Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

Abstract. In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties of X are implied by the amenability of K(X). IntroductionAmenability is a cohomological property of Banach algebras which was introduced in [J]. The definition is given below. It may be thought of as being, in some ways, a weak finiteness condition. For example, amenability of C*-algebras is equivalent to nuclearity, see [Haa]. Also, a group algebra, L 1 (G), is amenable if and only if the locally compact group, G, is amenable, see [J], and many theorems valid for finite or compact groups have weaker generalizations to amenable groups but to no larger class. This equivalence is the origin of the term for Banach algebras. However, in some situations amenability is not a finiteness condition. For example, a uniform algebra is amenable if and only if it is self-adjoint, see [Sh], and, for finite dimensional Banach algebras, amenability is equivalent to semisimplicity.The significance of amenability for some classes of Banach algebras suggests the question as to what it means for other Banach algebras. In this GAW partially supported by SERC grant GR-F-74332. denotes the algebra of compact operators on X and F (X) the algebra of approximable operators.) Relevant properties of Banach spaces, such as the approximation property, are now understood better than they were when [J] was written and so we are able to make more progress. We have not yet found such clear characterizations of amenability for the algebras of approximable and compact operators as are known for classes of algebras mentioned in the first paragraph. It does appear though that amenability of F (X) and K(X) may be equivalent to approximation properties for X. One immediate observation is that, since amenable Banach algebras have bounded approximate identities, if the algebra of compact operators on X is amenable, then, by [D, Theorem 2.6], X has the bounded compact approximation property and, if the algebra of approximable operators is amenable, then X has the bounded approximation property. Moreover, results in [G&W] and [Sa] show that, if K(X) is amenable, then X * has the bounded compact approximation property and, if F (X) is amenable, then X * has the bounded approximation property. It follows that, ifAmenability of F (X) is not equivalent to X or X * having the bou...

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Abstract.We give a direct transition from the existence of a bounded right approximate identity in the diagonal ideal for a weighted convolution algebra on a locally compact group to the existence of translation invariant means on an associated weighted L°°-space, thus giving a characterization of amenability for such an algebra. IntroductionA Banach algebra 21 is called amenable if 77 (21, X") = 0 for all Banach 2l-bimodules X. This concept was introduced by B. E. Johnson in [5]. In this paper Johnson showed that L (C7) is amenable if and only if G is amenable, i.e. if and only if there is a (left-)translation invariant mean on L°°(G). He also showed that amenable Banach algebras have bounded approximate identities and that L (C7) is amenable if and only if the augmentation ideal 70 = {/ e LX(G) | JGfi = 0} has a bounded right approximate identity. In the sequel we shall use the abbreviation 'BAP for 'bounded two-sided approximate identity', and 'BRAF and 'BLAF for the one-sided concepts.The importance of approximate identities for amenability was clarified by A. Ya. Khelemskiï and M. V. Sheinberg (see the survey article [6]). In a series of papers they developed a general homology theory for Banach algebras and obtained the following description of amenability. Given a Banach algebra 21, they considered the sequence 0 -► kerzt -Ú 2l®2lOÍ' i 21 -> 0, where ® is the projective tensor product, the Banach algebra Wfp is obtained by reversing the product in 21, the map n is given by n(a ®b) = ab, and the map i is the injection of the kernel of n. With the usual multiplication on 2l®2l0/7, the set ker n becomes a closed left-ideal of 2l2l0/', called the diagonal ideal.

This paper is concerned with two notions of cohomological triviality for Banach algebras, weak amenability and cyclic amenability. The first is defined within Hochschild cohomology and the latter within cyclic cohomology. Our main result is that H\(SF) = H\(sf) x H\(£),