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## Studia Mathematica

1997 | 123 | 2 | 117-134
Tytuł artykułu

### Higher-dimensional weak amenability

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Języki publikacji
EN
Abstrakty
EN
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).
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
117-134
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-04-01
poprawiono
1996-10-01
Twórcy
autor
• Department of Mathematics, The University of Newcastle, Newcastle upon Tyne, NE1 7RU England, b.e.johnson@ncl.ac.uk
Bibliografia
• [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.
• [2] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.
• [3] P. C. Curtis, Jr., and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. (2) 40 (1989), 89-104.
• [4] N. Grønbæk, Commutative Banach algebras, module derivations and semigroups, ibid., 137-157.
• [5] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).
• [6] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960.
Typ dokumentu
Bibliografia
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