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Some stability properties for the
Bishop–Phelps–Bollobás property for Lipschitz maps

We study the stability behavior of the Bishop–Phelps–Bollobás property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop–Phelps–Bollobás property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very close to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property from scalar functions to some vector-valued maps, getting some positive results related to the notions of Γ -flat operators and ACK structure. We get sharper results for the case of Lipschitz compact maps. The behavior of the property with respect to absolute sums of the target space is also studied. We also get results similar to the above for the density of strongly norm attaining Lipschitz maps and of Lipschitz compact maps.