Maximal Commutative Subalgebra of Unital Algebra is Unital

Theorem

Let $K$ be a field.

Let $A$ be a unital algebra over $K$.

Let $B$ be a commutative subalgebra of $A$ that is maximal with respect to set inclusion.

Then $B$ is unital.

We show that $B + K {\mathbf 1}_A$ is a subalgebra of $A$.

First we show that $B + K {\mathbf 1}_A$ is a vector subspace of $A$.

Let $x + t {\mathbf 1}_A, y + s {\mathbf 1}_A \in B + K {\mathbf 1}_A$ for $x, y \in B$ and $s, t \in K$.

Let $\lambda \in K$.

We have:

$\paren {x + t {\mathbf 1}_A} + \lambda \paren {y + s {\mathbf 1}_A} = {x + \lambda y} + \paren {t + \lambda s} {\mathbf 1}_A$

Since $B$ is a subalgebra of $A$, it is in particular a vector subspace of $A$.

Then $x + \lambda y \in B$.

So we have that $\paren {x + t {\mathbf 1}_A} + \lambda \paren {y + s {\mathbf 1}_A} \in B + K {\mathbf 1}_A$.

Hence $B + K {\mathbf 1}_A$ is a vector subspace of $A$.

We now show that $B + K {\mathbf 1}_A$ is closed under multiplication.

Let $x + t {\mathbf 1}_A, y + s {\mathbf 1}_A \in B + K {\mathbf 1}_A$ for $x, y \in B$ and $s, t \in K$.

We have:

$\paren {x + t {\mathbf 1}_A} \paren {y + s {\mathbf 1}_A} = \paren {x y + s x + t y} + t s {\mathbf 1}_A$

Since $B$ is a subalgebra of $A$, we have $x y \in B$.

Since $B$ is a vector subspace of $A$, we have that $x y + s x + t y \in B$.

So, we have:

$\paren {x + t {\mathbf 1}_A} \paren {y + s {\mathbf 1}_A} \in B + K {\mathbf 1}_A$

We conclude that $B + K {\mathbf 1}_A$ is a subalgebra of $A$.

Now note that we have:

$B = B + 0_K {\mathbf 1}_A \subseteq B + K {\mathbf 1}_A$

Since $B$ is maximal with respect to set inclusion, we have:

$B = B + K {\mathbf 1}_A$

Since ${\mathbf 0}_A \in B$, we have that ${\mathbf 1}_A \in B$.

$\blacksquare$