Isometries on extremely non-complex Banach spaces

0.27 MB·English

by  Piotr Koszmider

ISOMETRIES ON EXTREMELY NON-COMPLEX BANACH SPACES
PIOTRKOSZMIDER,MIGUELMART´IN,ANDJAVIERMER´I
Abstract. GivenaseparableBanachspaceE,weconstructanextremelynon-complexBanach
space(i.e.aspacesatisfyingthatkId+T2k=1+kT2kforeveryboundedlinearoperator T on
it)whosedualcontainsE∗ asanL-summand. Wealsostudysurjectiveisometriesonextremely
0
non-complex Banach spaces and construct an example of a real Banach space whose group of
1
surjectiveisometries reduces to±Id, butthe group of surjectiveisometries of its dual contains
0
thegroupofisometriesofaseparableinfinite-dimensionalHilbertspaceasasubgroup.
2

n
a
J
9 1. Introduction
2
All the Banachspaces in this paper will be real. Given a Banachspace X, we write X for the
∗
] topologicaldual,L(X)forthespaceofallboundedlinearoperators,W(X)forthespaceofweakly
A
compact operators and Iso(X) for the group of surjective isometries.
F
. A Banach space X is said to be extremely non-complex if the norm equality
h
t kId+T2k=1+kT2k
a
m holdsforeveryT ∈L(X). Thisconceptwasintroducedveryrecentlybythe authorsin[20], where
[ several different examples of C(K) spaces are shown to be extremely non-complex. For instance,
this is the case for some perfect compact spaces K constructed by the first author [19] such that
2
C(K) has few operators (in the sense that every operator is a weak multiplier). There are other
v
examples of extremely non-complex C(K) spaces which contain complemented copies of ℓ or
2
1 C(2ω)(andso,theydonothavefewoperators). ItistrivialthatX =Risextremelynon-com∞plex.
5 Theexistenceofinfinite-dimensionalextremelynon-complexBanachspaceshadbeenaskedin[16,
1 Question 4.11], where possible generalizations of the Daugavet equation were investigated. We
.
1 recall that an operator S defined on a Banach space X satisfies the Daugavet equation [4] if
0
kId+Sk=1+kSk
9
0 andthatthespaceX hastheDaugavet property [17]iftheDaugavetequationholdsforeveryrank-
:
v one operatoronX. We refer the readerto [2, 3, 17, 27] for backgroundonthe Daugavetproperty.
i Let us observe that X is extremely non-complex if the Daugavet equation holds for the square of
X
every (bounded linear) operator on X. Spaces X in which the square of every rank-one operator
r
a onX satisfies the Daugavetequationarestudied in [23], where it is shownthat their unit balls do
nothavestronglyexposedpoints. Inparticular,theunitballofanextremelynon-complexBanach
space (of dimension greater than one) does not have strongly exposed points and, therefore, the
space does not have the Radon-Nikody´m property, even the more it is not reflexive.
The name of extremely non-complex comes from the fact that a real Banach space X is said
to have a complex structure if there exists T ∈ L(X) such that T2 = −Id, so extremely non-
complex spaces lack of complex structures in a very strong way. Let us also comment that no
Date:April6th,2009. Revised: January19th,2010.
2000 Mathematics Subject Classification. Primary: 46B04. Secondary: 46B10, 46B20,46E15,47A99.
Keywordsandphrases. Banachspace;fewoperators;complexstructure;Daugavetequation;spaceofcontinuous
functions;extremelynon-complex;surjectiveisometries;duality.
The first author was partially supported by Polish Ministry of Science and Higher Education research grant
NN201386234. ThesecondandthethirdauthorswerepartiallysupportedbySpanishMECprojectno.MTM2006-
04837andJuntadeAndaluc´ıagrantsFQM-185andP06-FQM-01438.
1
2 PIOTRKOSZMIDER,MIGUELMART´IN,ANDJAVIERMER´I
hyperplane(actuallynofinite-codimensionalsubspace)ofanextremelynon-complexBanachspace
admits a complex structure. The existence of infinite-dimensional real Banach spaces admitting
no complex structure is known since the 1950’s, when J. Dieudonn´e [5] showed that this is the
case of the James’ space J. We refer the reader to the very recent papers by V. Ferenczi and
E. Medina Galego [10, 11] and references therein for a discussion about complex structures on
spaces and on their hyperplanes.
Our first goal in this paper is to present examples of extremely non-complex Banach spaces
whicharenot isomorphicto C(K) spaces. Namely, itis provedinsection3 that if K is a compact
space such that all operators on C(K) are weak multipliers, L is a closed nowhere dense subset of
K and E is a subspace of C(L), then the space
C (KkL)={f ∈C(K) : f| ∈E}
E L
isextremelynon-complex(seesection2forthedefinitionsandbasicfactsaboutthiskindofspaces).
