Nuclear Dimension

The nuclear dimension of a C^*-algebra is the least integer allowing approximation of its identity map on every finite set to any positive norm tolerance through finite-dimensional algebras using order-zero return pieces. Specifically, the intermediate algebra has a decomposition , the incoming map is completely positive and contractive, and each restriction of the completely positive return map is contractive and order zero. Order zero means that the map preserves orthogonality of positive elements. The approximation is on the finite set. If no finite works, the nuclear dimension is infinite. Finite-dimensional algebras have nuclear dimension zero.