Generalized inductive limits of finite-dimensional C * -algebras

Blackadar, Bruce; Kirchberg, Eberhard

doi:10.1007/s002080050039

Cited by 167 publications

(301 citation statements)

References 8 publications

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“…the remark after Definition 2.4.1 of [1]). Using (ii), (iii) and the fact that the u i commute with C(X)⊗1 D , one can even assume thatψ…”

Section: Proving the Main Result: The Second Approachmentioning

confidence: 99%

“…* -algebras by (1), and hence it is D-stable by [12,Thm. 4.3], since D is a separable unital strongly self-absorbing…”

Section: Proving the Main Result: The First Approachmentioning

confidence: 99%

“…maps ψ : A → C(X) ⊗ D and ϕ : C(X) ⊗ D → A which implement an asymptotic intertwining (in the sense of [1]) of A and C(X)⊗D as C(X)-algebras (Proposition 4.1).…”

Section: Proving the Main Result: The Second Approachmentioning

confidence: 99%

“…-Let n ∈ N, X ⊂ [0, 1] n a closed subset and A a unital C(X)-algebra. Let D be another unital C * -algebra.…”

mentioning

confidence: 99%

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

Dǎdǎrlat¹,

Winter²

2008

Bul. Soc. Math. France

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Abstract. -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. Résumé (Trivialisation de C(X)-algèbres à fibres fortement auto-absorbantes)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.

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“…the remark after Definition 2.4.1 of [1]). Using (ii), (iii) and the fact that the u i commute with C(X)⊗1 D , one can even assume thatψ…”

Section: Proving the Main Result: The Second Approachmentioning

confidence: 99%

“…* -algebras by (1), and hence it is D-stable by [12,Thm. 4.3], since D is a separable unital strongly self-absorbing…”

Section: Proving the Main Result: The First Approachmentioning

confidence: 99%

“…maps ψ : A → C(X) ⊗ D and ϕ : C(X) ⊗ D → A which implement an asymptotic intertwining (in the sense of [1]) of A and C(X)⊗D as C(X)-algebras (Proposition 4.1).…”

Section: Proving the Main Result: The Second Approachmentioning

confidence: 99%

“…-Let n ∈ N, X ⊂ [0, 1] n a closed subset and A a unital C(X)-algebra. Let D be another unital C * -algebra.…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Dǎdǎrlat¹,

Winter²

2008

Bul. Soc. Math. France

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“…The content of this section is based on a idea of Lin [28]. Lin assumes that A is nuclear and bases his arguments on a result of [3] according to which A is an inductive limit of residually finite dimensional C*-algebras. We show that the nuclearity assumption is not needed and make no use of residually finite dimensional C*-algebras.…”

Section: Finite Dimensional Approximations Of the Dimension Mapmentioning

confidence: 99%

Morphisms of Simple Tracially Af Algebras

Dǎdǎrlat

2004

Int. J. Math.

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Abstract. Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear * -homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite dimensional C*-algebras satisfying the UCT and reproves Huaxin Lin's theorem on the classification of nuclear tracially AF C*-algebras.

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Morphisms of Extensions of C*-Algebras: Pushing Forward the Busby Invariant

Eilers

Loring

Pedersen

1999

Advances in Mathematics

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We study completions of diagrams of extensions of C*-algebras of the form 0 wwÄ v wwÄ v wwÄ v wwÄ 0 0 wwÄ v wwÄ v wwÄ v wwÄ 0 in which all three C*-algebras in one of the rows and either the ideal or the quotient in the other are given, along with the three morphisms between them. We find universal solutions to all four of these problems under restrictions of varying severity, on the given vertical maps and describe the solutions in terms of push-outs and pull-backs of certain diagrams. Our characterization of the universal solution to one of the diagrams yields a concrete description of various amalgamated free products. This leads to new results about the K-theory of amalgamated free products, verifying the Cuntz conjecture in certain cases. We also obtain new results about extensions of matricial field C*-algebras, verifying partially a conjecture of Blackadar and Kirchberg. Finally, we show that almost commuting unitary matrices can be uniformly approximated by commuting unitaries when an index obstruction vanishes.

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Generalized inductive limits of finite-dimensional C * -algebras

Cited by 167 publications

References 8 publications

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

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

Morphisms of Simple Tracially Af Algebras

Morphisms of Extensions of C*-Algebras: Pushing Forward the Busby Invariant

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