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Injectivity of the Connecting Homomorphisms in Inductive Limits of Elliott–Thomsen Algebras

  • Zhichao Liu (a1)
Abstract

Let $A$ be the inductive limit of a sequence

$$\begin{eqnarray}A_{1}\xrightarrow[{}]{\unicode[STIX]{x1D719}_{1,2}}A_{2}\xrightarrow[{}]{\unicode[STIX]{x1D719}_{2,3}}A_{3}\longrightarrow \cdots\end{eqnarray}$$
with $A_{n}=\bigoplus _{i=1}^{n_{i}}A_{[n,i]}$ , where all the $A_{[n,i]}$ are Elliott–Thomsen algebras and $\unicode[STIX]{x1D719}_{n,n+1}$ are homomorphisms. In this paper, we will prove that $A$ can be written as another inductive limit
$$\begin{eqnarray}B_{1}\xrightarrow[{}]{\unicode[STIX]{x1D713}_{1,2}}B_{2}\xrightarrow[{}]{\unicode[STIX]{x1D713}_{2,3}}B_{3}\longrightarrow \cdots\end{eqnarray}$$
with $B_{n}=\bigoplus _{i=1}^{n_{i}^{\prime }}B_{[n,i]^{\prime }}$ , where all the $B_{[n,i]^{\prime }}$ are Elliott–Thomsen algebras and with the extra condition that all the $\unicode[STIX]{x1D713}_{n,n+1}$ are injective.

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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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