Covering Dimension and Quasidiagonality

Kirchberg, Eberhard; Winter, Wilhelm

doi:10.1142/s0129167x04002119

Cited by 146 publications

(227 citation statements)

References 27 publications

Supporting

Mentioning

227

Contrasting

Order By: Relevance

“…In view of the classification results of [42], [43] and [44] it also seems natural to rephrase the question in terms of the decomposition rank (cf. [22] and [41]):…”

Section: Proposition Letmentioning

confidence: 99%

Strongly self-absorbing $C^{*}$-algebras

Toms

Winter

2007

Trans. Amer. Math. Soc.

Self Cite

239

312

View full text Add to dashboard Cite

Abstract. Say that a separable, unital C * -algebra D C is strongly selfabsorbing if there exists an isomorphism ϕ : D → D ⊗ D such that ϕ and id D ⊗ 1 D are approximately unitarily equivalent * -homomorphisms. We study this class of algebras, which includes the Cuntz algebras O 2 , O ∞ , the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of O ∞ with UHF algebras of infinite type. Given a strongly self-absorbing C * -algebra D we characterise when a separable C * -algebra absorbs D tensorially (i.e., is D-stable), and prove closure properties for the class of separable D-stable C * -algebras. Finally, we compute the possible K-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C * -algebras.

show abstract

“…In view of the classification results of [42], [43] and [44] it also seems natural to rephrase the question in terms of the decomposition rank (cf. [22] and [41]):…”

Section: Proposition Letmentioning

confidence: 99%

Strongly self-absorbing $C^{*}$-algebras

Toms

Winter

2007

Trans. Amer. Math. Soc.

Self Cite

239

312

View full text Add to dashboard Cite

show abstract

“…In the K-theoretic classification program for simple unital separable stably finite nuclear C * -algebras, a great deal of progress has been made for those algebras which have stable rank one, real rank zero, and weak unperforation in the ordered K 0 -group (see, for example, [7], [13], [4], [1] and the last paragraph of [10]). One of the fundamental results in this direction is the work in [7], where Elliott and Gong classified (using K-theoretic invariants) all simple unital AH-algebras with bounded dimension growth and real rank zero.…”

Section: Introductionmentioning

confidence: 99%

Simple real rank zero algebras with locally Hausdorff spectrum

2006

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let A be a unital, simple, separable C * -algebra with real rank zero, stable rank one, and weakly unperforated ordered K 0 group. Suppose, also, that A can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then A is tracially approximately finite dimensional (i.e., A has tracial rank zero).Hence, A is an AH-algebra with bounded dimension growth and is determined by K-theoretic invariants.The above result also gives the first proof for the locally AH case.

show abstract

“…It generalizes the usual covering dimension of locally compact Hausdorff spaces to the realm of nuclear C * -algebras; its close cousin, the decomposition rank (see ref. 10), has already proved to be a very powerful tool in efforts to further Elliott's classification program, and there is much evidence to suggest that the nuclear dimension will be similarly important. An algebraic regularity property would typically provide enough space within the C * -algebra to decouple certain relations (or procedures) using only inner automorphisms.…”

mentioning

confidence: 99%

Minimal dynamics and the classification of C*-algebras

Toms

Winter²

2009

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

View full text Add to dashboard Cite

Let X be an infinite, compact, metrizable space of finite covering dimension and α : X → X a minimal homeomorphism. We prove that the crossed product C(X ) α Z absorbs the Jiang-Su algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered K-theory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms.

show abstract

Covering Dimension and Quasidiagonality

Cited by 146 publications

References 27 publications

Strongly self-absorbing $C^{*}$-algebras

Strongly self-absorbing $C^{*}$-algebras

Simple real rank zero algebras with locally Hausdorff spectrum

Minimal dynamics and the classification of C*-algebras

Contact Info

Product

Resources

About