Can anyone help me to solve the following problem.
We know if $B$ is a unital sub-algebra of a unital Banach algebra $A$ and if $a\in B$, then $\sigma _{A}[a] \subset \sigma _{B}[a] $(Here $\sigma _{T}$ denotes the spectrum with respect to the algebra $T$). What happens if we choose $B$ as an maximal sub-algebra. Can we say that corresponding spectrum will be equal?