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NON-EXPANSIVE BIJECTIONS TO THE UNIT BALL
OF `1-SUM OF STRICTLY CONVEX BANACH SPACES
V. KADETS AND O. ZAVARZINA
Abstract. Extending recent results by Cascales, Kadets, Ori-
huela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarz-
ina (2017) we demonstrate that for every Banach space Xand
every collection Zi, i ∈Iof strictly convex Banach spaces every
non-expansive bijection from the unit ball of Xto the unit ball of
sum of Ziby `1is an isometry.
1. Introduction
This article is motivated by the challenging open problem, posed
by B. Cascales, V. Kadets, J. Orihuela and E.J. Wingler in 2016 [2],
whether it is true that for every Banach space Xits unit ball BXis
Expand-Contract plastic, in other words, whether it is true that every
non-expansive bijective automorphism of BXis an isometry. It looks
surprising that such a general property, if true, remained unnoticed
during the long history of Banach space theory development. On the
other hand, if there is a counterexample, it is not an easy task to find it,
because of known partial positive results. Namely, in finite-dimensional
case the Expand-Contract plasticity of BXfollows from compactness
argument: it is known [5] that every totally bounded metric space
is Expand-Contract plastic. For infinite-dimensional case, the main
result of [2] ensures Expand-Contract plasticity of the unit ball of every
strictly convex Banach space, in particular of Hilbert spaces and of all
Lpwith 1 < p < ∞. An example of not strictly convex infinite-
dimensional space with the same property of the ball is presented in
[3, Theorem 1]. This example is `1or, more generally, `1(Γ), where the
same proof needs just minor modifications.
In this paper we “mix” results from [2, Theorem 2.6] and [3, Theorem
1] and demonstrate the Expand-Contract plasticity of the ball of `1-
sum of an arbitrary collection of strictly convex spaces. Moreover,
we demonstrate a stronger result: for every Banach space Xand every
collection Zi, i ∈Iof strictly convex Banach spaces we prove that every
2010 Mathematics Subject Classification. 46B20, 47H09.
Key words and phrases. non-expansive map; unit ball; Expand-Contract plastic
space.
The research of the first author is done in frames of Ukrainian Ministry of Science
and Education Research Program 0115U000481.
1
arXiv:1711.00262v1 [math.FA] 1 Nov 2017
2 KADETS AND ZAVARZINA
non-expansive bijection from the unit ball of Xto the unit ball of `1-
sum of spaces Ziis an isometry. Analogous results for non-expansive
bijections acting from the unit ball of an arbitrary Banach space to
unit balls of finite-dimensional or strictly convex spaces, as well as to
the unit ball of `1were established recently in [6].
Our demonstration uses several ideas from preceding papers men-
tioned above, but elaborates them substantially in order to overcome
the difficulties that appear on the way in this new, more general situ-
ation.
2. Notations and auxiliary statements
Before proving the corresponding theorem we will give the notations
and results which we need in our exposition.
In this paper we deal with real Banach spaces. As usual, for a
Banach space Ewe denote by SEand BEthe unit sphere and the
closed unit ball of Erespectively. A map F:U→Vbetween metric
spaces Uand Vis called non-expansive, if ρ(F(u1), F (u2)) ≤ρ(u1, u2)
for all u1, u2∈U, so in the case of non-expansive map F:BX→BZ
considered below we have kF(x1)−F(x2)k≤kx1−x2kfor x1, x2∈BX.
For a convex set M⊂Ewe denote by ext(M) the set of extreme
points of M. Recall that z∈ext(M) if for every non-trivial line seg-
ment [u, v] containing zin its interior, at least one of the endpoints
u, v should not belong to M. Recall also that a space Eis called
strictly convex when SE= ext(BE). In strictly convex spaces the
triangle inequality is strict for all pairs of vectors with different direc-
tions. That is, for every e1, e2∈Esuch that e16=ke2,k∈(0,+∞),
ke1+e2k<ke1k+ke2k.
Let Ibe an index set, and Zi, i ∈Ibe a fixed collection of strictly
convex Banach spaces. We consider the sum of Ziby `1and denote it
by Z. According to the definition, this means that Zis the set of all
points z= (zi)i∈I, where zi∈Zi, i ∈Iwith at most countable support
supp(z) := {i:zi6= 0}and such that Pi∈IkzikZi<∞. The space Z
is equipped with the natural norm
||z|| =k(zi)i∈Ik=X
i∈I
kzikZi.(2.1)
Remark, that even if Iis uncountable, the corresponding sum in (2.1)
reduces to an ordinary at most countable sum Pi∈supp(z)kzikZi, which
does not depend on the order of its terms, so there is no need to in-
troduce an ordering on Iand to appeal to any kind of definition for
uncountable sum, when we speak about our space Z.
