Small Combination of Slices inBanach Spaces

Recent Trends in Operator Theory and Applications

May 3rd to May 5th 2018University of Memphis

Sudeshna BasuGeorge Washington University

Notations

• X : Banach Space• !∗ Dual of X• #$ ={ x ϵ X : ||x|| ≤ 1}• %$ ={ x ϵ X : ||x||= 1}

Suppose , C⊆ X, D ⊆ #∗• Let f ∈ #∗ , and α >0 ,then S( C, f, α) = { x∈ C: f(x)> sup f(C) – α} is the open slice of C determined by f and α.• A point x∈ C , is called denting if the family of

open slices containing x forms a base for the norm topology at x( relative to C)

• If, D ⊆ #∗ and the slices are determined byfunctionals from X, we have 4∗-slices and 4∗ -denting points respectively.

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A generalization of denting points

• What if a slice is replaced by a “combination” of slices?

• A point x in a closed bounded convex set K in X is called a SCS ( small combination of slices)point of K, if for every !> 0, there exist slices "# of K, and a convex combination S =∑%# λ' "' such that x ∈S and diam (S)

• One can analogously define )∗-SCS points in the dual space.

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Denting points and SCS points

Denting Point ⇒PCPoint ⇒SCSPoint*∗ version of this result is also true

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Strongly Regular Sets

A closed bounded, convex set K ⊆ X is called strongly regular⇔ for every non empty subset C contained in K and #> 0 there are $% ,……….. $& of C suchthat diam ( ∑%& λ)$) )

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SCS points first introduced by N. Ghoussoub , G. Godefory , B. Maureyand W. Scachermeyer, as a ``slice generalisation" of the denting point and point of continuity points .

Later, Scachermeyer proved that a Banach space has Radon NikodymProperty (RNP) ⇔ X is strongly regular and it has the Krien Milman Property(KMP).

Subsequently, the concepts of SCS points was used by Rosenthal to investigate the structure of non dentable closed bounded convex sets in Banach spaces The "point version" of the results by Scachermeyer (i.e. charasterisation of RNP),was were proved by Hu and Lin

The "point version" of the results by Scachermeyer (i.e. charasterisationof RNP),was were proved by Hu and Lin .

Closed convex sets in Dual ( Hu and Lin) Let K he a closed bounded convex set in X* Then:w*-dent K =(w*-pc K) ∩ ext K =w*-SCS(K) ∩ext K.

( the first equality is proved in more generalityBy Lin, Lin and Troyanski we will refer to this result later )

Ball-SCSPA Banach space X is said to have Ball-Small Combination of Slices Property (BSCSP) ifthe unit ball has small combination of slices of arbitrarily small diameter. Analogously we can define !∗-BSCSP in a dual space.It is clear that if #$ has a SCS point, then it has BSCSP.

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!"-sum and SCS pointsSuppose X, Y, Z are Banach spaces such that Z = X ⊕"Y , suppose $%=( &%, (%) &% ≠ 0, (% ≠ 0, is a SCS-point of +,, then &% -./ (% are SCS point of +0and +1 respectively.

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!∗ −SCS pointsSuppose X, Y , Z are Banach spaces such that Z = X ⊕% Y , suppose &∗ = ('∗, (∗) ∈ *+∗ is a !∗ -SCS point of with both the components non-zero, then '∗ and (∗are !∗ −SCS points of *,∗ and *-∗respectively.

Example

Since in the sequence space !", any weakly open set has norm diameter 2, by taking X = #$, and Y = !% , Z = X ⊕"Y , any w∗-SCS point of()∗ has its second component 0

SCS points and !"-sums ( 1≤p

U-subspace and Ideal • A closed subspace Y of X is called a U subspace if for all &∗ϵ )∗, there exists a unique norm preserving extension of &∗ in *∗.

• A closed subspace Y of X is called an ideal if there exists a linear projection P:*∗→*∗ of norm one such that ker(P)= ),.

• An ideal M is an M ideal ifM is a U subspace and *∗= 9∗ ⊕; 9,

• An ideal Y is called a strict ideal if

SCS points in U-subspaces and M-ideals

Theorem :Suppose Y is an ideal which is also a U-subspace of X. If !∗ ∈ $%∗ is a &∗-SCS point of '(∗ , then y∗ is a w∗-SCS point of '%∗ .Corollary: Suppose M ⊆ X is an M-ideal in X. If *∗∈ $+∗ is &∗-SCS point of '(∗, then *∗∈ $+∗ is a &∗-SCS point of '+∗.

