Three Applications of the Cuntz Semigroup

Details Three Applications of the Cuntz Semigroup Three Applications of the Cuntz Semigroup

2007-04-30

arxiv.org/abs/0704.3988v1

Mathematics

Operator Algebras

10 pages

Scientific paper

Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: (i) for many simple C$^*$-algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott's classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; (iii) for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant.

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