Intersection of -Subalgebras is -Subalgebra

Theorem

Let $\tuple {A, \ast}$ be a $\ast$-algebra over $\C$.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of $\ast$-subalgebras of $A$.

Let:

$\ds B = \bigcap_{\alpha \mathop \in I} A_\alpha$

Then $B$ is a $\ast$-subalgebra of $A$.

It remains to show that for each $x \in B$ we have $x^\ast \in B$.

Let $x \in B$.

Then $x \in A_\alpha$ for each $\alpha \in I$.

Since each $A_\alpha$ is a $\ast$-subalgebra, we have $x^\ast \in A_\alpha$ for each $\alpha \in I$.

Hence we have:

$\ds x^\ast \in \bigcap_{\alpha \mathop \in I} A_\alpha$

So $x^\ast \in B$.

$\blacksquare$