Higher-dimensional weak amenability

Bade, Curtis and Dales have introduced the idea of weak amenability. A commutative Banach algebra U is weakly amenable if there are no non-zero continuous derivations from U to U*. We extend this by defining an alternating n-derivation to be an alternating n-linear map from U to U* which is a derivation in each of its variables. Then we say that U is n-dimensionally weakly amenable if there are no non-zero continuous alternating n-derivations on U. Alternating n-derivations are the same as alternating Hochschild cocycles. Since such a cocycle is a coboundary if and only if it is 0, the alternating n-derivations form a subspace of H-n(U, U*). The hereditary properties of n-dimensional weak amenability are studied; for example, if J is a closed ideal in U such that U/J is m-dimensionally weakly amenable and J is n-dimensionally weakly amenable then U is (m + n - 1)-dimensionally weakly amenable. Results of Bade, Curtis and Dales are extended to n-dimensional weak amenability. If U is generated by n elements then it is (n + 1)-dimensionally weakly amenable. If U contains enough regular elements a with \\a(m)\\ = o(m(n/(n+1))) as m --> +/-infinity then U is n-dimensionally weakly amenable. It follows that if U is the algebra lip(alpha)(X) of Lipschitz functions on the metric space X and alpha < n/(n + 1) then U is n-dimensionally weakly amenable. When X is the product of n copies of the circle then U is n-dimensionally weakly amenable if and only if alpha < n/(n + 1).