Let $A$ be a unital $C^\ast$ algebra and let $B$ be a $\ast$ subalgebra such that $B \oplus \mathbb C = A$ and such that the unit in $B$, $1_B$, is not equal to the unit in $A$. I am trying to show:
If $\lambda \in \mathbb C$ is non-zero then $b-\lambda\cdot 1_B$ is invertible in $B$ if and only if $b-\lambda \cdot 1_A$ is invertible in $A$.
(see Murphy's book at the top of page 45).
I started the proof like this: Let $b-\lambda\cdot 1_B$ be invertible in $B$. Then it is also invertible in $A$. Let $a \in A$ denote its inverse. Then
$$ ba - \lambda 1_B a = 1_A$$
Now the goal is to find $c \in A$ such that $(b -\lambda 1_A)c = 1_A$. Somehow I have to show that $1_B a = 1_A c$ but I can't seem to do it. So this leads nowhere.
Can someone help me prove this please?