In the sequel we will regard each Zias a subspace of Zin the following
natural way: Zi={z∈Z: supp(z)⊂ {i}}. It is well-known and easy
NON-EXPANSIVE BIJECTIONS TO THE UNIT BALL OF `1-SUM 3
to check that in this notation
ext(BZ) = [
i∈I
SZi.
Remark also that under this notation each z∈Zcan be written in
a unique way as a sum z=Pi∈Izi,zi∈Ziwith at most countable
number of non-zero terms, and the series converges absolutely.
Definition 2.1. Let Ebe a Banach space and H⊂Ebe a subspace.
We will say that a linear projector P:E→His strict if kPk= 1 and
for any x∈E\Hwe have kP(x)k<kxk.
Lemma 2.2. Every strict projector P:E→Hpossesses the following
property: for every x∈E\Hand every y∈Hwe have kP(x−y)k
<kx−yk.
Proof. If x /∈Hthen x−y /∈H, and since projector Pis strict we get
kP(x−y)k<kx−yk.
Consider a finite subset J⊂Iand an arbitrary collection z= (zi)i∈J,
zi∈SZi, i ∈J. For each of these zipick a supporting functional
z∗
i∈SZi∗, i.e. such a norm-one functional that z∗
i(zi) = 1. The strict
convexity of Ziimplies that z∗
i(x)<1 for all x∈BZi\ {zi},i∈J.
Denote z∗= (z∗
i)i∈Jand define the map Pz,z∗:Z→span{zi, i ∈J},
Pz,z∗((yi)i∈I) = X
i∈J
z∗
i(yi)zi.
Lemma 2.3. The map Pz,z∗is a strict projector onto span{zi, i ∈J}.
Proof. According to definition, we have to check that
(1) Pz,z∗is a projector on span{zi, i ∈J}.
(2) kPz,z∗k= 1.
(3) If (yi)i∈I/∈span{zi, i ∈J}then kPz,z∗((yi)i∈I)k<k(yi)i∈Ik.
Demonstration of (1). This is true since
P2
z,z∗((yi)i∈I) = Pz ,z∗ X
i∈J
z∗
i(yi)zi!=X
i∈J
z∗
i(z∗
i(yi)zi)zi
=X
i∈J
z∗
i(yi)z∗
i(zi)zi=X
i∈J
z∗
i(yi)zi=Pz,z∗((yi)i∈I).
Demonstration of (2). One may write
kPz,z∗((yi)i∈I)k=
X
i∈J
z∗
i(yi)zi
=X
i∈J
|z∗
i(yi)|
≤X
i∈J
kyik ≤ X
i∈I
kyik=k(yi)i∈Ik.(2.2)
Demonstration of (3). If there is N∈I\Jsuch that yN6= 0 the
item is obvious by the second line in (2.2). If yN= 0 for all N∈I\J
4 KADETS AND ZAVARZINA
then since y=Pi∈Jyi/∈span{zi, i ∈J}there is a j∈Jsuch that
yj/∈span{zj}and consequently |z∗
j(yj)|<kyjkfor this j. Thus, the
inequality (2.2) becomes strict when we pass from its first line to the
second one.
Proposition 2.4 (Brower’s invariance of domain principle [1]).Let U
be an open subset of Rnand f:U→Rnbe an injective continuous
map, then f(U)is open in Rn.
Proposition 2.5 ([3, Proposition 4]).Let Xbe a finite-dimensional
normed space and Vbe a subset of BXwith the following two properties:
Vis homeomorphic to BXand V⊃SX. Then V=BX.
Proposition 2.6 (P. Mankiewicz [4]).If X, Y are real Banach spaces,
A⊂Xand B⊂Yare convex with non-empty interior, then every
bijective isometry F:A→Bcan be extended to a bijective affine
isometry ˜
F:X→Y.
Proposition 2.7 (Extracted from [2, Theorem 2.3] and [6, Theorem
2.1]).Let F:BX→BYbe a non-expansive bijection. Then
(1) F(0) = 0.
(2) F−1(SY)⊂SX.
