On the classification problem for nuclear C^*-algebras

By Andrew S. Toms. 2005

211 citations recorded by the source corpus.

We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C ∗ -algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C ∗ -algebras are far-reaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of K-theoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e97e8fc0c63d7c876684430ea301a8fbde652e3

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In Algebra and Number Theory

1. W. Winter
2. Andrew S. Toms
3. G. Elliott
4. Stuart White
5. Yasuhiko Sato
6. Z. Niu
7. Aaron Tikuisis
8. Huaxin Lin

In Mathematical Physics

1. W. Winter
2. Andrew S. Toms
3. Yasuhiko Sato
4. Stuart White
5. G. Elliott
6. G'abor Szab'o
7. Z. Niu
8. Aaron Tikuisis

Citations in the corpus

• Comparison Theory and Smooth Minimal C*-Dynamics (2008, background): “ful regularity property necessary for the confirmation of G. A. Elliott’s K-theoretic rigidity conjecture (see [11] and [ 27 ]).”
• Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, background): “In the C∗algebra context, (i) is naturally replaced by Z-stability which, unlike the von Neumann situation, must be imposed as a hypothesis; the exotic examples of [50, 59, 62, 63] are not Z-stable.”
• Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra (2012, background): “In the quest to classify simple separable nuclear C∗-algebras, pioneered by George Elliott, it has become necessary to invoke some regularity property of the C∗-algebra, without which classification—at least by simple K-theoretical invariants—is known to be impossible, see for example [24] and [18].”
• FLAT DIMENSION GROWTH FOR C ∗ -ALGEBRAS (2005, background): “We establish their basic properties, show that they have natural connections to ordered K-theory, and prove that the range of the dimension–rank ratio is exhausted by simple algebras (this last result shows the class of simple, nuclear and non-Z-stable C∗-algebras to be uncountable).”
• Nuclear dimension of simple stably projectionless C∗-algebras (2019, background): “Finite nuclear dimension has proven to be a technically useful strengthening of nuclearity, that is both necessary for classification ([66, 53, 61, 28]) and a vital ingredient of the most recent classification theorems ([36, 48, 31, 22, 57, 69]).”
• Completeness of the isomorphism problem for separable C*-algebras (2013, background): “real rank zero algebras, AH algebras of slow dimension growth or separable, simple, purely infinite, nuclear algebras (modulo the universal coefficient theorem) [18,21–23,49,50,64] and there have been dramatic breakthroughs in the program, including the counterexamples to the general classification ”
• Localizing the Elliott conjecture at strongly self-absorbing C*-algebras (2007, background): “If one can arrange that u0 = 1B and hence (ut ∈ U0(B) for all t), then we say that ϕ and ψ are strongly asymptotically unitarily equivalent, written ϕ ≈sasu ψ. 1.2 The concept of strongly self-absorbing C∗-algebras was formally introduced in [54, Definition 1.3]: Definition: (i) A separable unital…”
• The Elliott conjecture for Villadsen algebras of the first type (2006, background): “In [16], Jiang and Su construct a C∗-algebra Z , which is the unique simple unital inductive limit of dimension drop intervals having K0 = Z , K1 = 0 and a unique normalised trace.”
• Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture (2009, background): “Z tensorially. (This condition is necessary for K-theoretic rigidity results akin to Theorem 0.1, both in general and in the case of tracial algebras (see [20] and [ 24 ], respectively).”
• A Remark on Invariants for C*-Algebras of Stable Rank One (2007, background): “In [5], Cuntz introduced an invariant for C*-algebras—very much analogous to the purely algebraic invariant K0—which has come to be known as the Cuntz semigroup, and which has recently received considerable attention (see [21], [2], [4], and [12]).”