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 . 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 as m → ±∞ then A is n-dimensionally weakly amenable. It follows that if A is the algebra 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).
@article{bwmeta1.element.bwnjournal-article-smv123i2p117bwm,
author = {B. Johnson},
title = {Higher-dimensional weak amenability},
journal = {Studia Mathematica},
volume = {122},
year = {1997},
pages = {117-134},
zbl = {0888.46033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv123i2p117bwm}
}
Johnson, B. Higher-dimensional weak amenability. Studia Mathematica, Tome 122 (1997) pp. 117-134. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv123i2p117bwm/
[00000] [1] W. G. Bade, P. C. Curtis, Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), 359-377. | Zbl 0634.46042
[00001] [2] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. | Zbl 0344.46071
[00002] [3] P. C. Curtis, Jr., and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. (2) 40 (1989), 89-104. | Zbl 0698.46043
[00003] [4] N. Grønbæk, Commutative Banach algebras, module derivations and semigroups, ibid., 137-157. | Zbl 0632.46046
[00004] [5] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). | Zbl 0256.18014
[00005] [6] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960. | Zbl 0095.09702