ISOMETRIES ON EXTREMELY NON-COMPLEX BANACH SPACES PIOTRKOSZMIDER,MIGUELMARTÍN,ANDJAVIERMERÍ 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,MIGUELMARTÍN,ANDJAVIERMERÍ 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. Dieudonné [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,MIGUELMARTÍN,ANDJAVIERMERÍ 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)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,MIGUELMARTÍN,ANDJAVIERMERÍ 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,MIGUELMARTÍN,ANDJAVIERMERÍ 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 Dieudonné-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 Dieudonné- 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 multipliers, 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,MIGUELMARTÍN,ANDJAVIERMERÍ 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).

Isometries on extremely non-complex Banach spaces

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