K-Theory for operator algebras. Classification of C${}^{\ast }$-algebras

Abstract

In this article we survey some of the recent goings-on in the classification programme of C${}^{\ast }$-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.

56 pages, notes from the Summer School on Operator Algebras and Applications, held in Santander, July 21-25 2008. Submitted to Contemporary Mathematics

References in corpus (2)

• A simple C*-algebra with a finite and an infinite projection
• Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem

Cited by in corpus (6)

• Nuclear dimension and Z-stability
• The Cuntz Semigroup and Comparison of Open Projections
• Tensor products and regularity properties of Cuntz semigroups
• Picard groups of certain stably projectionless C*-algebras
• Smooth lattice orbits of nilpotent groups and strict comparison of projections
• A unitary Cuntz semigroup for C*-algebras of stable rank one