THE BISHOP-PHELPS-BOLLOBÁS PROPERTY FOR OPERATORS BETWEEN SPACES OF CONTINUOUS FUNCTIONS

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1 THE BISHOP-PHELPS-BOLLOBÁS PROPERTY FOR OPERATORS BETWEEN SPACES OF CONTINUOUS FUNCTIONS MARÍA D. ACOSTA, JULIO BECERRA-GUERRERO, YUN SUNG CHOI, MACIEJ CIESIELSI, SUN WANG IM, HAN JU LEE, MARY LILIAN LOURENÇO, AND MIGUEL MARTÍN Abstract. We show that the space of bounded linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollobás property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an L 1 -space. 1. Introduction E. Bishop and R. Phelps proved in 1961 [7] that every (continuous linear) functional x on a Banach space X can be approximated by a norm attaining functional y. This result is called the Bishop-Phelps Theorem. Shortly thereafter, B. Bollobás [8] showed that this approximation can be done in such a way that, moreover, the point at which x almost attains its norm is close in norm to a point at which y attains its norm. This is a quantitative version of the Bishop-Phelps Theorem, known as the Bishop-Phelps-Bollobás Theorem. For a real or complex Banach space X, we denote by S X, B X and X the unit sphere, the closed unit ball and the dual space of X, respectively. Theorem 1.1 (Bishop-Phelps-Bollobás Theorem, [8, Theorem 1]). Let X be a Banach space and 0 < ε < 1/2. Given x B X and x S X with 1 x (x) < ε2 2, there are elements y S X and y S X such that y (y) = 1, y x < ε + ε 2 and y x < ε. We refer the reader to the recent paper [10] for a more accurate version of the above theorem. In 2008, M. D. Acosta, R. M. Aron, D. García and M. Maestre introduced the so-called Bishop-Phelps-Bollobás property for operators [1, Definition 1.1]. For our purposes, it will be useful to recall an appropriate version of this property for classes of operators defined in [2, Definition 1.3]. Given two Banach spaces X and Y, L(X, Y ) denotes the space of all (bounded Date: July 17th, Revised September 4th, Mathematics Subject Classification. Primary 46B04. ey words and phrases. Banach space, optimization, norm-attaining operators, Bishop-Phelps theorem, Bishop-Phelps- Bollobás theorem. First and eighth authors partially supported by Spanish MINECO and FEDER project no. MTM , Junta de Andalucía and FEDER grants FQM-185 and P09-FQM Second author supported by MTM and Junta de Andalucía grants FQM 0199 and FQM Third author partially supported by Basic Science Research Program through the National Research Foundation of orea (NRF) funded by the Ministry of Education, Science and Technology (No ), and also by Priority Research Centers Program through the National Research Foundation of orea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (No ). Sixth author partially supported by Basic Science Research Program through the National Research Foundation of orea(nrf) funded by the Ministry of Education, Science and Technology (2012R1A1A ). Seventh author partially supported by FAPESP (Project no 2012/ ). 1

2 2 ACOSTA, BECERRA, CHOI, CIESIELSI, IM, LEE, LOURENÇO, AND MARTÍN and linear) operators from X into Y. The subspace of L(X, Y ) of finite-rank operators F(X, Y ); (X, Y ) will denote the subspace of all compact operators. Definition 1.2. Let X and Y be Banach spaces and M a linear subspace of L(X, Y ). We say that M satisfies the Bishop-Phelps-Bollobás property if given ε > 0, there is η(ε) > 0 such that whenever T S M and x 0 S X satisfy that T x 0 > 1 η(ε), then there exist a point u 0 S X and an operator S S M satisfying the following conditions: Su 0 = 1, u 0 x 0 < ε, and S T < ε. In case that M = L(X, Y ) satisfies the previous property it is said that the pair (X, Y ) has the Bishop-Phelps-Bollobás property for operators (shortly BPBp for operators). Observe that the BPBp of a pair (X, Y ) means that one is able to approximate any pair of an operator and a point at which the operator almost attains its norm by a new pair of a norm-attaining operator and a point at which this new operator attains its norm. In particular, if a pair (X, Y ) has the BPBp, the set of norm-attaining operators is dense in L(X, Y ). The reverse result is far from being true: there are Banach spaces Y such that the pair (l 2 1, Y ) does not have the BPBp (see [1]). In [1] the authors provided the first version of the Bishop-Phelps-Bollobás Theorem for operators. Amongst them, a sufficient condition on a Banach space Y to get that for every Banach space X, the pair (X, Y ) has the BPBp for operators, which is satisfied, for instance, by Y = c 0 or Y = l. A characterization of the Banach spaces Y such that the pair (l 1, Y ) has the BPBp for operators is also given. There