Relatively weakly open convex combinations of slices

0.13 MB·

by  Trond A. Abrahamsen

RELATIVELY WEAKLY OPEN CONVEX COMBINATIONS
OF SLICES
TRONDA. ABRAHAMSENAND VEGARDLIMA
Abstract. We show that c0, and in fact C(K) for any scattered com-
7 pact Hausdorff space K, have the property that finite convex combina-
1
tions of slices of theunit ball are relatively weakly open.
0
2

n
a
J 1. Introduction

2
LetX bea(real orcomplex) Banach spacewithunitballB ,unitsphere
2 X
SX, and dual X∗. Given x∗ SX∗ and ε> 0 we define a slice of BX by
∈
]
A S(x∗,ε) := x B :Rex∗(x) > 1 ε ,
X
{ ∈ − }
F where Rex∗(x) denotes the real part of x∗(x).
.
h Recall the following successively stronger “big-slice concepts”, defined in
t
a [3]:
m
Definition 1.1. A Banach space X has the
[
(i) local diameter 2 property if every slice of BX has diameter 2.
1
(ii) diameter 2property ifeverynon-emptyrelativelyweaklyopensubset
v
9 of BX has diameter 2.
6 (iii) strong diameter 2property ifeveryfiniteconvexcombinationofslices
1
of B has diameter 2.
6 X
0
Since a slice is relatively weakly open, the diameter 2 property implies
.
1 the local diameter 2 property. For some spaces X the converse is also true.
0
7 For example, it is known that if every x ∈ SX is an extreme point of BX∗∗,
1 then every non-empty relatively weakly open subset of BX contains a slice
: by Choquet’s lemma (cf. e.g. Proposition 1.3 in [1]).
v
i By Bourgain’s lemma [5, Lemma II.1] every non-empty relatively weakly
X
open subset of B contains a finite convex combination of slices hence the
X
r
a strong diameter 2 property implies the diameter 2 property. On a particu-
larly sunny day at a conference at the University of Warwick in 2015, Olav
Nygaard asked if the converse is ever true. The aim of this short note is to
answerthisquestionaffirmatively byshowingthatbothc ,andinfactC(K)
0
for any scattered compact Hausdorff space K, have the much stronger prop-
erty that finite convex combinations of slices of the unit ball are relatively
weakly open. See Theorems 2.3 and 2.4 below.
Let us note that in general it is not true that finite convex combinations
of slices of the unit ball are relatively weakly open. The Banach space ℓ
2
Date: January 24, 2017.
2010 Mathematics Subject Classification. 46B04, 46B20.
Keywords andphrases. convexcombinationsofslices,relativelyweaklyopenset,scat-
tered compact, diameter two property.
1
2 T.A.ABRAHAMSENANDV.LIMA
is one example [5, Remark IV.5]. In their proof that the strong diameter
2 property is stronger than the diameter 2 property Haller, Langemets and
Põldvere [6] show that if Z is an ℓ -sum of two Banach spaces, Z = X Y
p p
⊕
with 1 < p < , then for every λ (0,1) there exists two slices S and S ,
1 2
∞ ∈
and a β > 0 such that λS +(1 λ)S (1 β)B .
1 2 Z
− ⊂ −
WeshouldalsoremarkthatthepositivepartoftheunitsphereofL [0,1],
1
F = f L [0,1] : f 0, f = 1 is another example of a closed convex
1
{ ∈ ≥ k k }
bounded subset of a Banach space that satisfies a converse to Bourgain’s
lemma in that finite convex combinations of slices of F are relatively weakly
open [5, Remark IV.5].
The notation and conventions we use are standard and follow e.g. [4].
2. Main result
Westartbyrecallingthefollowingdefinition(seee.g. [4,Definition14.19]).
Definition 2.1. A compact space K is said to bescattered compact if every
closed subset L K has an isolated point in L.
⊂
Let K be scattered compact and consider the Banach space C(K) of all
continuous functions on K with sup-norm. Rudin [9] showed that C(K)∗ =
ℓ (K) in this case. Pełczyński and Semadeni [8] showed that for a compact
1
Hausdorff space K we have C(K)∗ = ℓ (K) if and only if K is scattered (=
1
dispersed).
To prove the main result, we will need the following geometric lemma for
the unit circle in the complex plane.
Lemma 2.2. Let eiα and eiβ be distinct points on the unit circle with dis-
tance d = eiα eiβ . If 0 < µ < 1, then the point c = µeiα+(1 µ)eiβ on
| − | 2 −
the line segment between eiα and eiβ satisfies
d2µ
c 1 .
| | ≤ − 4
Proof. A straightforward calculation shows that d2 = 2 2cos(α β) and
− −
that c2 = µ2+(1 µ)2+µ(1 µ)2cos(α β). Hence c2 = 1 d2µ(1 µ).
