk_∞ = max{|1−y_α|,sup_β6=α|y_β|} ≤ 1 +|y_α| ≤ 2−δ. So φ_C(e_α)<2 =ke_αk_∞+ 1.

(b). IfCis a(∗)-subset and densely remotal, then by(a),M_γ = 1for allγ ∈Γ. Suppose there existsα∈Γsuch that for ally∈C,|y_α|<1.

CLAIM: Ifz∈Xsuch that|z_α|=kzk_∞>sup_γ6=α|z_γ|, thenz /∈R(C).

Otherwise, there exists y ∈ C, kz −yk_∞ = kzk_∞ + 1 = |z_α|+ 1. For any γ 6= α,

|z_γ−y_γ| ≤ |z_γ|+ 1. It follows thatsup_γ6=α|z_γ−y_γ| ≤sup_γ6=α|z_γ|+ 1<|z_α|+ 1. Therefore, we must have|y_α|= 1. This proves the claim.

Now, ifkz−e_αk_∞<1/3, thensup_γ6=α|z_γ|<1/3and|z_α|>2/3and hence, by the claim, z /∈R(C). HenceCcannot be densely remotal.

Conversely, if for allγ ∈Γ, there existsy ∈Csuch that|y_γ|= 1, then clearlyM_γ = 1for allγ ∈Γand this value is attained. Thus, by(a),Cis a(∗)-subset.

LetR ={x∈X :kxk_∞=|x_γ|for someγ∈Γ}. By Proposition3.2.2,R⊆R(B_Y).

IfX =c₀(Γ),R=c₀(Γ)and(c)follows. IfX =`_∞(Γ)orc(Γ), andx /∈R, letε >0.

Letkxk_∞ =m. There existsα ∈Γsuch thatm−ε <|x_α| ≤m. Definez= (z_γ)by the following

z_γ =

( x_γ ifγ 6=α sgn(x_α)⁻¹m ifγ =α

thenz∈Randkz−xk_∞=|x_α−z_α|=m− |x_α|< ε. Hence,Ris dense inX.

(d). Suppose1 ∈ C. Ifx /∈ R, then there exists a sequence{γ_n} ⊆Γsuch that|x_γ_n| → kxk_∞. Passing to a subsequence, if necessary, we may assume that{x_γ_n}is convergent, to

Ball remotality in Banach spaces and related topics

Ball Remotality In Banach Spaces And Related Topics

Author: Tanmoy P

Thesis submitted to the Indian Statistical Institute

in partial fulfillment of the requirements for the award of the degree of

Doctor of Philosophy

Ball Remotality In Banach Spaces And Related Topics

Dedicated to all those people who love Banach spaces . . .

Acknowledgments

Contents

C

1

Introduction

C

2

(∗)-subspaces and the farthest distance formula

2.1 Summary of results

2.2 The farthest distance formula

2.3 Characterization of (∗)-subsets

2.4 Strong unitaries and 1-dimensional (∗)-subspaces

C

3

Ball remotality in some classical Banach spaces

3.1 Summary of results

3.2 Ball remotality in c

(Γ), c(Γ) and `

(Γ)