Recasting the Elliott conjecture

By F. Perera, Andrew S. Toms. 2006

67 citations recorded by the source corpus.

Let A be a simple, unital, finite, and exact C*-algebra which absorbs the Jiang–Su algebra $${\mathcal{Z}}$$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—$${\mathcal{Z}}$$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of $${\mathcal{Z}}$$ -stable C*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that $${\mathrm{K}}$$ -theoretic invariants will classify separable and nuclear C*-algebras—with the recent appearance of counterexamples to its strongest concrete form.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8c2a42620b2a2861b2eed46688a64b7bf5814943

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In Algebra and Number Theory

1. Leonel Robert
2. F. Perera
3. W. Winter
4. Andrew S. Toms
5. M. Rørdam
6. N. Brown
7. P. Ara
8. A. Ciuperca

In Geometry and Topology

1. Andrew S. Toms
2. F. Perera
3. Leonel Robert
4. N. Brown
5. M. Rørdam
6. P. Ara
7. W. Winter
8. David Kerr

In Mathematical Physics

1. Leonel Robert
2. F. Perera
3. W. Winter
4. Andrew S. Toms
5. M. Rørdam
6. N. Brown
7. P. Ara
8. A. Ciuperca

Citations in the corpus

• Comparison Theory and Smooth Minimal C*-Dynamics (2008, background): “[ 21 ]). Let LAffb(T(A))++ denote the set of lower semicontinuous, affine, bounded, strictly positive functions on the tracial state space of A, and define a map ι : W(A) → LAffb(T(A))++ by ι(h ai )(τ) = dτ(a).”
• Divisibility properties for C*‐algebras (2011, background): “isibility properties are both necessary and sufficient conditions to ensure that the Cuntz semigroup of a simple, separable, unital, exact C∗-algebra A is (naturally) isomorphic to Aff(T(A)) ⊔ V(A), cf. [PT07], [BPT08], and [ERS]. The existence of simple C∗-algebras with bad comparability properties wa”
• The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras (2006, background): “It is proved in [14] that φ is both everywhere-defined and well-defined.”
• Isomorphism of Hilbert modules over stably finite C*-algebras (2008, background): “The equivalence of (3) and (4) above generalizes Proposition 2.8 in [ 6 ].”
• K-Theory for operator algebras. Classification of C$^*$-algebras (2009, background): “We first need a lemma, that appears in [50].”