It is also shown that there are extremely non-complex C (KkL) spaces which are not isomorphic
E
to C(K ) spaces. On the other hand, some spaces C (KkL) are isometric to spaces C(K ) for
′ E ′
some compact K . This can be used to note that the extremely non-complex spaces of the form
′
C (KkL) may have many operators besides weak multipliers (see Remark 3.14).
E
The next aim is to show that Iso(X) is a “discrete” Boolean group when X is extremely non-
complex. Namely, we show that T2 = Id for every T ∈ Iso(X) (i.e. Iso(X) is a Bolean group and
so, it is commutative) and that kT −T k ∈ {0,2} for every T ,T ∈ Iso(X). Next, we discuss
1 2 1 2
the relationship with the set of all unconditional projections on X and the possibility of this set
to be a Boolean algebra. This is the content of section 4. In section 5 we particularize these
results to spaces C (KkL) which are extremely non-complex, getting necessary conditions on the
E
elements of Iso C (KkL) . In particular, if C(K) is extremely non-complex, we show that the
E
only homeomorphism from K to K is the identity, obtaining that Iso C(K) is isomorphic to the
(cid:0) (cid:1)
Boolean algebra of clopen sets of K.
(cid:0) (cid:1)
Section 6 is devotedto apply all the results above to get anexample showing that the behavior
ofthe groupof isometrieswithrespectto duality canbe extremelybad. Namely, we showthat for
every separable Banach space E, there is a Banachspace X(E) such that Iso X(E) ={Id,−Id}
andX(E) =E ⊕ Z,soIso X(E) containsIso(E )asasubgroup. Todoso,wehavetomodify
∗ ∗ 1 ∗ ∗ (cid:0) (cid:1)
aconstructionofaconnectedcompactspaceK withfewopeeratorsgivenbytheefirstnamedauthor
(cid:0) (cid:1)
in [19e, §5] and use our construection of CE(KkL) for a nowhere dense L⊂K. For the special case
E = ℓ , we have Iso X(ℓ ) = {Id,−Id}, while Iso X(ℓ ) contains infinitely many uniformly
2 2 2 ∗
continuous one-parameter semigroups of surjective isometries. Let us comment that, in sharp
(cid:0) (cid:1) (cid:0) (cid:1)
contrast with the exameples above, when a Banach spaece X is strongly unique predual, the group
Iso(X )consistsexactlyoftheconjugateoperatorstotheelementsofIso(X). Quitealotofspaces
∗
are actually strong unique preduals. We refer the reader to [14] for more information.
We finish this introduction with some needed notation. If X is a Banach space, we write B
X
to denote the closed unit ball of X and given a convex subset A ⊆ X, ext(A) denotes the set of
extreme points of A. A closed subspace Z of X is an L-summand if X =Z⊕ W for some closed
1
subspaceW ofX,where ⊕ denotes the ℓ -sum. AclosedsubspaceY ofa BanachspaceX is said
1 1
to be an M-ideal of X if the annihilator Y of Y is an L-summandof X . We refer the reader to
⊥ ∗
[15] for backgroundon L-summands and M-ideals.
2. Notation and preliminary results on the spaces C (KkL)
E
All along the paper, K will be a (Hausdorff) compact (topological) space and L ⊆ K will
stand for a nowhere dense closed subset. Given a closed subspace E of C(L), we will consider the
subspace of C(K) given by
C (KkL)={f ∈C(K) : f| ∈E}.
E L
ISOMETRIES ON EXTREMELY NON-COMPLEX BANACH SPACES 3
This notation is compatible with the Semadeni’s book [25, II. 4] notation of
C (KkL)={f ∈C(K) : f| =0}.
0 L
ThisspacecanbeidentifiedwiththespaceC (K\L)ofthosecontinuousfunctionsf :K\L−→R
0
vanishing at infinity.
By the Rieszrepresentationtheorem,the dualspaceofC(K) isisometricto the spaceM(K)of
Radon measures on K, i.e. signed, Borel, scalar-valued, countably additive and regular measures.
More precisely, given µ∈M(K) and f ∈C(K), the duality is given by
µ(f)= fdµ.
Z
We recall that C (KkL) is an M-ideal of C(K) [15, Example I.1.4(a)], meaning that C (KkL)
0 0 ⊥
is an L-summand in C(K) . This fact allows to show the following well-known result.
∗
Lemma 2.1. C (KkL) ≡{µ∈M(K) : |µ|(L)=0}.
0 ∗
Proof. Since C (KkL) is an M-ideal in C(K), Proposition 1.12 and Remark 1.13 of [15] allow us
0
to identify C (KkL) with the subspace of C(K) =M(K) given by
0 ∗ ∗
C (KkL)# ={µ∈M(K) : |µ|(K)=|µ|(K\L)}={µ∈M(K) : |µ|(L)=0}. (cid:3)
0
WhenweconsiderthespaceC (KkL),itstillmakessensetotalkaboutfunctionalscorrespond-
E
ing to the measures on K, namely one understands them as the restriction of the functional from
C(K) to C (KkL). However, given a functional belonging to C (KkL) one may have several
E E ∗
measures on K associated with it. The next result describes the dual of a C (KkL) space for an
E
arbitraryE ⊆C(L). Itis worthmentioning thatits proofisanextensionofthatappearingin[21,
Theorem 3.3].
