Covering Dimension for Nuclear C*-Algebras II

Abstract

The completely positive rank is an analogue of topological covering dimension, defined for nuclear C*-algebras via completely positive approximations. These may be thought of as simplicial approximations of the algebra, which leads to the concept of piecewise homogeneous maps and a notion of noncommutative simplicial complexes. We introduce a technical variation of the completely positive rank and show that the two theories coincide in many important cases. Furthermore we analyze some of their properties; in particular we show that both theories behave nicely with respect to ideals and that they coincide with covering dimension of the spectrum for certain continuous trace C*-algebras.