Since|√| 1+x 1+−x for x −1 and µ(1− µ) µ for| µ| [0,−1] we ge−t
≤ 2 ≥ − − ≥ 2 ∈ 2
1 d2µ
c = 1 d2µ(1 µ) 1 d2µ(1 µ) 1
| | q − − ≤ − 2 − ≤ − 4
as desired. (cid:3)
Theorem 2.3. Let K be scattered compact. Then every finite convex com-
bination of slices of the unit ball of C(K) is relatively weakly open.
Proof. Let S(f ,ε ) k be slices of B and let λ > 0 with k λ = 1,
{ i i }i=1 C(K) i i=1 i
and consider the convex combination of these slices P
k
C = λ S(f ,ε ).
i i i
Xi=1
Let x = k λ z C with z S(f ,ε ). Our goal is to find a non-empty
i=1 i i ∈ i ∈ i i
relativelyPweakly open neighborhood of x that is contained in C.
Let d = min Ref (z ) (1 ε ) : 1 i k and let η > 0 be such that
i i i
{ − − ≤ ≤ }
η < d/3. Let E K be a finite set such that f (t) < η for 1 i k.
⊂ t∈/E| i | ≤ ≤
P
RELATIVELY WEAKLY OPEN CONVEX COMBINATIONS OF SLICES 3
Define
= y B : y(t) x(t) < δ,t E
C(K)
U n ∈ | − | ∈ o
where δ > 0. Next we specify how δ is chosen.
Let L = max 1 : i= 1,2,...,k . Let
{λi }
E = t E : there exists 1 i k such that z (t) <1 .
I { ∈ ≤ 0 ≤ | i0 | }
Define δ = (1+3L)−1min (1 z (t)). Let
I t∈EI −| i0 |
E = t E E : there exists i = j such that z (t) = z (t)
III I i j
{ ∈ \ 6 6 }
and define
D = min min z (t) z (t)2 .
i j
t∈EIzi(t)6=zj(t){| − | }
Choose 0 < ρ < min D/8,η/4L . Define δ = Dρ(4(1+3L))−1. Finally
III
{ }
we choose δ < min η/6L,δ ,δ .
I III
{ }
Let y . Define w(t) = y(t) x(t) for all t K. We will define z¯
i
∈ U − ∈ ∈
S(f ,ε ), i =1,2,...,k, and show that y can be written y = k λ z¯ C.
i i i=1 i i ∈
If t E is an isolated point let t = t . While for a nonP-isolated point
∈ V { }
t E we let t be a neighborhood of t chosen so that t t′ = for all
∈ V V ∩V ∅
t′ = t, t′ E, and z (t) z (s) < δ, 1 i k, x(t) x(s) < δ and
i i
6 ∈ | − | ≤ ≤ | − |
y(t) y(s) < δ for all s . In particular, we get x(s) y(s) < 3δ for
t
| − | ∈ V | − |
all s V .
t
∈
Definition of z¯ outside .
i t∈EVt
For s K t∈E t we dSefine z¯i(s) = y(s) for all 1 i k.
∈ \∪ V ≤ ≤
Definition of z¯ on .
i t∈EVt
Let t E. S
∈
Case I: If there exists i with z (t) < 1, then we choose by Urysohn’s
0 | i0 |
lemma a real-valued non-negative continuous function n S with
t C(K)
∈
n (t) = 1 such that n (s)= 0 off . Now, for s let
t t t t
V ∈ V
z¯ (s)= n (s)[z (s)+λ−1w(s)]+[1 n (s)]y(s)
i0 t i0 i0 − t
and for i= i we let
0
6
z¯(s) = n (s)z (s)+[1 n (s)]y(s).
i t i t
−
Then
k k
λ z¯(s) = n (s)w(s)+ λ n (s)z (s)+(1 n (s))y(s)
i i t i t i t
(cid:20) − (cid:21)
Xi=1 Xi=1
= n (s)(y(s) x(s))+n (s)x(s)+(1 n (s))y(s) = y(s).
t t t
− −
By the choice of δ
z (s)+λ−1w(s) z (t) + z (s) z (t) +L y(s) x(s)
| i0 i0 | ≤ | i0 | | i0 − i0 | | − |
z (t) +δ+3Lδ < 1,
≤ | i0 |
thus we have z¯(s) 1 for every 1 i k. Moreover, for each 1 i k
i
| | ≤ ≤ ≤ ≤ ≤
the function z¯ is continuous on K since z , y, x and n all are and since n
i i t t
is zero off so z¯ = y there.
t i
We willVneed that z¯ (t) z (t) λ−1 y(t) x(t) < Lδ < η and
| i0 − i0 | ≤ i0 | − |
z¯(t) z (t) = 0 for i= i .
