## On the asymptotic homotopy type of inductive limit $$C^*$$-algebras.(English)Zbl 0791.46047

Let $$X$$, $$Y$$ be compact, connected, metrisable spaces with base points $$x_ 0$$, $$y_ 0$$ and let $${\mathcal K}$$ denote the compact operators. It is shown that $$C_ 0(X\setminus x_ 0)\otimes {\mathcal K}$$ is asymptotically homotopic (or shape equivalent) to $$C_ 0(Y\setminus y_ 0)\otimes {\mathcal K}$$ if and only if $$X$$ and $$Y$$ have isomorphic $$K$$-groups. Similar results are obtained for certain inductive limits of nuclear $$C^*$$- algebras.

### MSC:

 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46L85 Noncommutative topology
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### References:

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