(3) If F(x)is an extreme point of BY, then F(ax) = aF (x)for all
a∈(−1,1).
Lemma 2.8 ([6, Lemma 2.3]).Let X, Y be Banach spaces, F:BX→
BYbe a bijective non-expansive map such that F(SX) = SY. Let V⊂
SXbe such a subset that F(av) = aF (v)for all a∈[−1,1],v∈V.
Denote A={tx :x∈V, t ∈[−1,1]}, then F|Ais a bijective isometry
between Aand F(A).
Lemma 2.9. Let X, Y be real Banach spaces, F:BX→BYbe a
bijective non-expansive map such that for every v∈F−1(SY)and every
t∈[−1,1] the condition F(tv) = tF (v)holds true. Then Fis an
isometry.
Proof. According to Proposition 2.7 F(0) = 0 and F−1(SY)⊂SX. Let
us first show that F(SX)⊂SY, that is F(SX) = SY.
For arbitrary x∈SXconsider the point y=F(x)
kF(x)k∈SYand define
ˆx=F−1(y). Then, denoting t=kF(x)kwe get
F(x) = ty =tF (ˆx) = F(tˆx).
By injectivity, this implies x=tˆx. Since kˆxk=1=kxk, we have that
kF(x)k=t= 1, that is F(x)∈SY.
Now we may apply Lemma 2.8 to V=F−1(SY) = SXand A={tx :
x∈SX, t ∈[−1,1]}=BX. Then F(A) = BY, so Lemma 2.8 says that
Fis an isometry.
NON-EXPANSIVE BIJECTIONS TO THE UNIT BALL OF `1-SUM 5
3. Main result
Theorem 3.1. Let Xbe a Banach space, Zi, i ∈Ibe a fixed collec-
tion of strictly convex Banach spaces, Zbe the `1-sum of the collection
Zi, i ∈I, and F:BX→BZbe a non-expansive bijection. Then Fis
an isometry.
The essence of the proof consists in Lemma 3.2 below which analyzes
the behavior of Fon some typical finite-dimensional parts of the ball.
Under the conditions of Theorem 3.1 consider a finite subset J⊂I,
|J|=nand pick collections z= (zi)i∈J,zi∈SZi, i ∈J,z∗= (z∗
i)i∈J,
where each z∗
i∈SZi∗is a supporting functional for the corresponding
zi. Denote xi=F−1(zi)∈SX. Denote by Unand ∂Unthe unit ball
and the unit sphere of span{xi}i∈Jrespectively. Let Vnand ∂Vnbe the
unit ball and the unit sphere of span{zi}i∈J.
Lemma 3.2. For every collection (ai)i∈Jof reals with Pi∈Jaixi∈Un
(3.1)
X
i∈J
aixi
=X
i∈J
|ai|,
(which means in particular that Unisometric to the unit ball of n-
dimensional `1), and
(3.2) F X
i∈J
aixi!=X
i∈J
aizi.
Proof. We will use the induction in n. Recall, that zi∈ext BZ. This
means that for n= 1, our Lemma follows from item (3) of Proposition
2.7. Now assume the validity of Lemma for index sets of n−1 elements,
and let us prove it for |J|=n. Fix an m∈Jand denote Jn−1=J\{m},
At first, let us prove that
(3.3) F(Un)⊂Vn.
To this end, consider r∈Un. If ris of the form amxmthe statement
follows from (3) of Proposition 2.7. So we must consider r=Pi∈Jaixi,
Pi∈J|ai| ≤ 1 with Pi∈Jn−1|ai| 6= 0. Denote the expansion of F(r) by
F(r)=(vi)i∈I. For the element
r1=X
i∈Jn−1
ai
Pj∈Jn−1|aj|xi
by the induction hypothesis
F(r1) = X
i∈Jn−1
ai
Pj∈Jn−1|aj|zi.
Moreover, on the one hand,
X
i∈J
aixi
≤X
i∈J
|ai|.
6 KADETS AND ZAVARZINA
On the other hand,
X
i∈J
aixi
=
X
i∈Jn−1
aixi−(−amxm)
≥
F
X
i∈Jn−1
aixi
−F(−amxm)
=
X
i∈Jn−1
aizi−amzm
=X
i∈J
|ai|.