are also positive results for operators from L 1 (µ) into L (ν) [5, 12], for operators from L 1 (µ) into L 1 (ν) [12], for certain ideals of operators from L 1 (µ) into another Banach space [11, 2], for operators from an Asplund space into C 0 (L) or into a uniform algebra [4, 9], and for operators from a uniformly convex space into an arbitrary Banach space [3, 16]. For some more recent results, see also [6]. Let us also point out that the set of norm attaining operators from L 1 [0, 1] into C[0, 1] is not dense in L(L 1 [0, 1], C[0, 1]) [19]. Our aim in this paper is to provide classes of Banach spaces satisfying a version of the Bishop-Phelps-Bollobás Theorem for operators. The first result, which is the content of section 2, states that given arbitrary compact Hausdorff topological spaces and S, the pair (C(), C(S)) satisfies the BPBp for operators in the real case. This result extends the one by J. Johnson and J. Wolfe [14] that the set of norm attaining operators from C() into C(S) is dense in L(C(), C(S)). In section 3, we prove that the space (C 0 (L), Y ) satisfies the Bishop- Phelps-Bollobás property whenever L is a locally compact Hausdorff topological space and Y is uniformly convex in both the real and the complex case. Let us remark that it was also proved in [14] that the set of norm-attaining weakly compact operators from C() into Y is dense in the space of all weakly compact operators. But, as commented above, there are Banach spaces Y such that the pair (l 2, Y ) does not satisfy the BPBp for operators (in the real case, l 2 l 2 1 ), so some assumption on Y is needed to get the Bishop-Phelps-Bollobás property. Finally, we devote section 4 to show that the space (X, Y ) has the Bishop-Phelps-Bollobás property when X is an arbitrary Banach space and Y is a predual of an L 1 -space in both the real and the complex case. This extends the result of [14] that the set of norm-attaining finite-rank operators from an arbitrary Banach space into a predual of an L 1 -space is dense in the space of compact operators. In particular, for Y = C 0 (L) for some locally compact Hausdorff topological space L, the result is a consequence of the already cited paper [4].

3 BISHOP-PHELPS-BOLLOBÁS PROPERTY IN SPACES OF CONTINUOUS FUNCTIONS 3 2. Operators between spaces of continuous functions Throughout this section, and S are compact Hausdorff topological spaces. Here C() is the space of real valued continuous functions on. M() denotes the space of regular Borel finite measures on, which identifies with the dual of C() by the Riesz representation theorem. For s S, we write δ s to denote the point measure concentrated at s. Lemma 2.1 ([13, Theorem 1, p. 490]). Let X be a Banach space and let S be a compact Hausdorff topological space. Given an operator A : X C(S), define µ : S X by µ(s) = A (δ s ) for every s S. Then the relationship [Ax](s) = µ(s)(x), x X, s S defines an isometric isomorphism between L(X, C(S)) and the space of w -continuous functions from S to X, endowed with the supremum norm, i.e. µ = sup{ µ(s) : s S}. Compact operators correspond to norm continuous functions. Lemma 2.2 ([14, Lemma 2.2]). Let µ : S M() be w -continuous. Let ε > 0, s 0 S and an open subset V of be given. Then there exists an open neighborhood U of s 0 such that if s U, then µ(s) (V ) µ(s 0 ) (V ) ε. The next result is a version of [14, Lemma 2.3] in which the main difference is that we start with an operator and a function in the unit sphere of C() where the operator almost attains its norm and construct a new operator and a new function, both close to the previous elements and satisfying additional restrictions. Condition iii) is the new ingredient that will be useful. Lemma 2.3. Let µ : S M() be a w -continuous function satisfying µ = 1 and 0 < δ < 1. Suppose that s 0 S and f 0 S C() satisfy f 0 dµ(s 0 ) > 1 δ2 12. Then there exist a w - continuous mapping µ : S M(), an open set U in S, an open set V of and h 0 C() satisfying the following conditions: i) µ (s) (V ) = 0 for every s U. ii) h 0 dµ (s) µ δ for every s U. iii) h 0 f 0 < δ. iv) h 0 = 1 and h 0 (t) = 1 t \ V. v) µ µ < δ. Proof. Let us write µ 0 := µ(s 0 ). By the Hahn decomposition theorem, there is a partition of into two measurable sets + and such that + is a positive set for µ 0 and is negative for µ 0. For every 0 < x < 1, consider two open subsets of given by and consider the set O + x := {t : f 0 (t) > x}, O x := {t : f 0 (t) < x}, D x := ( + O x + ) ( Ox ). Write α = δ2 12. By the assumption, we have 1 α < f 0 dµ 0 µ 0 (D x ) + x µ 0 ( \ D x ) Hence, µ 0 (D x ) + x(1 µ 0 (D x )) = (1 x) µ 0 (D x ) + x. (1) µ 0 (D x ) > 1 α 1 x.

