Unital Commutative Banach $*$-algebra has $1^*=1$

Question

I am wondering why $1^*=1$ in a unital commutative Banach $*$-algebra.

I guess I am having trouble with comprehending what the $*$-algebra is.

Answer 1

Recall that a unit is unique, and consider the computation: \begin{equation} a^{*} = (1a)^{*} = a^{*} 1^{*} = 1^{*} a^{*}, \end{equation} where $a$ is any element in the Banach algebra. We see that $1^{*}$ is a unit, and by uniqueness of the unit it follows that $1^{*} = 1$.