Thus, (3.1) is demonstrated and we may write the following inequali-
ties:
2 = kF(r1)−am
|am|zmk ≤
F(r1)−X
i∈J
vi
+
X
i∈J
vi−Fam
|am|xm
=kF(r1)−F(r)k+
F(r)−Fam
|am|xm
−2
X
i∈I\J
vi
≤ kF(r1)−F(r)k+
F(r)−Fam
|am|xm
≤
X
i∈Jn−1
ai
Pj∈Jn−1|aj|xi−X
i∈J
aixi
+
X
i∈J
aixi−am
|am|xm
≤X
i∈Jn−1
ai−ai
Pj∈Jn−1|aj|
+|am|+X
i∈Jn−1
|ai|+
am−am
|am|
=X
i∈Jn−1
|ai| 1 +
1−1
Pj∈Jn−1|aj|!+|am|1 +
1−1
|am|= 2.
So, all the inequalities in this chain are in fact equalities, which implies
that
F(r) = X
i∈J
viand kF(r1)−F(r)k+
F(r)−Fam
|am|xm
= 2.
Remind that our goal is to check that F(r)∈Vn. Suppose by contra-
diction that F(r) = Pi∈Jvi/∈Vnand denote for reader’s convenience
NON-EXPANSIVE BIJECTIONS TO THE UNIT BALL OF `1-SUM 7
by s=Pj∈Jn−1|z∗
j(vj)|. Then in notations of Lemma 2.3
2 =
F
X
i∈Jn−1
z∗
i(vi)
sxi
−F(r)
+
F(r)−Fz∗
m(vm)
|z∗
m(vm)|xm
=
X
i∈Jn−1z∗
i(vi)
szi−vi−vm
+
X
i∈Jn−1
vi+vm−z∗
m(vm)
|z∗
m(vm)|zm
>
Pz,z∗
X
i∈Jn−1z∗
i(vi)
szi−vi−vm
+
Pz,z∗
X
i∈Jn−1
vi+vm−z∗
m(vm)
|z∗
m(vm)|zm
=
X
i∈Jn−1z∗
i(vi)
s−z∗
i(vi)zi−z∗
m(vm)zm
+
X
i∈Jn−1
z∗
i(vi)zi+x∗
m(vm)−z∗
m(vm)
|z∗
m(vm)|zm
=X
i∈Jn−1
z∗
i(vi)−z∗
i(vi)
s
+|z∗
m(vm)|+X
i∈Jn−1
|z∗
i(vi)|+
z∗
m(vm)−z∗
m(vm)
|z∗
m(vm)|
=X
i∈Jn−1
|z∗
i(vi)|1 +
1−1
s+|z∗
m(vm)|1 +
1−1
|z∗
m(vm)|= 2.
Observe, that we have written the strict inequality in this chain be-
cause of Lemmas 2.3 and 2.2. The above contradiction means that our
assumption was wrong, that is
(3.4) F(Un)⊂Vn.
Further we are going to prove the inclusion
(3.5) ∂Vn⊂F(Un).
We will argue by contradiction. Let there is a point Pi∈Jti∈∂Vn\
F(Un) and denote τ=F−1(Pi∈Jti). Then || Pi∈Jti|| = 1 and τ /∈UN.
Rewrite
X
i∈J
ti=X
i∈J
ktikˆ
ti,ˆ
ti∈SZi.
Pick some supporting functionals ti∗in the points ˆ
ti,i∈Jand denote
t= (ˆ
ti)i∈Jand t∗= (ti∗)i∈J. Let us demonstrate that F(ατ )∈Vnfor
all α∈[0,1]. Indeed, if F(ατ)/∈Vnfor some α, denoting F(ατ ) =
8 KADETS AND ZAVARZINA
Pi∈Iwi, we deduce from Lemmas 2.3 and 2.2 the following contradic-
tion
1 = k0−ατk+kατ −τk ≥
0−X
i∈I
wi
+
X
i∈I
wi−X
i∈J
ti
= 2
X
i∈I\J
wi
+
X
i∈J
wi
+
X
i∈J
wi−X
i∈J
ti
>
Pt,t∗ X
i∈J
wi!
+
Pt,t∗ X
i∈J
wi!−X
i∈J
ti
=
X
i∈J
t∗
i(wi)ˆ
ti
+
X
i∈J
t∗
i(wi)ˆ
ti−X
i∈J
ti
=X
i∈J
|t∗
i(wi)|+X
i∈J
|ktik − t∗
i(wi)| ≥ X
i∈J
ktik= 1.
Note that F(Un) contains a relative neighborhood of 0 in Vn(here we
use item (1) of Proposition 2.7 and Proposition 2.4), so the continuous
curve {F(ατ) : α∈[0,1]}connecting 0 with Pi∈Jtiin Vnhas a non-
trivial intersection with F(Un). This implies that there is a a∈[0,1]
such that F(aτ)∈F(Un). Since aτ /∈Unthis contradicts the injec-
tivity of F. Inclusion (3.5) is proved. Now, inclusions (3.4) and (3.5)
together with Lemma 2.5 imply F(Un) = Vn. Observe, that Unand
Vnare isometric to the unit ball of n-dimensional `1, so they can be
considered as two copies of the same compact metric space. Hence
Expand-Contract plasticity of totally bounded metric spaces [5] im-
plies that every bijective non-expansive map from Unonto Vnis an
isometry. In particular, Fmaps Unonto Vnisometrically. Finally, the
application of Lemma 2.6 gives us that the restriction of Fto Unex-
tends to a linear map from span{xi, i ∈J}to span{zi, i ∈J}, which
evidently implies (3.2).
Proof of Theorem 3.1.Our aim is to apply Lemma 2.9. To satisfy the
conditions of the lemma, for every z∈SZwe must regard y=F−1(z)
and check that for every t∈[−1,1]
F(ty) = tz.(3.6)
To this end let us denote Jz= supp(z), and write
z=X
i∈Jz
zi=X
i∈Jz
kzik˜zi,
where ˜zi∈SZi. Let us also denote for all i∈Jz
xi:= F−1(˜zi)∈SX.
NON-EXPANSIVE BIJECTIONS TO THE UNIT BALL OF `1-SUM 9
For Jzbeing finite formula (3.2) of Lemma 3.2 implies that
y=F−1(z) = F−1 X
i∈Jz
kzik˜zi!=X
i∈Jz
kzikxi,and
F(ty) = F X
i∈Jz
tkzikxi!=X
i∈Jz
tkzik˜zi=tz,
which demonstrates (3.6) in this case. It remains to demonstrate (3.6)
for the case of countable Jz. In this case we can write Jz={i1, i2, . . .}
and consider its finite subsets Jn={i1, i2, . . . , in}. For these finite
subsets Pi∈Jnkzik ≤ 1, so Pi∈Jnkzikxi∈Un:= Bspan{xi}i∈Jn, and we
may deduce from Lemma 3.2 that
F X
i∈Jn
kzikxi!=X
i∈Jn
kzik˜zi.
Passing to limit as n→ ∞ we get
F X
i∈Jz
kzikxi!=X
i∈Jz
kzik˜zi=z, i.e. y=F−1(z) = X
i∈Jz
kzikxi.
One more application of formula (3.2) of Lemma 3.2 gives us
F X
i∈Jn
tkzikxi!=X
i∈Jn
tkzik˜zi,
which after passing to limit ensures (3.6):
F(ty) = F lim
n→∞ X
i∈Jn
tkzikxi!= lim
n→∞ X
i∈Jn
tkzik˜zi=X
i∈Jz
tkzik˜zi=tz.
This fact demonstrates applicability of Lemma 2.9 to our Fand thus
completes the proof of the theorem.
References
[1] Brouwer L.E.J. Beweis der Invarianz des n-dimensionalen Gebiets, Mathema-
tische Annalen, 71 (1912), 305–315.
[2] Cascales B., Kadets V., Orihuela J., Wingler E.J. Plasticity of the unit ball
of a strictly convex Banach space, Revista de la Real Academia de Ciencias
Exactas, F´ısicas y Naturales. Serie A. Matem´aticas, 110(2)(2016), 723–727.
[3] Kadets V., Zavarzina O. Plasticity of the unit ball of `1, Visn. Hark. nac. univ.
im. V.N. Karazina, Ser.: Mat. prikl. mat. meh., 83 (2017), 4–9.
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arXiv:1704.06961v2, to appear in Annals of Functional Analysis.
10 KADETS AND ZAVARZINA
(Kadets) School of Mathematics and Informatics, V.N. Karazin Kharkiv
National University, 61022 Kharkiv, Ukraine
ORCID: 0000-0002-5606-2679
E-mail address:v.kateds@karazin.ua
(Zavarzina) School of Mathematics and Informatics, V.N. Karazin
Kharkiv National University, 61022 Kharkiv, Ukraine
ORCID: 0000-0002-5731-6343
E-mail address:olesia.zavarzina@yahoo.com
