Injectivity of the connecting homomorphisms

Abstract

Let $A$ be the inductive limit of a sequence $$A_1\, \xrightarrow{\phi_{1,2}} \,A_2\,\xrightarrow{\phi_{2,3}} \,A_3\rightarrow\cdots$$ with $A_n=\oplus_{i=1}^{n_i}A_{[n,i]}$, where all the $A_{[n,i]}$ are Elliott-Thomsen algebras and $\phi_{n,n+1}$ are homomorphisms, in this paper, we will prove that $A$ can be written as another inductive limit $$B_1\,\xrightarrow{\psi_{1,2}} \,B_2\,\xrightarrow{\psi_{2,3}} \,B_3\rightarrow\cdots$$ with $B_n=\oplus_{i=1}^{n_i}B_{[n,i]}$, where all the $B_{[n,i]}$ are Elliott-Thomsen building blocks and with the extra condition that all the $\phi_{n,n+1}$ are injective.