4 4 ACOSTA, BECERRA, CHOI, CIESIELSI, IM, LEE, LOURENÇO, AND MARTÍN Next, consider the open subset W x of given by and observe that, since D x W x, we have W x := O + x O x = {t : f 0 (t) > x}, (2) µ 0 (W x ) µ 0 (D x ) 1 α 1 x. Write c := 1 δ 4 and choose real numbers a and b with 1 δ < a < b < c < 1. As the open subset W a contains O + b O b, there is u C() such that 0 u 1, u 1 on O+ b O b and supp u W a. Since the support of u is contained in W a (where f 0 is separated from 0), the function h 0 defined on by h 0 (t) = f 0(t) f 0 (t) u(t) + (1 u(t))f 0(t) if f 0 (t) 0, h 0 (t) = 0 otherwise, is continuous and, actually, h 0 B C(). We claim that h 0 f 0 < δ, which guarantees condition iii). Indeed, if t \ W a, then u(s) = 0 and so h 0 (t) = f 0 (t); if otherwise t W a, we have that h 0 (t) f 0 (t) = u(t) f 0 (t) f 0 (t) f 0(t) 1 f 0(t) < 1 a < δ, proving the claim. On the other hand, [ we know ] that u 1 in O + b O b, so h 0 = 1 on O + b O b. Therefore, if we write V := \ O + b O b, which is an open subset of, the second part of condition iv) is satisfied. Next, as the open subsets V and W c satisfy V W c =, there is a function f C() such that 0 f 1, f 1 on V and supp f \ W c. Since D c W c, we have that h 0 dµ 0 = h 0 dµ 0 + h 0 dµ 0 = µ 0 (D c ) + h 0 dµ 0 W c D c W c\d c W c\d c µ 0 (D c ) µ 0 (W c \ D c ) µ 0 (D c ) µ 0 ( \ D c ) 2 µ 0 (D c ) µ 0 (). Therefore, by using (1), we obtain that (3) h 0 dµ 0 > 1 2 α W c 1 c = δ. As a consequence, h 0 (1 f) dµ 0 = h 0 (1 f) dµ 0 + W c h 0 (1 f) dµ 0 \W c h 0 (1 f) dµ 0 µ 0 ( \ W c ) W c > 1 2 α 1 c ( 1 µ 0 (W c ) ) (by (3)) > 1 3 α = 1 δ (by (2)) 1 c

5 BISHOP-PHELPS-BOLLOBÁS PROPERTY IN SPACES OF CONTINUOUS FUNCTIONS 5 Now, in view of the w -continuity of µ, the previous inequality, condition (2) and Lemma 2.2, we get that there exists an open neighborhood U 0 of s 0 such that (4) h 0 (1 f)dµ(s) > 1 δ and µ(s) (W c ) > 1 δ, s U 0. We can also choose an open subset U of S such that s 0 U and U U 0, and a function g C(S) such that 0 g 1, g(u) = {1} and supp g U 0. Define µ : S M() by µ (s) = ( 1 g(s)f ) µ(s) (s S). That µ (s) is the unique Borel measure on satisfying ϕ dµ ( ) (s) = 1 g(s)f ϕ dµ(s) ϕ C(). It is clear that µ is w -continuous. If s U, g(s) = 1 and f(v ) = {1}, so condition i) is satisfied. Since 0 f, g 1, then µ µ = 1 and hence, in view of (4), for every s U we have that h 0 dµ ( ) (s) = 1 g(s)f h0 dµ(s) > 1 δ µ δ, so condition ii) is also satisfied. We only have to check condition v), that is, µ µ < δ. Indeed, if s S \ U 0, then g(s) = 0 and so µ (s) = µ(s); if, otherwise, s U 0, by (4) and the fact that f(w c ) = {0}, we obtain that ϕ d(µ(s) µ (s)) = ϕg(s)f dµ(s) µ(s) ( \ W c ) µ µ(s) (W c ) < δ for every ϕ B C(), proving the claim. Finally, since µ = 1, condition v) implies that µ 0 and, in view of i), we deduce that V, so \ V is not empty and h 0 = 1 since h 0 (t) = 1 for every t \ V. The last ingredient that we will use is the next iteration result due to Johnson and Wolfe. Lemma 2.4 ([14, Lemma 2.4]). Let µ : S M() be w -continuous and δ > 0. Suppose there is an open set U S, an open set V, s 0 U and h 0 