Lemma 2.2. Let K be a compact space, L a closed subset of K and E ⊆C(L). Then,
C (KkL) ≡C (KkL) ⊕ C (KkL) ≡C (KkL) ⊕ E .
E ∗ 0 ∗ 1 0 ⊥ 0 ∗ 1 ∗
Proof. We write P :C(K)−→C(L) for the restriction operator, i.e.
[P(f)](t)=f(t) (t∈L, f ∈C(K)).
Then, C (KkL)=kerP andC (KkL)={f ∈C(K) : P(f)∈E}. Since C (KkL)isanM-ideal
0 E 0
in C(K), it is a fortiori an M-ideal in C (KkL) by [15, Proposition I.1.17], meaning that
E
CE(KkL)∗ ≡C0(KkL)∗⊕1C0(KkL)⊥ ≡C0(KkL)∗⊕1 CE(KkL)/C0(KkL) ∗.
Now,itsufficestoprovethatthequotientCE(KkL)/C0(KkL)isi(cid:2)sometricallyisomorph(cid:3)ictoE. To
do so, we define the operator Φ:C (KkL)−→E given by Φ(f)=P(f) for every f ∈C (KkL).
E E
Then Φ is well defined, kΦk 6 1, and kerΦ = C (KkL). To see that the canonical quotient
0
operator Φ:C (KkL)/C (KkL)−→E is a surjective isometry, it suffices to show that
E 0
Φ {f ∈C (KkL) : kfk<1} ={g ∈E : kgk<1}.
E
e
Indeed, the left-hand sid(cid:0)e is contained in the right-h(cid:1)and side since kΦk61. Conversely,for every
g ∈ E ⊆ C(L) with kgk < 1, we just use Tietze’s extension theorem to find f ∈ C(K) such that
Φ(f)=f| =g and kfk=kgk. (cid:3)
L
If φ ∈ C (KkL) , by the above lemma we have φ = φ + φ with φ ∈ C (KkL) and
E ∗ 1 E 1 0 ∗
φ ∈C (KkL) ≡E . Observe that φ can be isometrically associated with a measure on K\L
E 0 ⊥ ∗ 1
by Lemma 2.1. whichwill be denotedφ| . Givena subset A⊆K satisfying A∩L=∅, φ| will
K L A
\
stand for the measure φ| restricted to A.
K L
\
The next result is a straightforwardconsequence of the above two lemmas.
4 PIOTRKOSZMIDER,MIGUELMART´IN,ANDJAVIERMER´I
Lemma 2.3. Let φ∈C (KkL) and x∈K\L. Then
E ∗
kφk= φ| + φ| +kφ k.
x K (L x ) E
{ } \ ∪{ }
(cid:13) (cid:13) (cid:13) (cid:13)
Thenexteasylemmadescribesth(cid:13)eseto(cid:13)fext(cid:13)remepoints(cid:13)intheunitballofthedualofCE(KkL)
and gives a norming set for C (KkL). We recall that a subset A of the unit ball of the dual of a
E
Banach space X is said to be norming if
kxk=sup{|φ(x)| : φ∈A} (x∈X).
Lemma 2.4. Let K be a compact space, L a nowhere dense closed subset and E ⊆ C(L). We
consider the set
A= θδ : y ∈K\L, θ ∈{−1,1} ⊂C (KkL) .
y E ∗
Then:
(cid:8) (cid:9)
(a) ext(BCE(K L)∗)=A∪ext(BE∗).
k
(b) A is norming for C (KkL).
E
(c) Therefore, A is weak∗-dense in ext(BCE(K L)∗).
k
Proof. By Lemma 2.2 and the description of the extreme points of the unit ball of an ℓ -sum of
1
Banach spaces [15, Lemma I.1.5], we have
ext(BCE(K L)∗)=ext(BC0(K L)∗)∪ext(BE∗).
k \
It suffices to recall that ext(BC0(K L)∗)=A (see [12, Theorem2.3.5] for instance) to get (a). The
\
fact that A is norming for C (KkL) is a direct consequence of the fact that K\L is dense in K.
E
Finally,everynormingsetisweak∗-denseinext(BCE(K L)∗)bytheHahn-Banachtheoremandthe
reversedKrein-Milman theorem. k (cid:3)
We introduce one more ingredient which will play a crucial role in our arguments. Given an
operator U ∈L C (KkL) , we consider the function
E ∗
gU(cid:0):K\L−→(cid:1)[−kUk,kUk], gU(x)=U(δx)({x}) x∈K\L .
