Slicely countably determined sets

Vladimir Kadets

Kharkiv V.N.Karazin National University

International conference“Banach Spaces and their Applications”

dedicated to 70th anniversary of A.M.PlichkoLviv, June 26 – 29, 2019

Outline

Motivation and some references

Slices and weak neighborhoods

SCD sets: definition and examples

Daugavet property and sets that are not SCD

The alternative Daugavet equation and numerical index 1.

Operations with SCD sets, spaces and hereditarily SCD sets

Open problems

Motivation

We are going to speak about SCD sets, SCD spaces and SCDoperators. The abbreviation “SCD set” comes from “Slicely Coun-tably Determined” which means that there is a countable familyof slices that determines the set in a sense that will be preci-sed a few slides later. This concept appeared 10 years ago inrelation with Daugavet property and spaces with numerical in-dex 1, and by now its major applications live in the same area ofresearch. I have a feeling that this property is of independent in-terest, and that there should be applications in other areas. Thisis the reason why I am speaking about this property today.

In the next two frames you will see a list of papers devoted toSCD circle of problems and their applications.

Papers that develop the SCD theory

A. AVILÉS, V. KADETS, M. MARTÍN, J. MERÍ,V. SHEPELSKA. Slicely countably determined Banachspaces, C. R., Math., Acad. Sci. Paris 347, no. 21–22,1277–1280 (2009).

A. Avilés, V. Kadets, M. Martı́n, J. Merı́, V. Shepelska.Slicely countably determined Banach spaces, Trans. Amer.Math. Soc. 362, 4871–4900 (2010).

V. KADETS AND V. SHEPELSKA. Sums of SCD sets andtheir application to SCD operators and narrow operators,Cent. Eur. J. Math. 8, no. 1, 129–134 (2010).

V. KADETS, A. PÉREZ, D. WERNER. Operations with slicelycountably determined sets, Funct. Approx. Comment.Math. 59, no. 1, 77–98 (2018).

Below, there will be no references for those results that comefrom the second paper (there are too many of them ,).

Sources where some applications are given

T. Bosenko (Ivashina). Strong Daugavet operators andnarrow operators with respect to Daugavet centers. Visn.Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 931, No. 62, 5–19(2010).

V. Kadets, M. Martı́n, J. Merı́, D. Werner. Lushness,numerical index 1 and the Daugavet property inrearrangement invariant spaces. Canad. J. Math. 65,331–348 (2013).

V. Kadets, M. Martı́n, J. Merı́, D. Werner. Lipschitz slicesand the Daugavet equation for Lipschitz operators. Proc.Amer. Math. Soc. 143, no. 12, 5281–5292 (2015).

V. Kadets, M. Martı́n, J. Merı́, A. Pérez. Spear operatorsbetween Banach spaces. Lecture Notes in Mathematics2205. Springer, 2018

Slices

Let X be a Banach space, A ⊂ X be a not empty subset. A sliceof A is a not empty part of A that is cut out by a hyperplane.

For x∗ ∈ X ∗ and ε > 0 denote the corresponding slice as

S(A, x∗, ε) = {x ∈ A : Re x∗(x) > sup Re x∗(A)− ε}.

Weak neighborhoods and slices

A slice of A that contains a point x ∈ A is a relative weakneighborhood of x . Moreover, finite intersection of slices forma base of relative weak neighborhoods of x .

Determining sequences and SCD sets

Let X be a Banach space and A 6= ∅ be a subset of X . A se-quence of nonempty subsets Vn ⊂ A, n ∈ N is called deter-mining for A if for every B ⊂ A intersecting all the Vn one hasconvB ⊃ A.

The set A is said to be an SCD set if A is bounded and there isa determining sequence of slices of A .

In other words, a bounded set A ⊂ X is SCD if there is asequence Sn of slices, such that for every selection of pointsxn ∈ Sn, n ∈ N one has conv{xn}n∈N = A. Evidently every SCDset is separable.

X is said to be an SCD space if all bounded subsets of X areSCD sets. A (linear bounded) operator T : X → Y is said to bean SCD operator, if T (BX ) is an SCD set.

Convexification

We can restrict ourselves to study bounded, closed and convexsets (bcc sets for short) because of the following result:

Proposition (K., Martı́n, Merı́, Pérez, 2018). Let X be a Banachspace. A bounded set A ⊂ X is SCD if and only if its convex hullconv(A) is SCD, and if and only if its closed convex hull conv(A)is SCD.

Denting points

A denting point of A is such a point that has slices of arbitrarilysmall diameter. On the picture below b is a denting point, and ais not.

Every denting point is an extreme point, but in infinite-dimensionalcase there are extreme points that are not denting. A typicalexample is point 1 ∈ BC[0,1].

Many denting points give SCD

Proposition. Let A be such a separable bcc set that the closedconvex hull of the set of its denting points is equal to A. Then Ais an SCD set.

