A Banach algebra 51 is amenable if each bounded derivation D from 51 to a dual Banach 5J-bimodule X* is inner. That is, if a bounded linear operator DM -* X* satisfies D(ab) = a • Db + (Da) • b, for each a, b e S&, then there exists x* e X* such that Da = ax* -x*-a. The module actions of 51 on X*, denoted by a x * and x*-a, are assumed to be the duals of appropriate bounded 5I-bimodule actions on X. If denotes the vector space of all bounded derivations from 51 to X* and denotes the space of inner derivations, then 51 amenable means that H\%X*) = Z 1 (M t X*)/B 1 ($L,X*), the first cohomology group of 51 with coefficients in X*, vanishes for each dual Banach 5l-module X*.Since Barry Johnson's original paper [17] where the notion of 'algebra amenability' was introduced, the literature in the West has been concerned mainly with applications to C*-algebras. However, these ideas have been explored by Soviet mathematicians, A. Ya Khelemskii, M. V. Sheinberg and others, with emphasis on applications to homological algebra. A recent translation of a survey article by Khelemskii [19] outlines the major results obtained. Perhaps the most striking of these results are two characterizations of amenability. Namely, a Banach algebra is amenable if and only if it has a bounded approximate identity, and one (and hence both) of the following hold:(i) the short exact sequence 0 -»• 51* -• (51 (g) 51)* -> K* -* 0 splits as an 51bimodule sequence;(ii) for any essential 5I-bimodule X, any admissible short exact sequence 0 -> ; i f * -> y -» Z -» 0 o f 5I-bimodules splits.Two important corollaries of these homological algebra results are first that a closed ideal / in an amenable Banach algebra 51 has a bounded approximate identity if and only if/ 1 = {Xe%*:X(J) = 0} is a topological direct summand in 51*, that is, / i s weakly complemented. Secondly if 51 is a uniform algebra on a compact space Q, then 51 = C(Q) if (and only if) 51 is amenable. The first of these results is due to Khelemskii and Sheinberg and the second to Sheinberg (see [19]).One of the purposes of this paper is to present new proofs of these results from homological algebra which avoid the substantial homological algebra machinery used in the original derivation. In particular no appeal will be made to the existence and properties of the Ext and Tor functors. Where possible, explicit constructions will be given for the splitting morphisms. Certainly some techniques from homological algebra are necessary since the conclusions involve the splitting of appropriate short exact sequences. However, the techniques we use to obtain this splitting are variations
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