$\Sigma \mathfrak{A}$ will denote the suspension $C_{0} ( {\ensuremath{\mathbb{R}}\xspace}) \otimes \mathfrak{A}$ of $\mathfrak{A}$ and $\Sigma^{j} \mathfrak{A}$ is defined recursively $\Sigma ( \Sigma^{j-1} \mathfrak{A}) )$.

$C^{*}$-algebras over topological spaces
----------------------------------------

Let $X$ be a topological space and let $\mathbb{O}( X)$ be the set of open subsets of $X$, partially ordered by set inclusion $\subseteq$. A subset $Y$ of $X$ is called *locally closed* if $Y = U \setminus V$ where $U, V \in \mathbb{O} ( X )$ and $V \subseteq U$. The set of all locally closed subsets of $X$ will be denoted by $\mathbb{LC}(X)$. For a $C^{*}$-algebra $\mathfrak{A}$, let $\mathbb{I} ( \mathfrak{A} )$ be the set of all closed two-sided ideals of $\mathfrak{A}$, partially ordered by $\subseteq$.

\[d:Xalgs\] Let $\mathfrak{A}$ be a $C^{*}$-algebra. Let $\mathrm{Prim} ( \mathfrak{A} )$ denote the *primitive ideal space* of $\mathfrak{A}$, equipped with the usual hull-kernel topology, also called the Jacobson topology.

Let $X$ be a topological space. A *$C^{*}$-algebra over $X$* is a pair $( \mathfrak{A} , \psi )$ consisting of a $C^{*}$-algebra $\mathfrak{A}$ and a continuous map ${  \psi  \colon  \mathrm{Prim} ( \mathfrak{A} )  \rightarrow  X  }$. A $C^{*}$-algebra over $X$, $( \mathfrak{A} , \psi )$, is *separable* if $\mathfrak{A}$ is a separable $C^{*}$-algebra. 