Cu-nuclearity implies LLP and exactness

摘要

The Cu-nuclearity property is an analogue of Skandalis's notion of K-nuclearity, adapted to the case of Cuntz semigroups of C*-algebras. We prove that this implies nuclearity, and we introduce a weaker form of the condition. We prove that the new condition weak Cu-nuclearity, for simple separable C*-algebras, implies exactness and the local lifting property (LLP). We also prove that if A is a simple C*-algebra with the weak Cu-nuclearity property, and B is any simple C*-algebra, then A ?min B = A ?max B. We prove that Cu-nuclearity does imply nuclearity, and that in some cases this is also true for weak Cu-nuclearity.