Bishop-Phelps-Bollob\'as property for positive operators when the domain is $C_0(L) $

Recently it was introduced the so-called Bishop-Phelps-Bollob{\'a}s property for positive operators between Banach lattices. In this paper we prove that the pair $(C_0(L), Y) $ has the Bishop-Phelps--Bollob{\'a}s property for positive operators, for any locally compact Hausdorff topological space $L$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. In case that the space $C_0(L)$ is separable, the same statement holds for any uniformly monotone Banach lattice $Y .$ We also show the following partial converse of the main result. In case that $Y$ is a strictly monotone Banach lattice, $L$ is a locally compact Hausdorff topological space that contains at least two elements and the pair $(C_0(L), Y )$ has the Bishop-Phelps--Bollob{\'a}s property for positive operators then $Y$ is uniformly monotone.


Introduction
Bishop-Phelps theorem [7] states that every continuous linear functional on a Banach space can be approximated (in norm) by norm attaining functionals. Bollobás proved a "quantitative version" of that result [8]. In order to state such result, we denote by B X , S X and X * the closed unit ball, the unit sphere and the topological dual of a Banach space X, respectively. If X and Y are both real or both complex Banach spaces, L(X, Y ) denotes the space of (bounded linear) operators from X to Y, endowed with its usual operator norm.
Recently in [4], the authors introduced a version of Bishop-Phelps-Bollobás property for positive operators between two Banach lattices. Let us mention that the only difference between this property and the previous one is that in the new property the operators appearing in Definition 1.1 are positive. In the same paper it is shown that the pairs (c 0 , L 1 (ν)) and (L ∞ (µ), L 1 (ν)) have the Bishop-Phelps-Bollobás property for positive operators for any positive measures µ and ν (see [4,Theorems 1.7 and 1.6]). The paper [5] contains some extensions of those results. More precisely, it is proved that the pair (c 0 , Y ) has the Bishop-Phelps-Bollobás property for positive operators whenever Y is a uniformly monotone Banach lattice (see [5,Corollary 3.3]). It is also shown that the pair (L ∞ (µ), Y ) has the Bishop-Phelps-Bollobás property for positive operators for any positive measure µ if Y is a uniformly monotone Banach lattice with a weak unit (see [5,Corollary 2.6] ).
The goal of this paper is to obtain a far reaching extension of those results. To be precise, we prove that the pair (C 0 (L), Y ) has the Bishop-Phelps-Bollobás property for positive operators, for any locally compact Hausdorff topological space L, whenever Y is a uniformly monotone Banach lattice with a weak unit. If C 0 (L) is separable, the same statement holds for any uniformly monotone Banach lattice Y. Further we show that these results are optimal in case that Y is strictly monotone. That is, if Y is a strictly monotone Banach lattice and L is a locally compact Hausdorff topological space that contains at least two elements, if the pair (C 0 (L), Y ) has the Bishop-Phelps-Bollobás property for positive operators then Y is uniformly monotone.

The results
. Let X and Y be Banach lattices. The pair (X, Y ) is said to have the Bishop-Phelps-Bollobás property for positive operators if for every 0 < ε < 1 there exists 0 < η(ε) < ε such that for every S ∈ S L(X,Y ) , such that S ≥ 0, if x 0 ∈ S X satisfies S(x 0 ) > 1−η(ε), then there exist an element u 0 ∈ S X and a positive operator T ∈ S L(X,Y ) satisfying the following conditions T (u 0 ) = 1, u 0 − x 0 < ε and T − S < ε.
In order to refine the result, we recall the following version of [2, Definition 1.3], which was introduced in [4]. The subspace M is said to have the Bishop-Phelps-Bollobás property for positive operators if for every 0 < ε < 1 there exists 0 < η(ε) < ε such that for every S ∈ S M , such that S ≥ 0, if x 0 ∈ S X satisfies S(x 0 ) > 1 − η(ε), then there exist an element u 0 ∈ S X and a positive operator T ∈ S M satisfying the following conditions We will use the notions of strictly monotone and uniformly monotone Banach lattice, that we recall now. Definition 2.3. Let X be a real Banach lattice. X is strictly monotone if for any x, y ∈ X + such that x ≤ y and x = y, it is satisfied that x < y . The Banach lattice X is uniformly monotone, if for every 0 < ε < 1, there is 0 < δ ≤ ε satisfying the following property x, y ∈ X + , x = 1, x + y ≤ 1 + δ ⇒ y ≤ ε.