i i 0
| − | 6
4 T.A.ABRAHAMSENANDV.LIMA
Case II: If for all 1 i,l k we have z (t) = z (t) with z (t) = 1, then
i l i
≤ ≤ | |
x(t) = z (t) and we can just let z¯(s) = y(s) for all 1 i k and s .
i i t
≤ ≤ ∈ V
Continuity of z¯ on K in this case is trivial.
i
We will need that z¯(t) z (t) = y(t) x(t) < δ <η.
i i
| − | | − |
Case III: If z (t) = 1 for all 1 i k, but not all z (t) are equal. Order
i i
| | ≤ ≤
theset argz (t) : 1 i k as an increasing sequence θ < θ < < θ
i 1 2 q
{ ≤ ≤ } { ··· }
and define θ = θ . We put A = i : argz (t) = θ and Λ = λ .
0 q p { i p} p i∈Ap i
With ρ as above we define for 1 p q P
≤ ≤
c = ρ(eiθp−1 eiθp).
p
−
Let s and define (for i A )
t p
∈ V ∈
c w(s)
p
z¯(s)= n (s) z (s)+ + +(1 n (s))y(s).
i t i t
(cid:20) Λ qΛ (cid:21) −
p p
We have
k q
λ z¯(s)= λ z¯(s)
i i i i
Xi=1 pX=1iX∈Ap
q q q
w(s)
= n (s) λ z (s)+ n (s)c + n (s) +(1 n (s))y(s)
t i i t p t t
q −
pX=1 iX∈Ap pX=1 pX=1
k
= n (s) λ z (s)+n (s)0+n (s)w(s)+(1 n (s))y(s)
t i i t t t
−
Xi=1
= n (s)x(s)+n (s)(y(s) x(s))+y(s) n (s)y(s)= y(s).
t t t
− −
With µ = ρ/Λ
p
c
z (t)+ p = eiθp +µ(eiθp−1 eiθp) =µeiθp−1 +(1 µ)eiθp.
i
Λ − −
p
so by Lemma 2.2
c eiθp−1 eiθp 2ρ Dρ Dρ
p
z (t)+ 1 | − | 1 < 1 <1 (1+3L)δ
i
| Λ | ≤ − 4Λ ≤ − 4Λ − 4 −
p p p
hence
c w(s) c w(s)
p p
z (s)+ + z (t)+ + z (s) z (t) +
(cid:12) i (cid:12) i i i (cid:12) (cid:12)
(cid:12) Λp qΛp (cid:12) ≤ | Λp| | − | (cid:12) qΛp (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) < 1 (1+3L)δ+δ+3Lδ = 1. (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
−
Thus we have z¯(s) 1. We will also need that
i
| | ≤
c w(t)
z¯(t) z (t) = p + ρeiθp−1 eiθp L+3δL 2Lρ+3Lδ η.
i i (cid:12) (cid:12)
| − | (cid:12)Λp qΛp (cid:12) ≤ | − | ≤ ≤
(cid:12) (cid:12)
(cid:12) (cid:12)
Also note that f(cid:12)or each 1 (cid:12)i k the function z¯i is continuous on K since
≤ ≤
z , y, x and n all are and since n is zero off so z¯ = y there.
i t t t i
V
Conclusion.
So far we have defined z¯ B and shown that y = k λ z¯. It only
i ∈ C(K) i=1 i i
remains to show that z¯ S(f ,ε ). We have P
i i i
∈
f (t)(z¯(t) z (t)) < η z¯ z 2η,
i i i i i
| − | k − k ≤
tX∈/E
RELATIVELY WEAKLY OPEN CONVEX COMBINATIONS OF SLICES 5
and
f (t)(z¯(t) z (t)) < f η < η.
i i i i
| − | k k
tX∈E
Hence f (z¯ z ) < 3η so that
i i i
| − |
Ref (z¯) Ref (z ) 3η > Ref (z ) d> 1 ε ,
i i i i i i i
≥ − − −
and we are done. (cid:3)
The above theorem applies to C[0,α] for any infinite ordinal α, and in
particular to c = C[0,ω]. It should be clear that the proof also works for
real scalars and that it proves the following result.
Theorem 2.4. Every finite convex combination of slices of the unit ball of
c is relatively weakly open.
0
3. Questions and remarks
We will end with some questions and remarks.
(i) WhichBanachspacessatisfythatfiniteconvexcombinationsofslices
of the unitball are relatively weakly open? Does this hold for spaces
with the strong diameter 2 property?
(ii) WhichBanachspacessatisfythatfiniteconvexcombinationsofslices
of the unit ball contain a non-empty relatively weakly open neigh-
borhood of some point in the combination?
(iii) WhichBanachspacessatisfythatfiniteconvexcombinationsofslices
of the unitball always have non-empty intersection with the sphere?