C() with h 0 = 1 such that a) if s U, then µ(s) (V ) = 0, b) h 0 dµ(s 0 ) µ δ, c) h 0 (t) = 1 for t \ V. Then, for any 2 3 < r < 1 there exist a w -continuous function µ : S M() and a point s 1 U such that a ) if s U, then µ (s) (V ) = 0, b ) h 0 dµ (s 1 ) µ rδ, c ) µ µ rδ. The next result improves [14, Theorem 1]. In the complex case, it is not known whether the same result is true or not. Theorem 2.5. Let and S be compact Hausdorff topological spaces. Then the pair (C(), C(S)) has the Bishop-Phelps-Bollobás property for operators in the real case. Moreover, the function η satisfying Definition 1.2 does not depend on the spaces and S (in fact one can take η(ε) = ε2 )

6 6 ACOSTA, BECERRA, CHOI, CIESIELSI, IM, LEE, LOURENÇO, AND MARTÍN Proof. Let us fix 2 3 T 0 S L(C(),C(S)) and f 0 S C() satisfy that T 0 (f 0 ) > 1 δ r < r < 1. Given 0 < ε < 2 let us choose 0 < δ < ε 2. Assume that. Then, there is an element s 1 S such that [ T 0 (f 0 ) ] (s 1 ) > 1 δ2 12. By using f 0 instead of f 0, if necessary, we may assume that T 0 (f 0 )(s 1 ) > 1 δ2 12. Therefore, we can apply Lemma 2.3 to the w -continuous function µ 0 : S M() associated with the operator T 0 (i.e. µ 0 (s) = T 0 (δ s) for every s S) to get that there exist a function h 0 S C(), an open set U in S, an open set V of and a w -continuous function µ 1 : S M() satisfying the following conditions: i) µ 1 (s) (V ) = 0 for every s U. ii) h 0 dµ 1 (s) µ 1 δ for every s U. iii) h 0 f 0 < δ. iv) h 0 = 1 and h 0 (t) = 1 t \ V. v) µ 1 µ 0 < δ. Now, by using Lemma 2.4, we inductively construct a sequence {µ n } of w -continuous functions from S into M() and a sequence {s n } in U satisfying µ n+1 µ n r n δ, µ n h 0 dµ n (s n ) + r n δ and µ n (s) (V ) = 0 for every s U and n N. If for every n N, we write T n L(C(), C(S)) to denote the bounded linear operator associated with the function µ n, we may rewrite i) and ii) as (5) T n+1 T n r n δ and T n T n (h 0 ) + r n δ. Since 0 < r < 1, the previous condition implies that {T n } is a Cauchy sequence, so it converges to an operator T L(C(), C(S)) satisfying T T 0 T k+1 T k r k 1 δ = δ 1 r < ε 2. k=0 k=0 By taking limit in the right-hand side of (5), we also have that T T (h 0 ) and, since h 0 S C(), T attains its norm at h 0. Finally, we have that so T 0, T 1 T T 0 T T 0 T < ε 2 < 1, T also attains its norm at h 0 and T T T 0 T T T + T T 0 = 1 T + T T0 < ε. As we already knew that h 0 f 0 < δ < ε, this shows that the pair (C(), C(S)) satisfies the Bishop-Phelps-Bollobás Theorem for operators with η = δ Compact operators from a space of continuous functions into a uniformly convex space Our purpose now is to prove the Bishop-Phelps-Bollobás property for compact operators. The following result due to im will play an essential role:

7 BISHOP-PHELPS-BOLLOBÁS PROPERTY IN SPACES OF CONTINUOUS FUNCTIONS 7 Lemma 3.1 ([15, Theorem 2.5]). Let Y be a uniformly convex space. For every 0 < ε < 1, there is 0 < γ(ε) < 1 with the following property: given n N, T S L(l n,y ) and x 0 S l n such that T x 0 > 1 γ(ε), there exist S S L(l n,y ) and x 1 S l n satisfying Sx 1 = 1, S T < ε and x 1 x 0 < ε. It is easy to show that the function γ in the previous result satisfies lim t 0+ γ(t) = 0. Let L be a locally compact Hausdorff topological space. As usual, C 0 (L) will be the space either of real or complex continuous functions on L with limit zero at infinity. We recall that C 0 (L) can be identified with the space M(L) of regular Borel measures on L by the Riesz representation theorem. For every f C 0 (L) and every (non-empty) set S L, we define Osc(f, S) by Osc(f, S) = sup f(x) f(y) x,y S The next result generalizes Lemma 3.1 and Proposition 3.2 of [14] to C 0 (L). actually based on the proof of these results. Its proof is Proposition 3.2. Let L be a locally compact Hausdorff topological space and let Y be a Banach space. For every ε > 0, T (C 0 (L), Y ) and f 0 C 0 (L), there exist a positive regular Borel measure µ, a non-negative integer m, pairwise disjoint compact subsets j of L and ϕ j C 0 (L) for 1 j m, satisfying the following conditions: (1) Osc(f 0, j ) < ε. (2) 0 ϕ j 1 and ϕ j 1 on j. (3) supp ϕ i supp ϕ j = for i j. (4) The operator P : C 0 (L) C 0 (L) given by m 1 ( ) P (f) := f dµ ϕ j, µ( j ) j f C 0 (L), is a norm-one projection from C 0 (L) onto the linear span of {ϕ 1,..., ϕ m } that also satisfies T T P < ε. Proof. Since T is a compact operator, the adjoint operator T is a compact operator from Y into C 0 (L), so we may take a finite ε 4 -net {µ 1,..., µ t } of T (B Y ) C 0 (L) M(L). We define the (finite regular) measure µ by µ = t i=1 µ i. For each 1 i t, we have that µ i µ, hence the Radon-Nikodým theorem allows us to find a function g i L 1 (µ) such that µ i = g i µ. Since the set of simple functions is dense in L 1 (µ), we may choose a set of simple functions {s i finite family (A j ) m 0 : i = 1,..., t} such that g i s i 1 < ε 12 for every 1 i t. Next, consider a of pairwise disjoint measurable sets such that for every 1 i t, there is αj i χ A j. Let M be a positive real number satisfying a family (α i j )m 0 of scalars such that s i = m 0 M max { αj i : 1 i t, 1 j m 0}. Since µ is regular, for each 1 j m0 we find ε a compact set C j A j such that µ(a j \ C j ) < 12m 0 M. As f 0 is continuous and each C j is compact, we may divide each C j into a family of Borel sets (B p j )n j p=1 such that Osc(f 0, B p j ) < ε 1 j m 0, 1 p n j.

8 8 ACOSTA, BECERRA, CHOI, CIESIELSI, IM, LEE, LOURENÇO, AND MARTÍN Applying the regularity again, for each j and p, there is a compact set p j B p j such that µ(b p j \ p j ) < ε 12m 0 n j M. Finally, choose suitable m N, a rearrangement ( j) m of the family { p j : 1 j m 0, 1 p n j, µ ( p ) j > 0} and scalars (β i j ) for j m and i t such that m m 0 ( n j ) βjχ i j =. Using Urysohn s lemma, we may choose a family (ϕ j ) m in C 0(L) satisfying that 0 ϕ j 1, ϕ j 1 on j for each j m and supp ϕ i supp ϕ j = for every i j. To finish the proof, we only have to check (4). Indeed, for i = 1,..., t, we write ν i = m βi j χ j µ M(L) = C 0 (L). By defining the operator P as in condition (4), it is easy to check that P is a norm one projection onto the linear span of {ϕ 1,..., ϕ m } and P ν i = ν i for each 1 i t. Therefore, for