Proof. At first, select a dense countable subset {dn}n∈N of theset of denting points of A. Then, conv{dn}n∈N = A. Next, forevery n ∈ N select slices Sn,m of A, m ∈ N such that dn ∈ Sn,mand diamSn,m 6 1m . Let us demonstrate that the collection Sn,m,n,m ∈ N is determining for A. Indeed, take arbitrary xn,m ∈ Sn,m.Then, limm→∞ xn,m = dn. Consequently, conv{xn,m}n,m∈N ⊃conv{dn}n∈N = A. 2

As a corollary we obtain that many “good” separable bcc sets,like compacts, weak compacts, or sets with the RNP, are SCDsets, and many “good” spaces like Lp with 1 < p

Countable π-base of relative weak topology gives SCD

Proposition. Let A ⊂ X be a bcc set, and (A, σ(X ,X ∗)) has acountable π-base, i.e. there is a sequence Un ⊂ A of relativelyweakly open subsets, such that for every relatively weakly opensubset U ⊂ A there is a Un ⊂ U. Then A is an SCD set.

Proof. For each n ∈ N select a finite collection of slices Sn,k ,1 6 k 6 mn such that some convex combination

∑mnk=1 λn,kSn,k

lies in Un (such a selection exists due to a Bourgain’s lemma).Then the countable collection of slices Sn,k , n ∈ N, 1 6 k 6 mnis determining. 2

This proposition implies that all separable spaces that do notcontain isomorphic copies of `1 (in particular, all spaces withseparable dual) are SCD.

Problem. Is it true that every bcc SCD set has a countable π-base of its relative weak topology?

A reformulation motivating the previous question

Proposition. For a bcc set A ⊂ X the following conditions areequivalent:

1. A is SCD.2. A possesses a “countable π-base of slices” in the following

sense: there is a sequence Sn ⊂ A, n ∈ N of slices, suchthat for every slice S ⊂ A there is an Sn ⊂ S.

Unconditional bases and SCD sets

Proposition (K., Martı́n, Merı́, and Werner, 2013). Let X be aspace with 1-unconditional basis. Then its closed unit ball BX isan SCD set.

Problem. Is it true that every Banach space with an unconditio-nal basis is an SCD space?

Daugavet equation

In frames of approximation theory it is often significant to know,whether for a given subspace Y of a Banach space X there isa norm-one linear projection P: X → Y . It is usually a goodexercise for students to find an example of Y ⊂ X where sucha norm-one projection does not exist. An easy solution is thesubspace Y ⊂ C[0,1] of those functions that f (0) = 0. In 1963I. K. Daugavet discovered the following effect: for every compactoperator T : C[0,1]→ C[0,1] the identity

‖Id + T‖ = 1 + ‖T‖,

called now the Daugavet equation, holds true. The proof can beeasily generalized to C(K ) on a perfect compact K . An evidentcorollary of this is that every projection on a finite - codimensio-nal subspace of C[0,1] has at least norm 2.

Daugavet property

A Banach space X has the Daugavet property if the Daugavetequation holds true for every operator T : X → X of rank one.We abbreviate this by writing X ∈ DPr.

Outside of C(K ) on a perfect compact K , the Daugavet propertyis possessed by the following important spaces: L1(Ω,Σ, µ) andL∞(Ω,Σ, µ) for non-atomic µ and analogous spaces of vector-valued functions C(K ,X ), L1(µ,X ) and L∞(µ,X ); the disc alge-bra and non-atomic C*-algebras.

The major application of SCD sets is the fact that in a space withthe Daugavet property every SCD operator T : X → X satisfiesthe Daugavet equation.

The Daugavet property and slices

Theorem (K., Shvidkoy, Sirotkin, Werner, 2000). For a Banachspace X the following conditions are equivalent:

I X ∈ DPr;I for every x ∈ SX , for every ε > 0, and for every slice S of

the unit ball there is some y ∈ S such that ‖x − y‖ > 2− ε.I for every x ∈ SX , for every ε > 0, and for every sequence

of slices Sn, n ∈ N of the unit ball there are xn ∈ Sn suchthat the sequence x , x1, x2, . . . is (1 + ε)-equivalent to thecanonical basis of `1.

This means in particular that every slice S of the unit ball of aspace X ∈ DPr has diamS = 2.

The Daugavet property and sets that are not SCD

The previous theorem implies that the unit ball of a space X ∈DPr cannot be SCD. Moreover, it satisfies a kind of “anti-SCD”condition: for every sequence (Sn) of slices of BX and everyx ∈ SX there is a set B = {xn : n ∈ N} with xn ∈ Sn, n = 1,2, . . . ,such that x /∈ convB.

So, in particular, the closed unit ball of C[0,1] is an example ofa separable bcc set that is not SCD.

