DERIVATIONS INTO ITERATED DUALS OF BANACH-ALGEBRAS

We introduce two new notions of amenability for a Banach algebra U. Th e algebra U is n-weakly amenable (for n is an element of N) if the fir st continuous cohomology group of U with coefficients in the nth dual space U-(n) is zero; i.e., H-1(U, U-(n)) = {0}. Further, U is permanen tly weakly amenable if U is n-weakly amenable for each n is an element of N. We begin by examining the relations between m-weak amenability and m-weak amenability for distinct m, n is an element of N. We then e xamine when Banach algebras in various classes are n-weakly amenable; we study group algebras, C-algebras, Banach function algebras, and al gebras of operators. Our results are summarized and some open question s are raised in the final section.