On admissible topologies in JB*‐triples

Given a complex JB*‐triple X , we define and study admissible topologies on X , i.e., locally convex topologies τ on X coarser than the norm topology, invariant under the group of surjective linear isometries of X , and such that the triple product is jointly τ − τ ‐continuous on bounded subsets of X . As a consequence of the joint τ − τ ‐continuity of the triple product, all holomorphic automorphisms g ∈ Aut ( B ) of the open unit ball B ⊂ X are homeomorphisms of ( B , τ | B ) and the natural action( a , z ) ↦ g a ( z )is jointly τ − τ ‐continuous on B × B .