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Injectivity of the Connecting Maps in AH Inductive Limit Systems

  • George A. Elliott (a1), Guihua Gong (a2) and Liangqing Li (a3)

Abstract

Let $A$ be the inductive limit of a system

$${{A}_{1\,}}\,\xrightarrow{{{\phi }_{1,\,2}}}\,{{A}_{2}}\,\xrightarrow{{{\phi }_{2,\,3}}}\,{{A}_{3}}\,\to \cdots $$

with ${{A}_{n}}\,=\,\oplus _{i=1}^{{{t}_{n}}}\,{{P}_{n,\,i}}{{M}_{\left[ n,\,i \right]}}(C({{X}_{n,\,i}})){{P}_{n,\,i}}$ , where ${{X}_{n,\,i}}$ is a finite simplicial complex, and ${{P}_{n,\,i}}$ is a projection in ${{M}_{[n,i]}}\,\left( C\left( {{X}_{n,i}} \right) \right)$ . In this paper, we will prove that $A$ can be written as another inductive limit

$${{B}_{1}}\,\xrightarrow{{{\psi }_{1,\,2}}}\,{{B}_{2}}\,\xrightarrow{{{\psi }_{2,\,3}}}\,{{B}_{3}}\,\to \cdots $$

with ${{B}_{n}}\,=\,\oplus _{i=1}^{{{s}_{n}}}\,{{Q}_{n,i}}{{M}_{\left\{ n,\,i \right\}}}(C({{Y}_{n,\,i}})){{Q}_{n,\,i}}$ , where ${{Y}_{n,\,i}}$ is a finite simplicial complex, and ${{Q}_{n,\,i}}$ is a projection in ${{M}_{\left\{ n,\,i \right\}}}(C({{Y}_{n,\,i}}))$ , with the extra condition that all the maps ${{\psi }_{n,n+1}}$ are injective. (The result is trivial if one allows the spaces ${{Y}_{n,\,i}}$ to be arbitrary compact metrizable spaces.) This result is important for the classification of simple $\text{AH}$ algebras. The special case that the spaces ${{X}_{n,\,i}}$ are graphs is due to the third author.

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References

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Injectivity of the Connecting Maps in AH Inductive Limit Systems

  • George A. Elliott (a1), Guihua Gong (a2) and Liangqing Li (a3)

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