Problem. Is it true that every separable Banach space that isnot an SCD space is isomorphic to a space with the Daugavetproperty?

The alternative Daugavet equation

Everybody knows the formula ‖T‖ = sup{|〈Tx , x〉| : x ∈ SH} forthe norm of a self-adjoint operator T in a Hilbert space H. Insome Banach spaces X an analogous formula

‖T‖ = sup{|x∗(Tx)| : x ∈ SX , x∗ ∈ SX∗ , x∗(x) = 1} (1)

holds true for EVERY operator T : X → X . Such spaces are saidto have numerical index 1. It can be shown that (1) is equivalentto the following alternative Daugavet equation

sup{‖Id + θT‖ : |θ| = 1} = 1 + ‖T‖, (2)

so n(X ) = 1 iff (2) holds true for every operator T : X → X .

More applications

Theorem. If in X all rank-1 operators satisfy the alternative Dau-gavet equation (this is called the alternative Daugavet property),then all SCD operators do the same.

Theorem. Every space with the alternative Daugavet propertywhose unit ball is an SCD set has numerical index 1. In parti-cular, every SCD-space with the alternative Daugavet propertyhas numerical index 1.

Operations with SCD spaces

SCD spaces are preserved (evidently) by passing to subspaces,but NOT preserved by passing to quotient spaces. This class isstable by finite direct sums, and by 1-unconditional infinite sumswith respect to shrinking bases and with respect to boundedlycomplete bases.

The following three space property also holds true:

Let X be a Banach space with a subspace Z such that Z andX/Z are SCD spaces. Then, X is also an SCD space.

Operations with SCD sets

For sets the SCD property is very unstable. Namely, in everyseparable Banach space with the Daugavet property (that is, inevery separable space where by now we know examples of non-SCD sets) there are

I convex closed bounded symmetric with respect to originSCD sets A, B such that A ∪ B is not SCD;

I convex closed bounded symmetric with respect to originSCD sets A, B such that A ∩ B is not SCD;

I convex closed bounded symmetric with respect to originSCD sets A, B such that A + B is not SCD.

(K., Pérez and Werner, 2018).

Hereditarily SCD sets

A bounded set is said to be hereditarily SCD (HSCD for short)if all its nonempty subsets are SCD. This concept appeared inconnection with narrow operators on spaces with the Daugavetproperty: for every operator T on a space X ∈ DPr, if T (BX )is HSCD, then T is narrow. Evidently, the intersection of HSCDsets is HSCD, but with respect to other natural operations theclass of HSCD sets does not behave good: there are

I convex closed bounded symmetric with respect to originHSCD sets A, B such that conv(A ∪ B) is not HSCD;

I convex closed bounded symmetric with respect to originHSCD sets A, B such that A + B is not HSCD.

(K. and Shepelska, 2010). It is an open problem if there existHSCD sets A, B such that A ∪ B is not SCD, and if there existHSCD sets A, B such that A + B is not SCD.

A possible line of research

The above instability results motivate the following question, thatto the best of my knowledge nobody approached yet:

What can be said about those sets A in a Banach space X thatfor every Banach space E , for every isomorphic copy à of A andevery SCD subset B of E the corresponding union Ã∪B is SCD?

HSCD-majorized operators

There are Banach spaces X and Y and two hereditarily SCDoperators T1,T2 : X → Y such that T1 + T2 is not SCD (K. andShepelska, 2010).

Such behavior is very inconvenient for applications. Fortunately,there is a natural way to fix this, enlarging the class of operators.

An operator T : X → Y is said to be HSCD-majorized if there isa Banach space Z and an HSCD operator T̃ : X → Z such that‖T (x)‖ 6 ‖T̃ (x)‖ for all x ∈ X .

I The class of HSCD-majorized operators is a two-sidedoperator ideal;

I If X ∈ DPr then all HSCD-majorized operators on X arenarrow. In particular, all HSCD-majorized operators thatact from X to X satisfy the Daugavet equation.

(K. and Shepelska, 2010).

Open problems

I have mentioned these questions during the talk, but it makessense to recall them and to make some comments.

I Is it true that every space with an unconditional basis is anSCD space?

I Is it true that every separable Banach space that is notSCD possess the Daugavet property in some equivalentnorm?

I Does the relative weak topology on a closed convexbounded SCD-set always have a countable π-base?

I Must the union of two hereditarily SCD subsets of aBanach space be an SCD set?

I Does there exist a pair U1, U2 of hereditarily SCD subsetsof a Banach space such that U1 + U2 is not SCD?

Thank you for your attention!

Motivation and some referencesSlices and weak neighborhoodsSCD sets: definition and examplesDaugavet property and sets that are not SCDThe alternative Daugavet equation and numerical index 1.Operations with SCD sets, spaces and hereditarily SCD setsOpen problems