# Higher-dimensional weak amenability

Studia Mathematica (1997)

• Volume: 123, Issue: 2, page 117-134
• ISSN: 0039-3223

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## Abstract

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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 ${H}^{n}\left(A,A*\right)$. 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 $\parallel {a}^{m}\parallel =o\left({m}^{n/\left(n+1\right)}\right)$ as m → ±∞ then A is n-dimensionally weakly amenable. It follows that if A is the algebra $li{p}_{\alpha }\left(X\right)$ 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).

## How to cite

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Johnson, B.. "Higher-dimensional weak amenability." Studia Mathematica 123.2 (1997): 117-134. <http://eudml.org/doc/216382>.

@article{Johnson1997,
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).},
author = {Johnson, B.},
journal = {Studia Mathematica},
keywords = {higher-dimensional weak amenability; alternating cocycle; commutative Banach algebra},
language = {eng},
number = {2},
pages = {117-134},
title = {Higher-dimensional weak amenability},
url = {http://eudml.org/doc/216382},
volume = {123},
year = {1997},
}

TY - JOUR
AU - Johnson, B.
TI - Higher-dimensional weak amenability
JO - Studia Mathematica
PY - 1997
VL - 123
IS - 2
SP - 117
EP - 134
AB - 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).
LA - eng
KW - higher-dimensional weak amenability; alternating cocycle; commutative Banach algebra
UR - http://eudml.org/doc/216382
ER -

## References

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1. [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. Zbl0634.46042
2. [2] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. Zbl0344.46071
3. [3] P. C. Curtis, Jr., and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. (2) 40 (1989), 89-104. Zbl0698.46043
4. [4] N. Grønbæk, Commutative Banach algebras, module derivations and semigroups, ibid., 137-157. Zbl0632.46046
5. [5] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). Zbl0256.18014
6. [6] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960. Zbl0095.09702

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