Let $A$ be a unital $C^*$-algebra and $C$ be a $C^*$-subalgebra containing the identity of $A,$ and let $a$ be a self-adjoint element in $C.$ Then $\sigma_A (a) = \sigma_C (a).$
It's quite clear that $\sigma_A (a) \subseteq \sigma_C (a).$ Now suppose $\lambda \in \sigma_C (a).$ This implies that $\lambda - a$ has no inverse in $C.$ From here how do I conclude that it has no inverse in $A$ as well? Any help in this regard would be much appreciated.
Thanks a bunch.