Do spectrum change on a maximal sub algebra? 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?

I cannot say much about the case requested, as I know very few examples of maximal subalgebras that are not C$^*$-subalgebras (and these preserve the spectrum). The case that is easy is the case where $B$ is maximal abelian. In that case, if $\lambda\in\rho_A[a]$, then there exists $c\in A$ with $c(a-\lambda 1)=1$. For any $x\in B$, we have
$$
xc=c(a-\lambda 1)xc=cx(a-\lambda 1)c=cx,
$$
so $c$ commutes with every $x\in B$ and so $c\in B$. This shows that $\lambda\in\rho_B[a]$, i.e. $\rho_A[a]\subset\rho_B[a]$, or $\sigma_B[a]\subset\sigma_A[a]$.