The following characterization of uniform monotonicity can be found in [5,Proposition 4.2]. Proposition 2.4. Let Y be a Banach lattice. The following conditions are equivalent.
. Let Y be a uniformly monotone Banach function space and 0 < ε < 1.
Assume that f 1 and f 2 are positive elements in Y such that where δ is the function satisfying the definition of uniform monotonicity for Y. Then there are two positive functions h 1 and h 2 in Y with disjoint supports satisfying that Proposition 2.6. Let L be a locally compact Hausdorff space and Y be a Banach lattice. Assume that S 1 and S 2 are positive operators in L(C 0 (L), Y ) and g 1 and g 2 are positive elements in C 0 (L). Then the operator U : Proof.
Lemma 2.7. Let L be a locally compact Hausdorff space and Y be a uniformly monotone Banach lattice. Let δ be the function satisfying the definition of uniform monotonicity for the Banach lattice Y and 0 < η < 1. Assume that f 0 ∈ S C 0 (L) and S ∈ S L(C 0 (L),Y ) , S ≥ 0 and .
Define the sets A 1 , A 2 , B 1 and B 2 by There are positive functions g 1 and g 2 in B C 0 (L) satisfying the following assertions a) g 1|A 1 = 1 and g 1|L\( Proof. We can define the functions g 1 and g 2 as follows It is immediate to check that g 1 and g 2 are positive functions in B C 0 (L) satisfying the conditions stated in a), b), c) and d). If h ∈ B C 0 (L) satisfies h |A 1 ∪B 1 = 0 we clearly have that |f 0 | + η 2 |h| ∈ S C 0 (L) . By using that S is a positive operator and the assumption we have that By using the uniform monotonicity of Y we obtain that S( η 2 |h|) ≤ η 2 and so S(h) ≤ S(|h|) ≤ 2η.
Finally notice that (g 1 + g 2 − 1)(A 1 ∪ B 1 ) = {0} and g 1 + g 2 − 1 ∞ ≤ 1. So from the previous part we conclude for every f ∈ C 0 (L), Theorem 2.8. The pair (C 0 (L), Y ) has the Bishop-Phelps-Bollobás property for positive operators, for any locally compact Hausdorff topological space L, whenever Y is a uniformly monotone Banach function space. The function η satisfying Definition 2.1 depends only on the modulus of uniform monotonicity of Y .
Proof. Assume that Y is a Banach function space on a measure space (Ω, µ). Let 0 < ε < 1 and δ be the function satisfying the definition of uniform monotonicity for the Banach function space Y. Choose a real number η such that 0 < η = η(ε) < ε 12 and satisfying also 1 1 + δ( ε 18 ) Assume that f 0 ∈ S C 0 (L) , S ∈ S L(C 0 (L),Y ) and S is a positive operator such that .
Hence we can apply Lemma 2.7 and so there are positive functions g 1 and g 2 in B C 0 (L) satisfying all the conditions stated in Lemma 2.7, therefore .
Since S is a positive operator and S(g 1 ) + S(g 2 ) ≤ 1, in view of (2.1) we can apply Lemma 2.5. Hence there are two positive functions h 1 and h 2 in Y satisfying the following conditions (by (2.2)).
Now we define the operator U : C 0 (L) −→ Y as follows Since Y is a Banach function space and S ∈ L(C 0 (L), Y ), U is well defined and belongs to L(C 0 (L), Y ). The operator U is positive since g 1 and g 2 are positive elements in C 0 (L) and S is a positive operator. For any f ∈ B C 0 (L) we have that .4), (2.5) and item e) of Lemma (2.7)). Hence Finally we define T = U U . Since U is a positive operator, T is also positive. Of course T ∈ S L(C 0 (L),Y ) and also satisfies by (2.7)).
The function f 1 given by belongs to S C 0 (L) and satisfies that We clearly have that U(f 1 ) = S(g 2 )χ sup h 2 − S(g 1 )χ sup h 1 . Since S ≥ 0 and g 1 and g 2 are positive functions, in view of Proposition 2.6 we have that Since h 1 and h 2 have disjoint supports, for each x ∈ Ω we obtain that Since Y is a Banach function space we conclude that By (2.9) and (2.8), since T attains its norm at f 1 , the proof is finished Now we will improve the statement in Theorem 2.8. For that purpose we use operators in an operator ideal, a notion that we recall below. Definition 2.9 ([11, Definition 9.1]). An operator ideal A is a subclass of the class L of all continuous linear operators between Banach spaces such that for all Banach spaces X and Y its components Y ) is a linear subspace of L(X, Y ) which contains the finite rank operators.
