Let $\mathcal{A}$ be a unital C*-algebra.
Is the following statement true?
$$A \text{ not invertible} \Leftrightarrow A^* \text{ not invertible}$$
Moreover, suppose $A\in \mathcal{A}$ has an inverse $A^{-1}\in \mathcal{A}$, what can I say about the inverse of the adjoint $A^*$?
It is enough to prove that $A$ is invertible iff $A^*$ is invertible.
Proof of implication $\Longrightarrow$. Let $B$ be inverse to $A$, then $AB=BA=1$. Apply adjoint to get $B^*A^*=A^*B^*=1$, hence $B^*=(A^*)^{-1}$.
Proof of implication $\Longrightarrow$. Let $B$ be inverse to $A^*$, then $A^*B=BA^*=1$. Apply adjoint to get $B^*A=AB^*=1$, hence $A^{-1}=B^*$.
These facts shows that $A$ is invertible iff $A^*$ is invertible and what is more $$ (A^{-1})^*=(A^*)^{-1} $$