Three Applications of the Cuntz Semigroup

Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: 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; for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant.