Produits finis de commutateurs dans les C*-algèbres
Annales de l'Institut Fourier, Tome 34 (1984) no. 4, pp. 169-202.

Soient A une C*-algèbre approximativement finie simple avec unité, GL1(A) le groupe des inversibles et U1(A) le groupe des unitaires de A. Nous avons défini dans un précédent travail un homomorphisme ΔT, appelé déterminant universel de A, de GL1(A) sur un groupe abélien associé à A. Nous montrons ici que, pour qu’un élément x dans GL1(A) ou dans U1(A) soit produit d’un nombre fini de commutateurs, il (faut et il) suffit que x∈Ker(ΔT). Ceci permet en particulier d’identifier le noyau de la projection canonique K1(A)→K1top(A). On établit aussi des résultats concernant les C*-algèbres stables et les C*-algèbres infinies simples avec unité.

Let A be a simple approximately finite dimensional C*-algebra with unit, let GL1(A) be the group of invertible elements and let U1(A) be that of unitaries in A. We have defined in a previous work a universal determinant ΔT of A, which is a homomorphism from GL1(A) onto an abelian group associated to A. We show here that in element x in GL1(A) or in U1(A) is a product of finitely many commutators if (and only if) x∈Ker(ΔT). In particular, one may thus characterize the kernel of the canonical projection K1(A)→K1top(A). Other results are established about stable C*-algebras and infinite simple C*-algebras with unit.

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Harpe, Pierre De La; Skandalis, Georges. Produits finis de commutateurs dans les $C^*$-algèbres. Annales de l'Institut Fourier, Tome 34 (1984) no. 4, pp. 169-202. doi : 10.5802/aif.993. http://www.numdam.org/articles/10.5802/aif.993/

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