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

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  07 January 2019

Zhichao Liu
Zhichao Liu*
Affiliation:
Postdoctoral Research Station of Mathematics, Hebei Normal University, Shijiazhuang, 050024, China Email: [email protected]
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Abstract

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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.

Keywords

injectiveinductive limitElliott–Thomsen algebra

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L35: Classifications of $C^*$-algebras
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 131 - 148
Copyright
© Canadian Mathematical Society 2018 

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