unitisation of suspension of a $C^*$ algebra

Question

Let $A$ be a $C^*$ algebra,the suspension of $A$ is defined by $SA=\{f\in C([0,1],A)|f(0)=f(1)=0\}$,then the unitisation of $SA$ is $\tilde{SA}=\{f\in C_0([0,1],A)|f(0)=f(1)\in \Bbb C\}$.

How to prove the above conclusion?

Answer 1

You should prove that $$ \widetilde{SA} / SA \cong \mathbb C. $$ The unitization of a C*-algebra is the unique unital C*-algebra with this property.