Higher-dimensional weak amenability
Volume 123 / 1997
Abstract
Bade, Curtis and Dales have introduced the idea of weak amenability. A commutative Banach algebra A is weakly amenable if there are no non-zero continuous derivations from A to A*. We extend this by defining an alternating n-derivation to be an alternating n-linear map from A to A* which is a derivation in each of its variables. Then we say that A is n-dimensionally weakly amenable if there are no non-zero continuous alternating n-derivations on A. 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 $H^n(A,A*)$. The hereditary properties of n-dimensional weak amenability are studied; for example, if J is a closed ideal in A such that A/J is m-dimensionally weakly amenable and J is n-dimensionally weakly amenable then A is (m+n-1)-dimensionally weakly amenable. Results of Bade, Curtis and Dales are extended to n-dimensional weak amenability. If A is generated by n elements then it is (n+1)-dimensionally weakly amenable. If A contains enough regular elements a with $∥a^m∥ = o(m^{n/(n+1)})$ as m → ±∞ then A is n-dimensionally weakly amenable. It follows that if A is the algebra $lip_α(X)$ of Lipschitz functions on the metric space X and α < n/(n+1) then A is n-dimensionally weakly amenable. When X is the product of n copies of the circle then A is n-dimensionally weakly amenable if and only if α < n/(n+1).
