The Elliott conjecture for Villadsen algebras of the first type

By Andrew S. Toms, W. Winter. 2006

74 citations recorded by the source corpus.

Abstract We study the class of simple C ∗ -algebras introduced by Villadsen in his pioneering work on perforated ordered K-theory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott's classification conjecture: two C ∗ -algebraic ( Z -stability and approximate divisibility), one K-theoretic (strict comparison of positive elements), and three topological (finite decomposition rank, slow dimension growth, and bounded dimension growth). The equivalence of Z -stability and strict comparison constitutes a stably finite version of Kirchberg's characterisation of purely infinite C ∗ -algebras. The other equivalences confirm, for Villadsen's algebras, heretofore conjectural relationships between various notions of good behaviour for nuclear C ∗ -algebras.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3b1ade05692e3affacd0097b4a5fed6a95b43f01

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In Algebra and Number Theory

1. W. Winter
2. Andrew S. Toms
3. G. Elliott
4. F. Perera
5. M. Rørdam
6. Leonel Robert
7. Stuart White
8. E. Ortega

In Mathematical Physics

1. W. Winter
2. Andrew S. Toms
3. F. Perera
4. G. Elliott
5. M. Rørdam
6. Leonel Robert
7. Yasuhiko Sato
8. David Kerr

Citations in the corpus

• Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “Establishing finite decomposition rank requires the full force of the classification theorems: a consequence of both theorems is that the algebras they cover are all in fact simple unital AH algebras of bounded”
• Nuclear dimension and -stability of pure C ∗ -algebras (2010, background): “In [35, 40, 43], some links between such regularity properties have been proven, and some others have been formally established as conjectures (see also [12]).”
• K-Theory for operator algebras. Classification of C$^*$-algebras (2009, background): “There has also been progress on this front, mainly due to Toms and Winter (see [ 68 ]).”
• The Corona Factorization property, Stability, and the Cuntz semigroup of a C*-algebra (2009, result): “It follows from a result of Toms and Winter, [ 21 ], that the Cuntz semigroup of any unital simple C -algebra with decomposition rank n has the n-comparison property (and hence also !-comparison and the Corona Factorization Property).”