(iv) If finite convex combinations of slices of both B and B are rel-
X Y
atively weakly open, is the same true for the unit ball of X Y
∞
⊕
and/or X Y?
1
⊕
It is not clear that there is a connection between having relatively weakly
open convex combinations of slices and the diameter two properties. But
we have the following observation.
Remark 3.1. Let X be a Banach space such that there exists a slice S =
1
S(x∗,ε) of B with diamS < 1, then there is a convex combination of
X 1
slices of B that is not relatively weakly open.
X
Define S = S( x∗,ε). Since x S if and only if x S we have that
2 1 2
− ∈ − ∈
S satisfies diamS = diamS < 1.
2 2 1
Let C = 1S + 1S . Fix x S . We have 1x + 1( x) = 0 C. If
2 1 2 2 ∈ 1 2 2 − ∈
1x + 1x C, then
2 1 2 2 ∈
1 1 1 1
x + x x x + x ( x) < 1
1 2 1 2
k2 2 k≤ 2k − k 2k − − k
hence C S = and C is not relatively weakly open.
X
∩ ∅
Regarding Question (iii) we have the following examples of spaces where
finite convex combinations of slices intersect the sphere.
Example3.2. FiniteconvexcombinationsofslicesoftheunitballofL [0,1]
1
always intersect the sphere. Here slices are given by functions g B .
i ∈ L∞[0,1]
We may assume that the g ’s are simple functions and find sets B [0,1]
i i
⊂
with B B = and χ g almost one. Thefunctions f = m(B )−1χ
i∩ j ∅ k Bi ik∞ i i Bi
does the job (m is Lebesgue measure).
6 T.A.ABRAHAMSENANDV.LIMA
Example 3.3. Let X be a Banach space such that whenever S = S(x∗,ε )
i i i
with x∗ S and ε > 0 for 1 i n, are n slices of B , then there exists
i ∈ X i ≤ ≤ X
x S S and y S such that x y = 1 and x +y S .
i i X X i i i
∈ ∩ ∈ k ± k ∈
Spaces that satisfy this condition include ℓc (Γ) for Γ uncountable since
∞
this space is almost square with ε = 0 [2, Remark 2.11]. It also includes
ℓ and C[0,1] since the slices there are defined by measures of bounded
∞
variation.
IfX isaspacewiththisproperty,thenfiniteconvex combinationsofslices
of B always intersect the sphere.
X
Indeed, let λ > 0 with n λ = 1 and let S = S(x∗,ε ) be slices of B
i i=1 i i i i X
with x∗i ∈SX∗ and εi > 0Pfor 1 ≤ i ≤ n.
Byassumption,thereexistsx S S andy S suchthat x y = 1
i i X X i
∈ ∩ ∈ k ± k
and x +y S .
i i
∈
Choose y∗ SX∗ such that y∗(y) = 1. Then
∈
1 = x y y∗(y) y∗(x ) = 1 y∗(x )
i i i
k ± k ≥ ± ±
hence y∗(x ) = 0. Now n λ (x +y) n λ S and
i i=1 i i ∈ i=1 i i
P P
n n
λ (x +y) λ y∗(y)= 1.
i i i
k k ≥
Xi=1 Xi=1
Example 3.4. If X has the Daugavet property, then finite convex com-
binations of weak∗-slices of BX∗ intersect the sphere SX∗. To see this let
1 i n, xi SX, εi > 0, and let S(xi,εi) be slices of BX∗.
≤ ≤ ∈
Let x∗1 ∈ S(x1,ε1)∩Sx∗ and V1 = span(x∗1). By [7, Lemma 2.12] there
exists x∗2 ∈S(x2,ε2)∩SX∗ such that kx∗2+v∗k = 1+kv∗k for all v∗ ∈V1.
Let V = span(x∗,x∗,...,x∗). Again by [7, Lemma 2.12] there exists
k 1 2 k
x∗k+1 ∈ S(xk+1,εk+1)∩SX∗ such that kx∗k+1+v∗k= 1+kv∗k for all v∗ ∈ Vk.
Ifλ > 0with n λ = 1,thenitfollowsthat n λ x∗ n λ S(x ,ε )
i i=1 i i=1 i i ∈ i=1 i i i
and P P P
n n−1 n
λ x∗ =λ + λ x∗ = = λ = 1.
k i ik n k i ik ··· i
Xi=1 Xi=1 Xi=1
Acknowledgments. We thank Stanimir Troyanski and Olav Nygaard for
fruitful conversations.
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(T.A.Abrahamsen)Departmentof Mathematics, University of Agder, Post-
boks 422, 4604 Kristiansand, Norway.
E-mail address: [email protected]
URL:http://home.uia.no/trondaa/index.php3
(V. Lima) NTNU, Norwegian University of Science and Technology, Aale-
sund, Postboks 1517, N-6025 Ålesund Norway.
E-mail address: [email protected]