Spectral permanence for C-star algebras (${\sigma }_A(a)={\sigma }_B(a)$)

If $A$ is a closed C-star algebra inside a unital C-star algebra $B$ and the two algebras share their identity, an element of $A$ is invertible in $A$ exactly when it is invertible in $B$. Consequently its spectrum of an element does not depend on which of these two algebras is used.