SMOOTHNESS PROPERTIES OF THE UNIT BALL IN A JB-ASTERISK-TRIPLE

An element a of norm one in a JB-triple A is said to be smooth if the re exists a unique element x in the unit ball A: of the dual A of A a t which a attains its norm, and is said to be Frechet-smooth if, in ad dition, any sequence (x(n)) of elements in A(1) For which (x(n)(a)) c onverges to one necessarily converges in norm to x. The sequence (a(2n +1)) of odd powers of a converges in the weak-topology to a tripotent u(a) in the JBW-envelope A** of A. It is shown that a is smooth if a nd only if u(a) is a minimal tripotent in A* and a is Frechet-smooth if and only if, in addition, u(a) lies in A.