every 1 i t we have that α i j p=1 χ p j µ i P ν i = g i µ ν i g i µ s i µ + s i µ ν i g i s i 1 + s m 0 iµ αjχ i m0 Cj µ + αjχ i Cj µ < ε m α i j < ε 12 + ε m µ ( ) m 0 ( A j \ C j + α i j n j p=1 µ ( B p j \ ) p j n j αj i χ B p j p=1 m βjχ i j µ ) m 0 ( µ n j αj i p=1 χ p j ) µ < ε 12 + ε 12 + ε 12 = ε 4. Since {µ 1,..., µ t } is a ε 4 -net of T (B Y ), the above inequality shows that {ν 1,..., ν t } is a ε 2 -net of T (B Y ). Now, given y B Y, we can choose i t satisfying ν i T y < ε 2 and observe that T y P T y T y ν i + ν i P T y = T y ν i + P ν i P T y 2 T y ν i < ε. Hence, we have T T P = T P T < ε, as desired. The following result shows that (C 0 (L), Y ) satisfies the Bishop-Phelps-Bollobás property for every locally compact Hausdorff topological space L and every uniformly convex space Y, and that the function η(ε) involved in the definition of the property does not depend on L. Theorem 3.3. Let Y be a uniformly convex Banach space. For every 0 < ε < 1 there is 0 < η(ε) < 1 such that for any locally compact Hausdorff topological space L, if T S (C0 (L),Y ) and f 0 S C0 (L) satisfy T f 0 > 1 η(ε), there exist S S (C0 (L),Y ) and g 0 S C0 (L) such that Sg 0 = 1, S T < ε and g 0 f 0 < ε. Proof. Given 0 < ε < 1, we choose 0 < δ < ε 4 such that 0 < γ(δ) < ε 4, where γ(δ) satisfies the statement of Lemma 3.1. We also consider α such that 0 < α < min { δ, γ(δ) } 2 and η(ε) := α > 0. Fix 0 < ε < 1, T S (C0 (L),Y ) and f 0 S C0 (L) with T f 0 > 1 η(ε) = 1 α. Applying Proposition 3.2, we get a positive regular Borel measure µ on L, a non-negative integer m, pairwise disjoint compact subsets j of L and ϕ j C 0 (L) (1 j m) such that

9 BISHOP-PHELPS-BOLLOBÁS PROPERTY IN SPACES OF CONTINUOUS FUNCTIONS 9 (1) Osc(f 0, j ) < α, (2) For every 1 j m, 0 ϕ j 1 and ϕ j 1 on j, (3) supp ϕ i supp ϕ j = for i j, (4) T T P < α, where P L(C 0 (L)) is given by m 1 ( ) ( P (f) := f dµ ϕ j f C0 (L) ), µ( j ) j and it is a norm-one projection onto the linear span of {ϕ 1,..., ϕ m }. Now, if t j for some j = 1,..., m, we obtain that [P (f 0 )](t) f 0 (t) 1 ( = f0 (s) f 0 (t) ) dµ(s) µ( j ) j 1 f 0 (s) f 0 (t) dµ(s) α. µ( j ) j Hence { (6) max [P (f 0 ) f 0 ](t) : t m We also have that j } α. (7) T P (f 0 ) T (f 0 ) T T P > 1 2α > 1 γ(δ), and this implies that 1 2α T P 1 and 1 2α P (f 0 ). Since the functions {ϕ j : 1 j m} have pairwise disjoint support, the linear operator Φ : lin{ϕ 1,..., ϕ m } l m satisfying Φ(ϕ j ) = e j for every j = 1,..., m is an onto linear isometry (where {e 1,..., e m } is the natural basis of l m ). We define U 1 := T Φ 1 : l m Y and observe that, clearly, U 1 1. On the other hand, the element x 0 := Φ(P (f 0 )) B l m satisfies that U 1 (x 0 ) = T P (f 0 ) so, in view of (7), U 1 (x 0 ) > 1 γ(δ) and so U 1, x 0 > 1 γ(δ) > 0. ( ) We consider U = U 1 U 1 and apply Lemma 3.1 to the pair x U, 0, to get an operator V : l m x 0 Y with V = 1 and x 1 S l m with (8) V U < δ, x1 x0 x 0 < δ, and V ( x 1) = 1. We clearly have that (9) U U 1 = U 1 U 1 U 1 = 1 U 1 γ(δ) and, also, (10) As a consequence, x 1 x 0 x 1 x0 x 0 + x 0 x 0 x0 δ + 1 x0 δ + γ(δ). V U 1 V U + U U 1 δ + γ(δ).

