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A suspension theorem for continuous trace $ C\sp *$-algebras

Author: Marius Dădărlat
Journal: Proc. Amer. Math. Soc. 120 (1994), 761-769
MSC: Primary 46L85; Secondary 19K99, 46L80, 55P99
MathSciNet review: 1166354
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Abstract: Let $ \mathcal{B}$ be a stable continuous trace $ {C^{\ast}}$-algebra with spectrum $ Y$. We prove that the natural suspension map $ {S_{\ast}}:[{C_0}(X),\mathcal{B}] \to [{C_0}(X) \otimes {C_0}({\mathbf{R}}),\mathcal{B} \otimes {C_0}({\mathbf{R}})]$ is a bijection, provided that both $ X$ and $ Y$ are locally compact connected spaces whose one-point compactifications have the homotopy type of a finite CW-complex and $ X$ is noncompact. This is used to compute the second homotopy group of $ \mathcal{B}$ in terms of $ K$-theory. That is, $ [{C_0}({{\mathbf{R}}^2}),\mathcal{B}] = {K_0}({\mathcal{B}_0})$, where $ {\mathcal{B}_0}$ is a maximal ideal of $ \mathcal{B}$ if $ Y$ is compact, and $ {\mathcal{B}_0} = \mathcal{B}$ if $ Y$ is noncompact.

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