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.
Categories: math.OA