Weak Amenability of C -algebras and a Theorem of Goldstein
نویسنده
چکیده
A Banach algebra A is weakly amenable provided that every bounded derivation from A to its dual A is inner. In H1], the rst-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 simpliied and uniied 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 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 .
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