Transitivity of the norm on Banach spaces having a Jordan structure

We study transitivity conditions on the norm of J B*-triples, C*-algebras, J B-algebras, and their preduals. We show that, for the predual X of a JBW* -triple, each one of the following conditions i) and ii) implies that X is a Hilbert space, i) The closed unit ball of X has some extreme point and th e norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results ar e applied to obtain partial affirmative answers to the open problem whether every JB*-triple with transitive norm is a Hilbert space. We extend to arb itrary C*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C*-algebras. Moreov er, we prove that the Calkin algebra has convex transitive norm. We also pr ove that, if X is a J B-algebra, and if either the norm of X is convex tran sitive or X has a predual with convex transitive norm, then X is associativ e. As a consequence, a J B-algebra with almost transitive norm is isomorphi c to the field of real numbers.