  Motivated by the philosophy that $C^*$-algebras reflect noncommutative
topology, we investigate the stable homotopy theory of the (opposite) category
of $C^*$-algebras. We focus on $C^*$-algebras which are non-commutative
CW-complexes in the sense of [ELP]. We construct the stable $\infty$-category
of noncommutative CW-spectra, which we denote by $\mathtt{NSp}$. Let
$\mathcal{M}$ be the full spectral subcategory of $\mathtt{NSp}$ spanned by
"noncommutative suspension spectra" of matrix algebras. Our main result is that
$\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on
$\mathcal{M}$.
  To prove this we first prove a general result which states that any compactly
generated stable $\infty$-category is naturally equivalent to the
$\infty$-category of spectral presheaves on a full spectral subcategory spanned
by a set of compact generators. This is an $\infty$-categorical version of a
result by Schwede and Shipley [ScSh1]. In proving this we use the language of
enriched $\infty$-categories as developed by Hinich [Hin2,Hin3].
  We end by presenting a "strict" model for $\mathcal{M}$. That is, we define a
category $\mathcal{M}_s$ strictly enriched in a certain monoidal model category
of spectra $\mathtt{Sp^M}$. We give a direct proof that the category of
$\mathtt{Sp^M}$-enriched presheaves $\mathcal{M}_s^{op}\to\mathtt{Sp^M}$ with
the projective model structure models $\mathtt{NSp}$ and conclude that
$\mathcal{M}_s$ is a strict model for $\mathcal{M}$.