10 10 ACOSTA, BECERRA, CHOI, CIESIELSI, IM, LEE, LOURENÇO, AND MARTÍN Finally, we define the operator S : C 0 (L) Y given by S(f) := V (Φ(P (f))) for every f C 0 (L), which is clearly a compact operator and satisfies S 1. Consider the element f 1 = m x1 (j)ϕ j B C0 (L). It is clear that P (f 1 ) = f 1, Φ(f 1 ) = x 1 and that S S(f 1 ) = V ΦP (f 1 ) = V ( x 1) = 1. We deduce that f 1 S C0 (L) and that S = S(f 1 ) = 1. This implies that S attains its norm at f 1. Next, we estimate the distance between S and T as follows On the other hand, { max [f 1 f 0 ](t) : t m S T = V ΦP T V ΦP T P + T P T V ΦP UΦP + UΦP T P + α V U + UΦP T Φ 1 ΦP + α < δ + U T Φ 1 + α (by (8)) 2δ + U U 1 2δ + γ(δ) < ε (by (9)). j } = max max x 1 (j) f 0 (t) 1 j m t j max 1 j m { x 1 (j) x 0 (j) + max t j x 0 (j) f 0 (t) x 1 x 0 + max 1 j m max t j [P (f0 ) f 0 ](t) δ + γ(δ) + α < 2δ + γ(δ) (by (10) and (6)). Hence, there exists an open set O L such that m (11) j O, [f1 f 0 ](t) ( ) < 3δ + γ(δ) t O. By Urysohn s Lemma again, there is ψ C 0 (L) such that 0 ψ 1, ψ 1 on m k=1 j and supp ψ O. We write g 0 := ψf 1 + (1 ψ)f 0 B C0 (L) and we claim that S attains its norm at g 0 and that f g 0 < ε, which finishes the proof. Indeed, on the one hand, it is clear that the restriction of g 0 to m k=1 j coincides with f 1. It follows that P (g 0 ) = P (f 1 ) and so S(g 0 ) = S(f 1 ) and S attains its norm at g 0. On the other hand, for t L \ O we have g(t) = f 0 (t). If, otherwise, t O, condition (11) gives that g 0 (t) f 0 (t) = ψ(t) ( f 1 (t) f 0 (t) ) < 3δ + γ(δ) < ε. 4. Compact operators into a predual of an L 1 (µ)-space Our goal is to show that the space of compact operators from an arbitrary Banach space into an isometric predual of an L 1 -space has the Bishop-Phelps-Bollobás property in both the real and the complex case. We need a preliminary result which follows easily from the Bishop-Phelps-Bollobás theorem. It is also a very particular case of [1, Theorem 2.2]. Lemma 4.1. For every 0 < ε < 1, there is 0 < η (ε) < 1 such that for every positive integer n and every Banach space X, the pair (X, l n ) has the BPBp for operators with this function η (ε). More concretely, given an operator U S L(X,l n ) and an element x 0 S X such that U(x 0 ) > 1 η (ε), there exist V S L(X,l n ) and z 0 S X satisfying V z 0 = 1, z 0 x 0 < ε and V U < ε. }

11 BISHOP-PHELPS-BOLLOBÁS PROPERTY IN SPACES OF CONTINUOUS FUNCTIONS 11 Theorem 4.2. For every 0 < ε < 1 there is η(ε) > 0 such that if X is any Banach space, Y is a predual of an L 1 -space, T S (X,Y ) and x 0 S X satisfy T x 0 > 1 η(ε), then there exist S S F(X,Y ) and z 0 S X with Sz 0 = 1, z 0 x 0 < ε and S T < ε. Proof. For any 0 < ε < 1 we take η(ε) = min{ ε 4, η (ε/2)}, where η is the function provided by the previous lemma. Fix 0 < ε < 1, T S (X,Y ) and x 0 S X satisfying T x 0 > 1 η(ε). Let us choose a positive number δ with δ < 1 4 min{ ε 4, T (x 0) 1 + η ( ε 2)} and let {y1,..., y n } be a δ-net of T (B X ). In view of [17, Theorem 3.1] and [18, Theorem 1.3], there is a subspace E Y isometric to l m for some natural number m and such that dist (y i, E) < δ for every i n. Let P : Y Y be a norm one projection onto E. We will check that P T T < 4δ. In order to show that we fix any element x B X and so T x y i < δ for some i n. Let e E be any element satisfying e y i < δ. Then we have T (x) P T (x) T (x) y i + y i e + e P T (x) 2δ + P (e) P T (x) 2δ + e T (x) 2δ + e y i + y i T (x) < 4δ. So P T > T 4δ = 1 4δ > 0. As a consequence we also obtain that P T (x 0 ) > T (x 0 ) 4δ > 1 η ( ε ). 