Three Applications of the Cuntz Semigroup

Abstract

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