This obviously extends to operators on CE(KkL) by passing to the(cid:0)adjoint, th(cid:1)at is, for T ∈
L CE(KkL) one canconsidergT∗ :K\L−→[−kTk,kTk]. Thisis ageneralizationofatoolused
in [26] under the name “stochastic kernel”. One of the results in that paper can be generalizedto
(cid:0) (cid:1)
the following.
Lemma 2.5. Let K be a compact space, L a nowhere dense closed subset of K, E ⊆ C(L), and
T ∈ L CE(KkL) . If the set {x ∈ K \L : gT∗(x) > 0} is dense in K \L, then T satisfies the
Daugavet equation.
(cid:0) (cid:1)
Proof. We use Lemmas 2.2, 2.3 and 2.4 to get
(1) kId+T k> sup kδ +T (δ )k
∗ x ∗ x
x K L
∈ \
= sup |1+T∗(δx)({x})|+kT∗(δx)|(K (L x ))k+kT∗(δx)|Ek
x K L \ ∪{ }
∈ \
= sup |1+T∗(δx)({x})|−|T∗(δx)({x})|+kT∗(δx)k
x K L
∈ \
> sup 1+T∗(δx)({x})−|T∗(δx)({x})|+kT∗(δx)k.
x K L
∈ \
Now, we claim that the set {x∈K\L : kT (δ )k>kTk−ε} is nonempty and open in K\L for
∗ x
every ε>0. Indeed, take a norm one function f ∈C (KkL) such that kT(f)k>kTk−ε and use
E
the fact that K\L is dense in K to find x∈K\L satisfying
kT∗(δx)k>|T∗(δx)(f)|=|T(f)(x)|>kTk−ε.
ISOMETRIES ON EXTREMELY NON-COMPLEX BANACH SPACES 5
To show that {x∈K\L : kT (δ )k>kTk−ε} is open we prove that the mapping
∗ x
x7−→kT∗(δx)k (x∈K\L)
is lower semicontinuous. To do so, since T is weak continuous and k·k is weak lower semicon-
∗ ∗ ∗
tinuous, it suffices to observe that the mapping which sends x to δ is continuous with respect to
x
the weak topologyonC (KkL) . Butthisis sosince,fora∈R andf ∈C (KkL),the preimage
∗ E ∗ E
of the subbasic set {φ∈C (KkL) : φ(f)<a} in this topology is {x∈K\L : f(x)<a} which
E ∗
is open in K\L.
To finish the proof, we use the hypothesis to find x ∈K\L satisfying
0
kT∗(δx0)k>kTk−ε and gT∗(x0)=T∗(δx0)({x0})>0
and we use it in (1) to obtain
kId+T∗k>1+kT∗(δx0)k>1+kTk−ε
which implies that kId+Tk=kId+T k>1+kTk since ε was arbitrary. (cid:3)
∗
3. Spaces C (KkL) when C(K) has few operators
E
Let us start fixing some notation and terminology that will be used throughout the section. If
g : K −→ R is a bounded Borel function, we will consider the operator gId : C(K) −→ C(K)
∗ ∗
which sends the functional which is the integration of a function f ∈ C(K) with respect to a
measure µ to the functional which is the integration of the product fg with respect to µ.
Definition 3.1. LetK be acompactspaceandT ∈L(C(K)). We saythatT is aweak multiplier
if T =gId+S where g :K −→R is a bounded Borel function on K and S ∈W C(K) .
∗ ∗
(cid:0) (cid:1)
This definition was given in [19] in an equivalent form (see [19, Definition 2.1] and [19, Theo-
rem 2.2]).
Definition 3.2. We say that an open set V ⊆ K is compatible with L if and only if L ⊆ V or
L∩V = ∅. In the first case, the notation C (VkL) has the same meaning as in the previous
E
section. If L∩V = ∅, we will write C (VkL) just to denote C(V). Let us also observe that if
E
L⊆V, then
C (VkL) ≡C (VkL) ⊕ C (VkL) ≡C (VkL) ⊕ E
E ∗ 0 ∗ 1 0 ⊥ 0 ∗ 1 ∗
since Lemma 2.2 applies to V.
Given an open set V ⊆ K compatible with L, we consider the restriction operator PV :
C (KkL) −→C (VkL) given by
E ∗ E ∗
PV(φ)=φ| +φ
V L E
\
for φ = φ| +φ where φ| ∈ C (KkL) and φ ∈ C (KkL) ≡ E . Observe that φ
K L E K L 0 ∗ E 0 ⊥ ∗ E
\ \
can also be viewed as an element of C (VkL) since the spaces C (VkL) and C (KkL) are
0 ⊥ 0 ⊥ 0 ⊥
isometrically isomorphic (both coincide with E ).
