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Weakly Amenable


The notion of weak amenability was first introduced by Bade et al. (1987), who termed a commutative Banach algebra A "weakly amenable" if every continuous derivation from A into a symmetric Banach A-bimodule is zero. But this is equivalent to H^1(A,A^*)=0 ,and one may apply this latter condition as the definition of weak amenability for an arbitrary Banach algebra. So a Banach algebra A is said to be weakly amenable if every bounded derivation from A into its dual A^* is inner (Helemskii 1989).

It is known that every C^*-algebra is weakly amenable (Haagerup 1983).


See also

Amenable, Banach Algebra

This entry contributed by Mohammad Sal Moslehian

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References

Bade, W. G.; Curtis, P. C.; and Dales, H. G. "Amenability and Weak Amenability for Beurling and Lipschitz Algebras." Proc. London Math. Soc. 55, 359-377, 1987.Haagerup, U. "All Nuclear C^*-Algebras Are Amenable." Invent. Math 74, 305-319, 1983.Helemskii, A. Ya. The Homology of Banach and Topological Algebras. Dordrecht, Netherlands: Kluwer, 1989.

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Weakly Amenable

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Moslehian, Mohammad Sal. "Weakly Amenable." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/WeaklyAmenable.html

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