  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.
