Trivialization of 𝒞(X)-algebras with strongly self-absorbing fibres
[Trivialisation de 𝒞(X)-algèbres à fibres fortement auto-absorbantes]
Bulletin de la Société Mathématique de France, Tome 136 (2008) no. 4, pp. 575-606.

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)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)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.

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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     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},
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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://archive.numdam.org/articles/10.24033/bsmf.2567/

[1] B. Blackadar & E. Kirchberg - « Generalized inductive limits of finite-dimensional C * -algebras », Math. Ann. 307 (1997), p. 343-380. | MR | Zbl

[2] E. Blanchard & E. Kirchberg - « Global Glimm halving for C * -bundles », J. Operator Theory 52 (2004), p. 385-420. | MR | Zbl

[3] M. Dadarlat - « Continuous fields of C * -algebras over finite dimensional spaces », preprint, arXiv:math.OA/0611405, 2006. | MR | Zbl

[4] -, « Fiberwise KK-equivalence of continuous fields of C * -algebras », preprint, arXiv:math.OA/0611408, 2006.

[5] M. Dadarlat & W. Winter - « On the KK-theory of strongly self-absorbing C * -algebras », preprint, arXiv:0704.0583, to appear in Math. Scand., 2007. | MR | Zbl

[6] J. Dixmier & A. Douady - « Champs continus d’espaces hilbertiens et de C * -algèbres », Bull. Soc. Math. France 91 (1963), p. 227-284. | Numdam | MR | Zbl

[7] I. Hirshberg, M. Rørdam & W. Winter - « 𝒞 0 (X)-algebras, stability and strongly self-absorbing C * -algebras », Math. Ann. 339 (2007), p. 695-732. | MR | Zbl

[8] W. Hurewicz & H. Wallman - Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, 1941. | MR | Zbl

[9] G. G. Kasparov - « Equivariant KK-theory and the Novikov conjecture », Invent. Math. 91 (1988), p. 147-201. | MR | Zbl

[10] E. Kirchberg - « Central sequences in C * -algebras and strongly purely infinite algebras », in Operator Algebras: The Abel Symposium 2004, Abel Symp., vol. 1, Springer, 2006, p. 175-231. | MR | Zbl

[11] M. Rørdam - Classification of nuclear C * -algebras, Encyclopaedia Math. Sci, vol. 126, Springer, 2002. | MR | Zbl

[12] A. S. Toms & W. Winter - « Strongly self-absorbing C * -algebras », Trans. Amer. Math. Soc. 359 (2007), p. 3999-4029. | MR | Zbl

[13] W. Winter - « Localizing the Elliott conjecture at strongly self-absorbing C * -algebras », preprint, arXiv:0708.0283, 2007. | MR

[14] -, « Simple C * -algebras with locally finite decomposition rank », J. Funct. Anal. 243 (2007), p. 394-425. | MR | Zbl

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