Trivialization of $\mathcal {C}(X)$-algebras with strongly self-absorbing fibres

Dadarlat, Marius; Winter, Wilhelm

doi:10.24033/bsmf.2567

Trivialization of

𝒞 (X)

-algebras with strongly self-absorbing fibres
[Trivialisation de

𝒞 (X)

-algèbres à fibres fortement auto-absorbantes]

Dadarlat, Marius ; Winter, Wilhelm

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

Résumé
Abstract

Soit $A$ une $𝒞 (X)$ -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 $𝒞 (X) \otimes 𝒟$ sont isomorphes en tant que $𝒞 (X)$ -algèbres. Ce resultat est connu pour ne pas s’étendre au cas des espaces de dimension infinie.

Suppose $A$ is a separable unital $𝒞 (X)$ -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 $𝒞 (X) \otimes 𝒟$ are isomorphic as $𝒞 (X)$ -algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

MR Zbl

DOI : 10.24033/bsmf.2567

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

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     author = {Dadarlat, Marius and Winter, Wilhelm},
     title = {Trivialization of $\mathcal {C}(X)$-algebras with strongly self-absorbing fibres},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {575--606},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {136},
     number = {4},
     year = {2008},
     doi = {10.24033/bsmf.2567},
     mrnumber = {2443037},
     zbl = {1170.46051},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/bsmf.2567/}
}

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Dadarlat, Marius; Winter, Wilhelm. Trivialization of $\mathcal {C}(X)$-algebras with strongly self-absorbing fibres. Bulletin de la Société Mathématique de France, Tome 136 (2008) no. 4, pp. 575-606. doi : 10.24033/bsmf.2567. http://www.numdam.org/articles/10.24033/bsmf.2567/

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