In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space Y satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × Y is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space C(K) of continuous functions on a compact Hausdorff topological space K and the space K(H) of compact operators on a Hilbert space H. On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × L 1 (μ) fails for any infinite-dimensional L 1 (μ), a result that was known only when L 1 (μ) = 1 .

We prove that every JBW * -triple M with rank one or rank bigger than or equal to three satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of M onto the unit sphere of another Banach space Y extends to a surjective real linear isometry from M onto Y . We also show that the same conclusion holds if M is not a JBW * -triple factor, or more generally, if the atomic part of M * * is not a rank two Cartan factor.

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.

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