∗
Given an open set V ⊆ K compatible with L and h : K −→ [0,1] a continuous function
constant on L with support included in V, we denote by P : C (KkL) −→ C (VkL) and
V E E
I :C (VkL)−→C (KkL) the operators defined by
h,V E E
P (f)=f| and I (f)=hf
V V h,V
for f ∈C (KkL)and f ∈C (VkL) respectively. We observethat I is well defined, that is, hf
E E e eh,V
isafunctioninC(K)withhf| ∈E (indeed, hf iscontinuousinV asaproductoftwocontinuous
L
functions and it is conetinuous in K \Supp(h) as a constant function, since these two sets forme
an open cover of K we haveethat hf is continueous in K; being h constant on L, it is clear that
hf| ∈E). Finally, If V ⊆K is an open set compatible with L satisfying V ⊆V we will also use
L 1 1
e
e
6 PIOTRKOSZMIDER,MIGUELMART´IN,ANDJAVIERMER´I
the notation P for the restriction operator from C (V kL) to C (VkL). In the next result we
V E 1 E
gather some easy facts concerning these operators.
Lemma 3.3. Let V and h be as above. Then, the following hold:
(a) P (φ)(f)=φ(f| ) for φ∈C (VkL) and f ∈C (KkL).
V∗ V E ∗ E
(b) I (φ)(f)=φ(fh) for φ∈C (KkL) and f ∈C (VkL).
h∗,V E ∗ E
(c) If V is an open set such that V ⊆ V and h| ≡ 1, then (I P )(µ) = µ for every
0 e e 0 e V0 h∗,V V∗
µ∈C(K) with Supp(µ)⊆V .
∗ 0
(d) If E = C(L), then C (KkL) = C(K), C (VkL) = C(V), and PVP (µ) = µ for every
E E V∗
µ∈C(V) .
∗
Proof. (a) and (b) are obvious from the definitions of the operators. To prove (c) we fix f ∈
C (VkL), µ∈C(K) with Supp(µ)⊆V and we observe that
E ∗ 0
e
(Ih∗,V PV∗)(µ)(f)=PV∗(µ) Ih,V(f) =PV∗(µ)(fh)
(cid:0) (cid:1)
e =µ (fh)| =e (fh)| dµe= fdµ=µ(f).
V V
(cid:16) (cid:17) Z ZV0
e e e e
The first two assertions of (d) are obvious. For the third one, given µ ∈C(V) and f ∈C(V),
∗
use the regularity of the measure P (µ) to find an open set V ⊆K satisfying
V∗ n
e
1
(2) V ⊆Vn and PV∗(µ) (Vn\V)< n
for every n∈N. Next, take fn ∈C(K) satisfying(cid:12)(cid:12) (cid:12)(cid:12)
f | ≡f, f | ≡0, and kf k=kfk
n V n K Vn n
\
for every n∈N, and observe that
e e
PVPV∗(µ)(f)= PV∗(µ) |V(f)= fd PV∗(µ) |V
ZV
(cid:0) (cid:1) (cid:0) (cid:1)
e = fndPV∗(µe)− e fndPV∗(µ)
ZK ZVn\V
=P (µ)(f )− f dP (µ)=µ(f)− f dP (µ).
V∗ n n V∗ n V∗
ZVn\V ZVn\V
Therefore, using (2) and letting n→∞, it follows that PVP (µe)(f)=µ(f). (cid:3)
V∗
Our first application uses the above operators in the simple caseein whiceh E =C(L).
Proposition 3.4. Let K be a compact space, let V , V , and V be open nonempty subsets of K
0 1 2
such that V ⊆ V , and let R : C(V )−→ C(V ) be a linear operator. Suppose that all operators
0 1 2 1
on C(K)areweak multipliers. Then, therearea Borel function g :V −→Rwith support included
1
in V ∩V and a weakly compact operator S :C(V ) −→C(V ) such that
1 2 1 ∗ 2 ∗
R (µ)=gµ+S(µ)
∗
for every µ∈C(K) with Supp(µ)⊆V .
∗ 0
Proof. We fix a continuous function h : K −→ [0,1] satisfying h| ≡ 1 and h| ≡ 0 and we
V0 (K\V1)
define R ∈L(C(K)) by
0
(3) R (f)=I RP (f)
0 h,V1 V2
ISOMETRIES ON EXTREMELY NON-COMPLEX BANACH SPACES 7
for f ∈ C(K). Hence, there are a bounded Borel function g : K −→ R and a weakly compact
operator S ∈L(C(K) ) such that R (µ)=gµ+S(µ) for µ∈C(K) , which allows us to write
∗ 0∗ ∗
b
(4) b PV2R0∗PV∗1 =PVb2gIdCb(K)∗PV∗1 +PV2SPV∗1.
