Weak amenability of C*-algebras and a theorem of Goldstein

A Banach algebra A is weakly amenable provided that every bounded derivation from A to its dual A is inner. In [H1], the first-named author, building on earlier work of J. W. Bunce and W. L. Paschke [BP], proved that every C -algebra is weakly amenable. We give a simplified and unified proof of this theorem. B. E. Johnson has proved that every bounded Jordan derivation from a C -algebra A to any Banach A -bimodule is a derivation [Jo]. We present a new proof of this theorem. As an application of these results, we give an elementary proof of the following theorem of S. Goldstein [Go]. For each bounded bilinear form V : A \Theta A ! C on a C -algebra A , the following assertions are equivalent: (a) V (a; b) = 0 whenever a; b 2 A are self-adjoint and satisfy ab = 0; (b) there are functionals '; / 2 A for which V (a; b) = '(ab) + /(ba) for all a; b 2 A . Moreover, the functionals in (b) can be chosen to be positive if and only if V (c; c ) 0 for each c 2 A . 1991 Mathe..