Inverse of Star of Element in Unital *-Algebra/Corollary

Corollary

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

Let $a \in A$.

Then $a$ is invertible if and only if $a^\ast$ is invertible.

Proof

From Inverse of Star of Element in Unital *-Algebra:

if $a$ is invertible then $a^\ast$ is invertible.

Hence:

if $a^\ast$ is invertible then $a^{\ast \ast}$ is invertible.

Since $\ast$ is an involution, we have $a^{\ast \ast} = a$ and hence:

if $a^\ast$ is invertible then $a$ is invertible.

We conclude:

$a$ is invertible if and only if $a^\ast$ is invertible.

$\blacksquare$