A WEDDERBURN THEOREM FOR NON-ASSOCIATIVE COMPLETE NORMED ALGEBRAS ANTONIO FERNANDEZ LOPEZ AND ANGEL RODRIGUEZ PALACIOS Introduction A well-known theorem of Wedderburn states that a finite-dimensional semisimple associative algebra is the direct sum of ideals which are simple algebras. We recall that Wedderburn's theorem has been extended by Zorn [27] to the alternative case, by Albert and Jacobson [17] to the Jordan case, and finally by Albert (see [1,17]) to the general non-associative case, all these extensions being in a finite-dimensional frame. In terms of lattices, one may say that for an algebra as above, the greatest element of the lattice R of its ideals is the join (least upper bound) of the set of the atoms of R. It seems natural to look for conditions under which the (complete) lattice of the closed ideals of a non-associative normed algebra has the above property. In other words, when is a non-associative normed algebra the closure of the sum of its minimal closed ideals? Since even for semisimple Banach (associative) algebras, the answer to this question is negative in general, additional hypotheses of algebraic- topological or even geometric content must be required. In this direction there are various extensions of the Wedderburn theorem mentioned above to associative normed algebras. We can cite, for example, the paper of Ambrose [2] on .//""-algebras and other works dealing with dual algebras [19,20] and annihilator algebras [6]. Finally, we must cite some works dealing with certain non-associative extensions of the results of Ambrose (see [4,28,30,9]), Kaplansky [7] and Bonsall and Goldie [14,10]. It is easy to see that, if a non-associative normed algebra A with zero annihilator is the closure of the direct sum of its minimal closed ideals (in short, A is decomposable), then A is necessarily semiprime, and for every proper closed ideal / of A there exists a non-zero closed ideal / such that IJ = 0 = JI. (In short, A is a generalized annihilator non-associative normed algebra.) In the reverse direction, Civin and Yood [8] have proved that every semisimple generalized annihilator (associative) Banach algebra is the closure of the direct sum of its minimal closed ideals, and that these are topologically simple Banach algebras. The assumption of semisimplicity in the theorem of Civin and Yood is almost essential. For if there exists a non-semisimple decomposable Banach algebra with zero annihilator, then one can easily obtain a topologically simple radical Banach algebra, thus answering one of the most famous unsolved problems in the theory of Banach algebras. A non-associative extension of the Civin and Yood theorem requires a suitable notion of semisimplicity for infinite-dimensional non-associative algebras. A similar situation appears in relation with the non-associative extension of Johnson's theorem Received 18 December 1984; revised 18 July 1985. 1980 Mathematics Subject Classification 46H20. J. London Math. Soc. (2) 33 (1986) 328-338