(2) The ideal property: If S ∈ A(X 0 , Y 0 ), R ∈ L(X, X 0 ) and T ∈ L(Y 0 , Y ), then the composition T SR is in A(X, Y ).
We used in the previous proof that for a Banach function space Y on a measure space (Ω, µ), for any measurable set A ⊂ Ω, the operator h → hχ A is linear and bounded. So in case that I is some operator ideal and the operator S in the proof of Theorem 2.8 belongs to I(C 0 (L), Y ), then the operator U also belongs to I(C 0 (L), Y ). Hence we obtain the following result.  , where the same result was proved in case that the domain space is a L ∞ or c 0 space. Our purpose now is to obtain a version of Theorem 2.8 for some abstract Banach lattices. In order to get this result, we use that every uniformly monotone Banach lattice is order continuous (see [6,Theorem 21,p. 371] and [12, Proposition 1.a.8]). Also any order continuous Banach lattice with a weak unit is order isometric to a Banach function space (see [12,Theorem 1.b.14]). From Theorem 2.8 and the previous argument we deduce the following result.
Corollary 2.11. The pair (C 0 (L), Y ) has the Bishop-Phelps-Bollobás property for positive operators, for any locally compact Hausdorff topological space L whenever Y is a uniformly monotone Banach lattice with a weak unit. Moreover, the function η satisfying Definition 2.1 depends only on the modulus of uniform monotonicity of Y .
By using the previous result and the same argument of [5,Corollary 3.3] we obtain the next statement: Corollary 2.12. For any locally compact Hausdorff topological space L such that C 0 (L) is separable, The pair (C 0 (L), Y ) has the Bishop-Phelps-Bollobás property for positive operators whenever Y is a uniformly monotone Banach lattice. Moreover, the function η satisfying Definition 2.1 depends only on the modulus of uniform monotonicity of Y .
Our intention now is to provide a class of Banach lattices for which the previous result is in fact a characterization. The proof of the next result is a refinement of the arguments used in [5,Proposition 4.3], where it is assumed that the domain has a non-trivial M-summand.
Proposition 2.13. Let Y be a strictly monotone Banach lattice and L a locally compact Hausdorff topological space that contains at least two elements. If the pair (C 0 (L), Y ) has the BPBp for positive operators then Y is uniformly monotone.
Proof. It suffices to show that Y satisfies condition 2) in Proposition 2.4 in case that the pair (C 0 (L), Y ) has the BPBp for positive operators.
Let 0 < ε < 1. Let us take elements u and v in Y such that where η is the function satisfying the definition of BPBp for positive operators for the pair (C 0 (L), Y ).
Since L contains at least two elements there are t i ∈ L and f i ∈ S C 0 (L) for i = 1, 2 such that Define the operator S from C 0 (L) to Y given by Since v − u and u are positive elements in Y, S is a positive operator from C 0 (L) to Y . By using that v − u and u are positive elements in Y, for any element f ∈ B C 0 (L) we have that Since we also have that f 1 + f 2 ∈ S C 0 (L) and S(f 1 + f 2 ) = v ∈ S Y , we obtain that S ∈ S L(C 0 (L),Y ) . Notice also that S(f 1 ) = v − u > 1 − η(ε). By using that the pair (C 0 (L), Y ) has the BPBp for positive operators and [5, Remark 2.2], there exist a positive operator T ∈ S L(C 0 (L),Y ) and a positive element g ∈ S C 0 (L) satisfying T (g) = 1, T − S < ε and g − f 1 < ε.
In view of Proposition 2.4 we proved that Y is uniformly monotone.
Taking into account the previous result and Corollary 2.11 we obtain the following characterization.
Corollary 2.14. Let Y be a strictly monotone Banach lattice and L a locally compact Hausdorff topological space that contains at least two elements. If the pair (C 0 (L), Y ) has the BPBp for positive operators then Y is uniformly monotone. In case that Y has a weak unit the converse is also true.