2 Hence the operator R = P T P T satisfies R(x 0) > 1 η ( ε 2). Since E is isometric to l m, by Lemma 4.1 there exist an operator S L(X, E) L(X, Y ) with S = 1 and z 0 S X satisfying that S R < ε 2, z 0 x 0 < ε 2, and Sz 0 = 1. Finally, we have that S T S R + R P T + P T T < ε + 1 P T + 4δ 2 < ε + 8δ < ε. 2 Acknowledgment. The authors wish to express their thanks to the anonymous referee whose careful reading and suggestions have improved the final form of the paper. References [1] M.D. Acosta, R.M. Aron, D. García and M. Maestre, The Bishop-Phelps-Bollobás theorem for operators, J. Funct. Anal. 254 (11) (2008), [2] M.D. Acosta, J. Becerra-Guerrero, D. García, S.. im and M. Maestre, Bishop-Phelps-Bollobás property for certain spaces of operators, preprint [3] M.D. Acosta, J. Becerra-Guerrero, D. García and M. Maestre, The Bishop-Phelps- Bollobás Theorem for bilinear forms, Trans. Amer. Math. Soc. (to appear). Available at [4] R.M. Aron, B. Cascales and O. ozhushkina, The Bishop-Phelps-Bollobás Theorem and Asplund operators, Proc. Amer. Math. Soc. 139 (10) (2011), [5] R.M. Aron, Y.S. Choi, D. García and M. Maestre, The Bishop-Phelps-Bollobás theorem for L(L 1 (µ), L [0, 1]), Adv. Math. 228 (1) (2011), [6] R.M. Aron, Y.S. Choi, S.. im, H.J. Lee and M. Martín, The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B, preprint Available at

12 12 ACOSTA, BECERRA, CHOI, CIESIELSI, IM, LEE, LOURENÇO, AND MARTÍN [7] E. Bishop and R.R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. (N.S.) 67 (1961), [8] B. Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), [9] B. Cascales, A.J. Guirao and V. adets, A Bishop-Phelps-Bollobás type theorem for uniform algebras, Adv. Math. 240 (2013), [10] M. Chica, V. adets, M. Martín, S. Moreno-Pulido and F. Rambla-Barreno, Bishop-Phelps-Bollobás moduli of a Banach space, preprint Available at [11] Y.S. Choi and S.. im, The Bishop-Phelps-Bollobás theorem for operators from L 1 (µ) to Banach spaces with the Radon-Nikodým property, J. Funct. Anal. 261 (6) (2011), [12] Y.S. Choi, S.. im, H.J. Lee and M. Martín, The Bishop-Phelps-Bollobás theorem for operators on L 1 (µ), preprint Available at [13] N. Dunford and J.T. Schwartz, Linear operators, Volume I, Interscience, New York, [14] J. Johnson and J. Wolfe, Norm attaining operators, Studia Math. 65 (1) (1979), [15] S.. im, The Bishop-Phelps-Bollobás Theorem for operators from c 0 to uniformly convex spaces, Israel J. Math. (to appear). Available at [16] S.. im and H.J. Lee, Uniform convexity and Bishop-Phelps-Bollobás property, Canadian J. Math. (to appear). Available at [17] A.J. Lazar and J. Lindenstrauss, Banach spaces whose duals are L 1 spaces and their representing matrices, Acta Math. 126 (1971), [18] N.J. Nielsen and G.H. Olsen, Complex preduals of L 1 and subspaces of l n (C), Math. Scand. 40 (2) (1977), [19] W. Schachermayer, Norm attaining operators on some classical Banach spaces, Pacific J. Math. 105 (2) (1983), (Acosta, Becerra and Martín) Universidad de Granada, Facultad de Ciencias. Matemático, Granada (Spain) address: dacosta@ugr.es, juliobg@ugr.es, mmartins@ugr.es Departamento de Análisis (Choi) Department of Mathematics, POSTECH, Pohang ( ), Republic of orea address: mathchoi@postech.ac.kr (Ciesielski) Poznań University of Technology, Piotrowo 3A, Poznań, Poland address: maciej.ciesielski@put.poznan.pl (im) School of Mathematics, orea Institute for Advanced Study (IAS), 85 Hoegiro, Dongdaemun-gu, Seoul , Republic of orea address: lineksk@gmail.com (Lee) Department of Mathematics Education, Dongguk University - Seoul, Seoul, Republic of orea address: hanjulee@dongguk.edu (Lourenço) Universidade de Sao Paulo, Instituto de Matemática e Estatística, CP CEP: São Paulo, Brazil address: mllouren@ime.usp.br