# Higher-dimensional weak amenability

Studia Mathematica (1997)

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

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topJohnson, 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

top- [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] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. Zbl0344.46071
- [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] N. Grønbæk, Commutative Banach algebras, module derivations and semigroups, ibid., 137-157. Zbl0632.46046
- [5] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). Zbl0256.18014
- [6] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960. Zbl0095.09702

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