We claim that, considering the weakly compabct operator givenbybS =PV2SPV∗1 and defining the
functions g˘:K −→R and g :V −→R by
1
b
g(x) if x∈V ∩V
1 2
g˘(x)= and g =g˘|
(0 if x∈/ V1∩V2 V1
b
the following holds for µ∈C(V ) :
1 ∗
(5) PV2R P (µ)=gµ+S(µ).
0∗ V∗1
Indeed, for µ∈C(V ) and f ∈C(cid:0)(V ) we obs(cid:1)erve that
1 ∗ 2
PV2gIdC(K)∗PV∗1 (µ)(f)=(gPV∗1(µ))|V2(f)= gfdPV∗1(µ)
ZV2
(cid:0) (cid:1)
b = b gfdP (µ)= b g˘fdP (µ)= g˘fdP (µ)
V∗1 V∗1 V∗1
ZV1∩V2 ZV1∩V2 ZK
and, for n ∈ N, we use Lusin’s Theorem (bsee [22, Theorem 21.4], for instance) to find a compact
set K ⊆K such that
n
1 1
(6) g˘| is continuous on K , |P (µ)|(K \K )< , and |µ| V \(V ∩K ) < .
Kn n V∗1 n n 1 1 n n
Using Tietze’s extension Theorem we may find a continuous function g(cid:0) :K −→R sat(cid:1)isfying
n
g | =g˘| and kg k=kg˘| k6kg˘k
n Kn Kn n Kn
for every n∈N. Now it is easy to check that
g˘fdPV∗1(µ)= gnfdPV∗1(µ)+ (g˘−gn)fdPV∗1(µ)
ZK ZK ZK\Kn
=P (µ)(g f)+ (g˘−g )fdP (µ)
V∗1 n n V∗1
ZK\Kn
=µ (g f)| + (g˘−g )fdP (µ)
n V1 n V∗1
(cid:0) (cid:1) ZK\Kn
= gnfdµ+ (g˘−gn)fdPV∗1(µ)
ZV1 ZK\Kn
= gfdµ+ (g −g˘)fdµ+ (g˘−g )fdP (µ)
n n V∗1
ZV1 ZV1\(V1∩Kn) ZK\Kn
=µ(gf)+ (gn−g˘)fdµ+ (g˘−gn)fdPV∗1(µ)
ZV1\(V1∩Kn) ZK\Kn
which, letting n→∞ and using (6), implies that
g˘fdP (µ)=µ(gf)
V∗1
ZK
and, therefore,
PV2gIdC(K)∗PV∗1 (µ)(f)=µ(gf).
This, together with (4) and the(cid:0)definition of S, fin(cid:1)ishes the proof of the claim. On the other hand
by (3), we can write b
PV2R0∗PV∗1 =PV2PV∗2R∗Ih∗,V1PV∗1.
8 PIOTRKOSZMIDER,MIGUELMART´IN,ANDJAVIERMER´I
So, if the support of µ is included in V , by Lemma 3.3, parts (c) and (d) we obtain
0
PV2R P (µ)=R (µ)
0∗ V∗1 ∗
and, consequently, R (µ)=gµ+S(µ) follows from (5). (cid:3)
∗
Remark 3.5. The result above shows that if every operator on C(K) is a weak multiplier then,
in the above sense, there are also few operators on C(V) for V open (since for such a closed set
it is possible to define an appropriate function h as in the proof). In general, one cannot replace
closures of open sets by general closed sets: it is shown in [9] that under CH, there are compact
K’s as above which contain βN (and, of course, there are many operators on C(βN) ≡ ℓ ). On
∞
the other hand, using the set-theoretic principle ♦, it is also shown in [9] that there are K’s such
that for every infinite closed K ⊆K, all operators on the space C(K ) are weak multipliers.
′ ′
Corollary 3.6. Let K be a compact space, let V , V , and V be open nonempty subsets of K such
0 1 2
that V ⊆ V and V ∩V = ∅, and let R : C(V ) −→ C(V ) be a linear operator. Suppose that
0 1 1 2 2 1
all operators on C(K) are weak multipliers. Then, P R is weakly compact.
V0
Proof. By Proposition 3.4 there is a weakly compact operator S : C (V ) −→ C (V ) such that
∗ 1 ∗ 2
R (µ)=S(µ)foreverymeasureµwithsupportincludedinV . Inotherwords,(P R) =R P
∗ 0 V0 ∗ ∗ V∗0
is weakly compact and so, by Gantmacher theorem, P R is weakly compact. (cid:3)
V0
The following result is an easy consequence of the Dieudonn´e-Grothendieck theorem which we
state for the sake of clearness.
