A suspension theorem for continuous trace $C^ *$-algebras

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.