Continuity of functors with respect to generalized inductive limits

Abstract

Let (Ai, ϕi, i+1) be a generalized inductive system of a sequence (Ai) of unital separable C*-algebras, with A = limi→∞(Ai, ϕi, i+1). Set ϕj, i = ϕi−1,i ∘ ⋯ ∘ ϕj+1,j+2 ∘ ϕj, j+1 for all i > j. We prove that if ϕj, i are order zero completely positive contractions for all j and i > j, and $$\sigma \left({{\phi _{j,i}}\left({{1_{{A_j}}}} \right)} \right)$$ for all j and i > j > 0, where $$\sigma \left({{\phi _{j,i}}\left({{1_{{A_j}}}} \right)} \right)$$ is the spectrum of $${\phi _{j,i}}\left({{1_{{A_j}}}} \right)$$ , then limi→∞(Cu(Ai), Cu(ϕi, i+1)) = Cu(A), where Cu(A) is a stable version of the Cuntz semigroup of C*-algebra A. Let (An, ϕm, n) be a generalized inductive system of C*-algebras, with the ϕm, n order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of An is no more than some integer k for each n, then the decomposition rank (nuclear dimension) of A is also no more than k.