Lemma 3.7. Let K be a compact space, X a Banach space and S : X −→ C(K) a weakly
∗ ∗
compact operator. Then, for every bounded subset B ⊆X , the set
∗
x∈K : ∃φ∈B so that S(φ)({x})6=0
is countable. (cid:8) (cid:9)
Proof. Suppose that the set
x∈K : ∃φ∈B so that S(φ)({x})6=0
is uncountable for some bou(cid:8)nded set B ⊆X∗. Then, there is ε>0 s(cid:9)o that the set
x∈K : ∃φ∈B so that |S(φ)({x})|>ε
is infinite, which contradicts(cid:8)the fact of being S(B) relatively weakly(cid:9)compact by the Dieudonn´e-
Grothendieck theorem (see [6, Theorem VII.14], for instance). (cid:3)
Lemma 3.8. Let K be a compact space, let V , V , and V be open nonempty subsets of K
0 1 2
compatible with L such that V ⊆ V and V ∩L = ∅, and let T : C (KkL) −→ C (KkL) be a
0 1 1 E E
linear operator. Then, there exists an operator R:C(V )−→C (V kL) such that
1 E 2
R (φ)| =(T P )(φ)|
∗ V0 ∗ V∗2 V0
for all φ∈C (V kL) .
E 2 ∗
Proof. Takeacontinuousfunctionh:K −→[0,1]satisfyingh| ≡1andh| ≡0,anddefine
V0 (K\V1)
the operatorR=P TI . Givenφ∈C (V kL) andf ∈C(V )withSupp(f)⊆V ,by parts
V2 h,V1 E 2 ∗ 1 0
(b) and (a) of Lemma 3.3 and using the facts h| ≡1 and Supp(f)⊆V , we can write
V0 0
R∗(φ)(f)=Ih∗,V1T∗PV∗2(φ)(f)=T∗PV∗2(φ)(fh)=T∗PV∗2(φ)(f)
which finishes the proof. (cid:3)
We are ready to state and prove the main result of the section.
ISOMETRIES ON EXTREMELY NON-COMPLEX BANACH SPACES 9
Theorem 3.9. Let K be a perfect compact space such that all operators on C(K) are weak multi-
pliers, let L⊆K be closed and nowhere dense, and E a closed subspace of C(L). Then, C (KkL)
E
is extremely non-complex.
Proof. Fixed T ∈ L C (KkL) , we have to show that its square satisfies the Daugavet equation.
E
ByLemma2.5,itis(cid:0)enoughtop(cid:1)rovethattheset{x∈K\L : g(T2)∗(x)>0}isdenseinK\L. To
do so, we proceed ad absurdum: suppose that there is an open set U ⊆K such that U ∩L=∅
1 1
and g(T2)∗(x) < 0 for each x ∈ U1. By going to a subset, we may w.l.o.g. assume that U1 is a
G set. Therefore, we can find open sets W ⊆ K such that W = U , W ⊆ W , and
δ n n N n 1 n+1 n
K\W is the closure of an open set containing L for every n∈N∈. Next, we fix a nonempty open
n
T
set U ⊆ K with U ⊆ U , and we observe that it is uncountable (since K is perfect) so, there is
0 0 1
ε>0 such that the set
A={x∈U0 : g(T2)∗(x)<−ε}
is uncountable. Moreover,we claim that there is n ∈N such that the set
0
ε
B = x∈A : |T (δ )|(W \U ))<
∗ x n0 1 2kTk
(cid:26) (cid:27)
is uncountable. Indeed, fixed x ∈ A, the regularity of the measure T (δ ) implies that there is
∗ x
n∈N so that |T (δ )|(W \U ))< ε which gives us
∗ x n 1 2 T
k k
ε
A= x∈A : |T∗(δx)|(Wn\U1))<
2kTk
n N(cid:26) (cid:27)
[∈
and the uncountability of A finishes the argument.
For x∈B, we write
φx =T∗(δx)|K\(L∪Wn0)+T∗(δx)E ∈CE (K\Wn0)kL ∗
and we can decompose T (δ ) as follows: (cid:0) (cid:1)
∗ x
T∗(δx)=T∗(δx)|U1 +T∗(δx)|Wn0\U1 +φx.
Hence, for every x∈B, we get
(7) −ε> (T )2(δ ) ({x})=T T (δ )| +T (δ )| +φ {x}.
∗ x ∗ ∗ x U1 ∗ x Wn0\U1 x
However,the following(cid:2)claims sh(cid:3)ow that this(cid:2)is impossible. (cid:3)
Claim 1. T T (δ )| <ε/2.
∗ ∗ x Wn0\U1
Proof of c(cid:13)laim(cid:2)1. It follows o(cid:3)b(cid:13)viously from the choice of n0.
(cid:13) (cid:13)
Claim2. Thefunctionx7−→T (T (δ )| )({x})isnon-negativeforallbutcountablymanyx∈B.
