Extending biholomorphic automorphisms of the open unit ball of a complex Banach space to the bidual

Abstract

Let X be a complex Banach space. Dineen (1986) proved that each biholomorphic automorphism of the open unit ball of X extends to a biholomorphic automorphism of the open unit ball of the bidual X″ of X. As far as we know, the question of the uniqueness of the extension remained unanswered, however. In this paper we show that the uniqueness of the extension occurs if and only if each surjective linear isometry on X can be uniquely extended to a surjective linear isometry on X″. Since this last condition is fulfilled whenever X is a JB*-triple (thanks to the results by Barton-Timoney (1986)), we rediscover Isidro's result (2018) on the uniqueness of the extension in the JB*-triple case.