Injectivity of the connecting homomorphisms

TL;DR

This paper proves that an inductive limit of Elliott-Thomsen algebras can be re-expressed with injective connecting homomorphisms, simplifying the structure for analysis.

Contribution

It demonstrates that any inductive limit of Elliott-Thomsen algebras can be represented with injective connecting maps, enhancing the understanding of their structural properties.

Findings

Rewrites inductive limits with injective connecting homomorphisms.

Maintains the algebraic structure while ensuring injectivity.

Facilitates further analysis of Elliott-Thomsen algebra limits.

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.