∗ ∗ x U1
Proofof claim 2. ByLemma3.8appliedtoV =U ,V =W andV =U ,weobtainanoperator
0 1 1 n0 2 1
R:C(W )−→C(U ) such that
n0 1
R (ψ)| =T P (ψ)|
∗ U1 ∗ U∗1 U1
for ψ ∈ C(U ) . On the other hand, by Proposition 3.4 applied to R and V = U , V = U ,
1 ∗ 0 0 1 1
V = W , we get a bounded Borel function g : U −→ R and a weakly compact operator
2 n0 1
S :C(U ) −→C(W ) such that
1 ∗ n0 ∗
R (µ)=gµ+S(µ)
∗
for every µ with support in U . In particular, for x∈B we have
0
T∗(δx)|U1 =T∗PU∗1(δx)|U1 =R∗(δx)|U1 =gδx+S(δx)|U1 =g(x)δx+S(δx)|U1
10 PIOTRKOSZMIDER,MIGUELMART´IN,ANDJAVIERMER´I
and using this twice, we get
T (T (δ )| )({x})=T g(x)δ +S(δ )| ({x})=g(x)T (δ )({x})+T S(δ )| ({x})
∗ ∗ x U1 ∗ x x U1 ∗ x ∗ x U1
=g(xh)T∗(δx)|U1({x})+iR∗ S(δx)|U1 ({x}) (cid:2) (cid:3)
=g(x)2+g(x)S(δx)|U1({x}(cid:2))+ gS(δ(cid:3)x)|U1 ({x})+S S(δx)|U1 ({x})
=g(x)2+(2gPU1S)(δ )({x})+(cid:0)(SPU1S)((cid:1)δ )({x}) (cid:2) (cid:3)
x x
for every x∈B. Finally, since gPU1S and SPU1S are weakly compact operators, we conclude by
Lemma 3.7 that (gPU1S)(δ )({x}) and (SPU1S)(δ )({x}) are zero for all but countably many x,
x x
completing the proof of the claim.
Claim 3. T (φ )({x})=0 for all but countably many x∈U .
∗ x 0
Proof of claim 3. By Lemma 3.8 applied to V = U , V = U , and V = K \W we obtain
0 0 1 1 2 n0+1
an operator R : C(U ) −→ C (K \W )kL such that R (φ)({x}) = T P (φ)({x}) for
1 E n0+1 ∗ ∗ K∗\Wn0
x∈U0 and φ∈CE (K\Wn0+1(cid:0))kL ∗. We denot(cid:1)e J :CE (K\Wn0+1)kL −→C(K\Wn0+1) the
inclusionoperatorandweapplyCorollary3.6fortheoperatorJRandtheopensetsV =K\W ,
(cid:0) (cid:1) (cid:0) (cid:1) 0 n0
V =K\W , and V =U to obtain that the operator P JR is weakly compact.
1 n0+1 2 1 K\Wn0
Besides, we recall that φx ∈ CE (K \Wn0kL) ∗ and that it can be viewed as an element of
CE (K \Wn0+1kL) ∗ by just exten(cid:0)ding it by zer(cid:1)o outside K \Wn0. For x ∈ U0 we take φx a
Hahn-Banach extension of φ to C(K \W ) and we observe that J P (φ ) = φ . Indeed,
(cid:0) (cid:1) x n0 ∗ K∗\Wn0 x x e
for f ∈C (K\W )kL we have that
E n0+1
e
(cid:0) J P (cid:1) (φ )(f)=φ P (Jf) =φ P (f)
∗ K∗\Wn0 x x K\Wn0 x K\Wn0
=φ (cid:0)f| =(cid:1)φ f| (cid:0) =φ ((cid:1)f).
e ex K\Wn0 x eK\Wn0 x
Therefore, for x∈U , we can write (cid:0) (cid:1) (cid:0) (cid:1)
0 e
T∗(φx)({x})=T∗PK∗\Wn0(φx)({x})=R∗(φx)({x})
=R J P (φ )({x})=(P JR) (φ )({x})
∗ ∗ K∗\Wn0 x K\Wn0 ∗ x
whereweareidentifying φ withits extensionbyzerotoK. Nowtheproofoftheclaimisfinished
x e e
by just applying Lemma 3.7 to the operator P JR.
K\Wn0
Finally, the claims obviously contradict (7) completing the proof of the theorem. (cid:3)
When E = {0}, we get a sufficient condition to get that a space of the form C (K \ L) is
0
extremely non-complex.
Corollary 3.10. Let K be a compact space such that all operators on C(K) are weak multipliers.
Suppose L⊆K is closed and nowhere dense. Then, C (K \L) is extremely non-complex.
0
To show that there are extremely non-complex spaces of the form C (KkL) which are not
E
isomorphic to the C(K ) spaces, we need the following (well-known) result which allows us to
′
constructspacesC (KkL)foreveryperfectseparablecompactspaceKandeveryseparableBanach
E
space E.
Lemma 3.11. Let K be a perfect compact space. Then:
(a) There is a nowhere dense closed subset L ⊂ K such that L can be continuously mapped
onto the Cantor set.
(b) Therefore, every separable Banach space E is (isometrically isomorphic to) a subspace of
C(L).