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Recently it was introduced the so-called BishopÃ?ÃÂ¢??PhelpsÃ?ÃÂ¢??Bollobas property for positive operators between Banach lattices. In this paper we prove that the pair (C_0(L), Y ) has the BishopÃ?ÃÂ¢??PhelpsÃ?ÃÂ¢??Bollobas 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 BishopÃ?ÃÂ¢??PhelpsÃ?ÃÂ¢??Bollobas property for positive operators then Y is uniformly monotone.
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