I'm trying to understand the connection between suspension of a topological space and suspension of a C$^*$-algebra. The suspension of a topological space $X$ is the product space $X \times I$ (where $I$ is the unit interval) and identify $X \times \{0\} $ to a point and also identify $X \times \{1\} $ to a point, and suspension of a C$^*$-algebra $A$ is $$S(A)=\{ f \in C_{0}([0,1],A) | f(0)=f(1)=0\}.$$
If $A$ is a unital abelian C$^*$-algebra then $A$ isomorphic to $C(X)$ for some compact $X$ , my question is: Is $S(A)$ isomorphic to $C(S(X))$? I have the same question for mapping cone of topological space and mapping cone of C$^*$-algebra.
No, if $A=C(X)$, we do not have $SA\cong C(SX)$. Consider $X=\{p\}$, a one-point space. Then $A=C(X)\cong\mathbb C$ and $SX\cong[0,1]$, so $SA\cong C_0(\mathbb R)$, but $C(SX)\cong C([0,1])$.
Using the same space, we also don't have the desired isomorphism with cones.