K-Theory for operator algebras. Classification of C$^*$-algebras
By P. Ara, F. Perera, Andrew S. Toms. 2009
117 citations recorded by the source corpus.
In this article we survey some of the recent goings-on in the classification programme of C$^*$-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott conjecture does not hold at its boldest. We review the construction of this object both by means of positive elements and via its recent interpretation using countably generated Hilbert modules (due to Coward, Elliott and Ivanescu). The passage from one picture to another is presented with full, concise, proofs. We indicate the potential role of the Cuntz semigroup in future classification results, particularly for non-simple algebras.
Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9d4c0d6ef700a8c11a67954655a94f746cce58de
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In Algebra and Number Theory
- W. Winter
- Stuart White
- Aaron Tikuisis
- Leonel Robert
- Andrew S. Toms
- Yasuhiko Sato
- N. Brown
- G'abor Szab'o
In Applied Mathematics
- W. Winter
- Aaron Tikuisis
- Stuart White
- Leonel Robert
- Andrew S. Toms
- Yasuhiko Sato
- B. Blackadar
- N. Brown
In Mathematical Physics
- W. Winter
- Aaron Tikuisis
- Stuart White
- Leonel Robert
- G'abor Szab'o
- Yasuhiko Sato
- Andrew S. Toms
- N. Brown
Citations in the corpus
- Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, background): “See [3] for more about the Cuntz semigroup.”
- An algebraic approach to the radius of comparison (2010, background): “The semigroup W (A) is then the subsemigroup of Cu(A) of Cuntz classes with a representative in ⋃ n Mn(A)+. Alternatively, Cu(A) can be defined to consist of equivalence classes of countably generated Hilbert modules over A.”
- Quasidiagonality of nuclear C*-algebras (2015, methodology): “By construction ν0|C0(( 13 , 23 ),A) = (φ ⊕(N−i+1) ⊕ ψ⊕(i−1))|C0(( 13 , 23 ),A),(5.49) ν1|C0(( 13 , 23 ),A) = (φ ⊕(N−i) ⊕ ψ⊕i)|C0(( 13 , 23 ),A).”