Trivialization of $𝒞\left(X\right)$-algebras with strongly self-absorbing fibres

Dadarlat, Marius; Winter, Wilhelm

Bulletin de la Société Mathématique de France, Tome 136 (2008) no. 4, pp. 575-606

Suppose $A$ is a separable unital $𝒞\left(X\right)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_1$-injective $C^{*}$-algebra $𝒟$. We show that $A$ and $𝒞\left(X\right)\otimes 𝒟$ are isomorphic as $𝒞\left(X\right)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Soit $A$ une $𝒞\left(X\right)$-algèbre séparable unital dont chaque fibre est isomorphe à une même $C^{*}$-algèbre $𝒟$ $K_1$-injective et fortement auto-absorbante. Nous montrons que si l’espace compact et Hausdorff $X$ est de dimension finie, alors $A$ et $𝒞\left(X\right)\otimes 𝒟$ sont isomorphes en tant que $𝒞\left(X\right)$-algèbres. Ce resultat est connu pour ne pas s’étendre au cas des espaces de dimension infinie.

Classification : 46L05, 47L40
Keywords: strongly self-absorbing $C^*$-algebra, asymptotic unitary equivalence, continuous field of $C^{*}$-algebras
Mots-clés : $C^{*}$-algèbre fortement auto-absorbante, équivalence unitaire asymptotique, champ continu de $C^{*}$-algèbres

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