Nuclearity and ${\mathrm {CPC}^*}$-Systems

Abstract We write arbitrary separable nuclear $\mathrm {C}^*$ -algebras as limits of inductive systems of finite-dimensional $\mathrm {C}^*$ -algebras with completely positive connecting maps. The characteristic feature of such ${\mathrm {CPC}^*}$ -systems is that the maps become more and more orthogonality preserving. This condition makes it possible to equip the limit, a priori only an operator space, with a multiplication turning it into a $\mathrm {C}^*$ -algebra. Our concept generalizes the NF systems of Blackadar and Kirchberg beyond the quasidiagonal case.