The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras

By N. Brown, F. Perera, Andrew S. Toms. 2006

117 citations recorded by the source corpus.

Abstract We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple C*-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among -stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C*-algebras. We also prove in passing that the Cuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for a large class of simple unital C*-algebras.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4bd5eaf04a74185b265796e0f7d432e7dba74fe3

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Citations in the corpus

K-Theory for operator algebras. Classification of C$^*$-algebras (2009, unknown): “O”
Pullbacks, $C(X)$-algebras, and their Cuntz semigroup (2011, methodology): “orphic C∗-algebrasthat agreeon theElliott invariant andseveral possible extensions ofit; see[32]. For wellbehavedsimplealgebras, thissemigroup canbe recovered from the classical Elliott invariant (see[5], [6]), and in the non-simple case it has already been used to prove actual classiﬁcation results”
C*-algebras of stable rank one and their Cuntz semigroups (2018, unknown): “…was ﬁrst raised by N. Brown, and has been solved in a number of instances: For simple C*-algebras that tensorially absorb the Jiang-Su algebra Z , the problem is solved in [BPT08, Theorem 5.5] in the exact, unital case, and in [ERS11, Corollary 6.8] dropping both exactness and existence of a unit.”
Nuclear dimension and -stability of pure C ∗ -algebras (2010, unknown): “[7] or [5]; it follows from [3] that in fact x ∈ W(A) ⊂ W(A ⊗ K).”
Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “2—it is not an addition at all! Recent work of Brown, Perera, and the second named author shows that for a simple unital separable amenable C-algebra which absorbs Z tensorially, there is a functor which recovers the Cuntz semigroup from the Elliott invariant ([5], [45]).”
The cone of lower semicontinuous traces on a C*-algebra (2008, methodology): “As another corollary of Theorem 6.6, let us give a computation of the Cuntz semigroup of a simple C*-algebra absorbing the Jiang-Su algebra (this computation was previously obtained in [ 2 ] with the additional assumptions that the algebra was unital, exact, and of stable rank one).”
Comparison Theory and Smooth Minimal C*-Dynamics (2008, background): “cture of the Cuntz semigroup. The Cuntz semigroup is a sensitive invariant in the matter of distinguishing simple separable amenable C∗-algebras, and has recently received considerable attention (see [3], [4], [5], [7], [8], [11], [27], and [30]). It is, however, very difﬁcult to compute in general—”