Suppose $A$ is a $C^*$-algebra, then the suspension $SA$ is given by
$SA=\{f\in C(\Bbb T,A):f(1)=0\}$.
I saw the following conclusions from Olsen's book (page 136) Denote the unitization of $SA$ by $\tilde{SA}$, then
$\tilde{SA}=\{f\in C(\Bbb T,A^+):f(1)=\lambda \in \Bbb C,\pi_{\Bbb C}f(z)=\lambda,\forall z \in \Bbb T\}$.
How to verify the above result?
Define $$\varphi:\tilde{SA}\to\{f\in C(\mathcal T,\tilde A):f(1)=\lambda,\pi_\mathbb Cf(z)=\lambda\}$$ by $$(\varphi(f,\lambda))(z)=(f(z),\lambda)$$ for $f\in SA$, $\lambda\in\mathbb C$. Show that this map is a $*$-isomorphism.