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SCS point in a Strict ideal

Suppose Y is a strict ideal of X. If !∗ ∈ $%∗ is a&∗-SCS point of $%∗, then !∗is a &∗ -SCS point of $'∗.

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M ideals

Theorem: Suppose M is an M ideal in X. If !∗ has #∗-BSCSP, then $∗ also has #∗-BSCSP.

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Corollaries

Corollary: Suppose X = ⊕" #$. If #$∗ has the &∗-BSCSP for some i, then #∗ has the &∗ -BSCSP.Corollary :Suppose K is a compact Hausdorffspace with an isolatedpoint. If then #∗ has the &∗-BSCSP, then has '(), #)∗ the &∗-BSCSP.

BSCSP in a Strict ideal

Proposition : Suppose Y is a strict ideal of X. If !∗ has &∗ -BSCSP, then '∗ also has &∗ -BSCSP.

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!"-Sum( 1≤ p ≤∞)Theorem :Suppose X = ⊕%& '(. X has -BSCSP if and only if '( has the BSCSP for some i.

(i) The )∗- version of this result is also true.(ii)The above arguments extend easily to vector-valued continuous functions

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Density Properties

Let us consider the following densities of !∗-SCS points of #$∗(i) All points of %$∗ are !∗-SCS points of #$∗.(ii) The !∗-SCS points of #$∗. are dense in %$∗ .(iii) #$∗ is the closure of !∗-SCS points of #$∗(iv) #$∗ is the closed convex hull of !∗-SCS points of #$∗(v) X is the closed linear span of !∗-SCS points of #$∗(SSCSP)

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Equivalence

• (Lin, Lin and Troyanski)• A point in the unit ball is denting if and only if it is PC and extreme

Can one characterize denting points in terms of SCS points ?

X is said to have SSCSP ( span small combination of slices property) if

!∗ = $%&' ()∗-SCS points of *+∗)

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Stability

• SSCSP is stable under !" sums• SSCSP is stable under #$ sums

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!" (#,X)Let X be a Banach space, # the Lebesguemeasure on [0,1], and 1

C(K)• For C(K) TFAEi)C(K) is Nicely Smoothii) C(K) has BGP, iii) C(K) has SSCSP, iv) C(K) has Property (II) v)K is finite.

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What happens in C(K,X)? It turns out that C(K,X) has SSCSP P if and only if X has SSCSP and K is finite.

• Stability of SSCSP under !" −sums.( whenever that is true)

• The set A = { δ(k) × %∗: k ∈ K, %∗ ∈ ()∗} a subset of the unit ball of the dual of C(K,X) turns out to be a norming set and does the job.

P cannot be ANPI,II’and MIP 27

If X εY i.e. the injective tensorproduct of X and Y has BGP(NS),then X and Y also has BGP(NS).

INJECTIVE TENSOR PRODUCTS

Question : Does similar result for SSCSP?

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Converse

If X* and Y* has RNP, TFAEa) X and Y are nicely smoothb) X !Y is nicely smoothc) X and Y has BGPd) X ε Y has BGPe) X and Y has SSCSPf) X εY has SSCSP

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Projective Tensor Product

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Proposition 19. Suppose X, Y are Banach space with RNP, such that X hasthe approximation property and L(X, Y* ) = K(X, Y* ) i.e. any bounded linear operator from X to Y* is compact then, TFAE(i) K(X, Y* ) has SSCSP.(ii) K(X, Y *) has BGP.(iii) K(X, Y *) is nicely smooth.(iv) X, Y are reflexive.(v) X ! Y is reflexive.Ä

Open Questions(i) How can each of these properties be realised as a ball separation property?(ii) What stability results will hold for these properties?(iii) D(2P)-properties is a recent topic which generated a lot of interest in the study of Banach spaces, it is known that Banach spaces wiith Daugavetproperties have these properties.

It is also know that Banach spaces with Daugavet properties do not have RNP. In fact one can conclude easily that Banach Spaces with Daugavet propertiesdo not have SSCSP and do not have MIP. But there are examples of spaces with D-2P which has SSCSP ,evenStronger properties . So it will be interesting to examine where all ballseparation stand in the context of D-2P properties.

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