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Commutativité de certaines algèbres de Banach. (Commutativity of certain Banach algebras). (French) Zbl 0583.46048
In the paper we solve five questions remained open after the survey of V. A. Belfi and R. S. Doran [Enseign. Math., II. Ser. 26, 103-130 (1980; Zbl 0441.46039)] and give a partial answer to a sixth one.
Among the problems considered there is the question of wether the condition (c): \(\| x\cdot y\| \leq \alpha \cdot \| y\cdot x\|\) for an \(\alpha >0\) and every x,y, implies the commutativity in the non unitary case. It is shown that this is true when the algebras possess an approximate unit (bounded or not). As corollaries we get the results of Lepage and of Baker and Pym. We have now obtained (O. H. Cheikh and the author) that the answer is false in general [Preprint, E. N. S. Takaddoum, B.P. 5118, Rabat, Maroc].
Another question considered in the paper is whether a normed algebra (unitary or not) is commutative when subjected to the condition: \(E.x^ 2=E.x\), for every x. It is shown that the answer is positive when it is complete and admits an approximate unit and also when it is normed but unitary. In the Banach case, the question is now completely solved [J. Esterle and the author, Proc. Am. Math. Soc. 96, 91-94 (1986)] by the following
Theorem: Let E be a Banach algebra: Then \(E.x=E.x^ 2\), for every x, if, and only if, \(E=F\oplus R\) where R is the radical of E, \(E.R=\{0\}\) and F is isomorphic to \({\mathbb{C}}^ n\) for some integer n.

46J40 Structure and classification of commutative topological algebras
46H05 General theory of topological algebras
46H20 Structure, classification of topological algebras