Show that if $B$ is a maximal abelian subalgebra of a unital Banach algebra $A$.Then $B$ is closed and contains the unit.

Question

Show that if $B$ is a maximal abelian subalgebra of a unital Banach algebra $A$.

Then $B$ is closed and contains the unit.

How to approach this. $\bar B$ is closed abelian subalgebra which contain $B$. also we have $$B_1=\{b+\lambda1: b\in B ,\lambda\in\mathbb{C}\}$$ which also contain $B$ and also contain unity. But how to connect them to $B$ and show $B$ contain unity and is closed.

Any suggestion.Thanks in Advanced

Answer 1

You don't have to prove both at once; take things one step at a time.

Let $B$ be a maximal abelian subalgebra of $A$. Assume $B$ is not closed. Then it's closure is abelian, and properly contains $B$, contradicting maximality of $B$.

Now assume $B$ is not unital. Then $\{b+\lambda1:b\in B,\lambda\in\mathbb C\}$ is abelian, and properly contains $B$, again contradicting maximality of $B$.