In the context of C-star algebras, a suspension of a C-star algebra $A$ is defined as $$ SA := \left\{f:\left[0,\,1\right]\to A\,\text{continious}\,\mid\,f\left(0\right)=f\left(1\right)=0_A\right\} $$
On the other hand, to define $\pi_1\left(A, a_0\right)$, we take equivalence classes of continuous maps $S^1\to A$ which map some base-point of $S^1$ (say 1) to a chosen base point, say, $a_0\in A$. Later on we say that if $a_0$ and $a_1$ are in the same path-connected component of $A$, then $\pi_1\left(A, a_0\right)$ and $\pi_1\left(A, a_1\right)$ are isormorphic as groups.
My question is: what is the importance of $0_A$ in the definition of suspension of a C-star algebra? Does it play a similar role as the base point in $\pi_1$, namely, any two points in the same path-connected component will give rise to the same C-star algebra? Or does $0_A$ somehow play a special role? Is the space of loops in $A$ with a fixed base-point equal to the suspension? And what happens if the base-point is allowed to vary (we know that then for the homotopy group we don't get a group structure)