Decomposition rank and Z-stability

By W. Winter. 2008

120 citations recorded by the source corpus.

We show that separable, simple, unital C*-algebras with finite decomposition rank absorb the Jiang-Su algebra Z tensorially. This has a number of consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. In particular, it completes the classification of C*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a859e00faff8b3688cbe35d75ceee96151b3b664

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Citations in the corpus

Dimension, comparison, and almost finiteness (2017, methodology): “In parallel with [52], we simply refer to it as comparison, and also define the useful higher-order notions of m-comparison for integers m ≥ 0, with comparison representing the case m = 0 (Definition 3.”
Sequentially split ∗-homomorphisms between C*-algebras (2015, background): “These properties play a key role in Toms–Winter’s regularity conjecture, which asserts that these properties should be equivalent on a large scale [15, 67, 68].”
Classification of inductive limits of 1-dimensional NCCW complexes (2010, methodology): “In the case that A⊗ K contains a projection, this is essentially a consequence of Winter’s Z-stability result [34] for simple C∗-algebras of finite decomposition rank, and of the computation of Cu(A) obtained by Brown and Toms in [5] for simple unital Z-stable A (Z denotes the Jiang-Su algebra).”
Cartan subalgebras in C*-algebras. Existence and uniqueness (2017, background): “Classification of C*algebras has seen tremendous advances recently [66, 67, 34, 35, 55, 22, 14, 62].”
Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture (2009, methodology): “In this section we combine ideas from [3] and [27] with those of the preceding section to prove our Theorem 0.”
Rokhlin dimension for actions of residually finite groups (2014, background): “In view of the astonishing progess of C∗-algebra classification over the last few years [18, 13, 12, 73], which is largely driven by the discovery of various regularity properties (see for example [83, 81, 82, 59, 48, 36, 75, 65, 50, 84, 66, 5]), a natural and important question appears, which can b”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, background): “Moreover, the implications (i) =⇒ (ii) and (ii) =⇒ (iii) have been established in general by the third named author [42, 44] (see [34] for an extension to the stably projectionless case) and Rørdam [29] respectively; recent progress of Matui and the first named author [23] establishes (iii) =⇒ (ii) ”
Nuclear dimension of simple $$\mathrm {C}^*$$-algebras (2019, background): “amenable factors due to Connes and Haagerup [13, 29]. A major task at this point is to identify simple nuclear C∗-algebras of ﬁnite nuclear dimension, through establishing the Toms–Winter conjecture ([21, 72]; the precise statement is [79, Conjecture 9.3], and an overview is given in [77, Section 5]”
A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, unknown): “5 of [118], A is Z-stable.”
Nuclear dimension and -stability of pure C ∗ -algebras (2010, methodology): “The essential difference is that in [40] strong tracial m-comparison was provided by [40, Corollary 3.”
STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN (2017, background): “that an inﬁnite exact C∗-algebra is Z-stable if and only if it is O∞-stable; see [45]. In the ﬁnite case, (ii) =⇒ (iii) was shown by Rørdam in [45]. I showed ﬁrst (i’) =⇒ (ii) and then (i) =⇒ (ii) in [60] and [61], respectively. In [36], Matui and Sato showed (iii) =⇒ (ii) when Ahas only one tracial”
Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, methodology): “∗-algebras enjoy this property. The following conjecture of Toms and WW predicts the answer: ﬁnite nuclear dimension should coincide with two other regularity properties of very diﬀerent natures (cf. [30, 95]; the precise form stated here is given in [100, Conjecture 9.3]). Conjecture A (Toms-Winter”