Identity Element in Unital *-Algebra is Hermitian

Theorem

Let $\struct {A, \ast}$ be a unital $\ast$-algebra.

Let ${\mathbf 1}_A$ be the identity element of $A$.

Then we have:

${\mathbf 1}_A^\ast = {\mathbf 1}_A$

Proof

From Product of Element in *-Star Algebra with its Star is Hermitian, we have:

$\paren { {\mathbf 1}_A {\mathbf 1}_A^\ast}^\ast = {\mathbf 1}_A {\mathbf 1}_A^\ast$

We have, since $\ast$ is an involution:

$\paren { {\mathbf 1}_A {\mathbf 1}_A^\ast} = {\mathbf 1}_A^{\ast \ast} = {\mathbf 1}_A$

and:

${\mathbf 1}_A {\mathbf 1}_A^\ast = {\mathbf 1}_A^\ast$

Hence we conclude:

${\mathbf 1}_A^\ast = {\mathbf 1}_A$

$\blacksquare$

Sources

• 1990: Gerard J. Murphy: C*-Algebras and Operator Theory ... (previous) ... (next): $2.1$: $C^\ast$-Algebras