Inverse of Star of Element in Unital *-Algebra

Theorem

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

Let $a \in A$ be invertible.

Then $a^\ast$ is invertible and:

$\paren {a^\ast}^{-1} = \paren {a^{-1} }^\ast$

Corollary

Let $a \in A$.

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

Proof

We have:

$a a^{-1} = a^{-1} a = {\mathbf 1}_A$

From $(C^\ast 3)$ in the definition of an involution, we have:

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

From Identity Element in Unital *-Algebra is Hermitian, we therefore have:

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

Hence we have:

$\paren {a^\ast}^{-1} = \paren {a^{-1} }^\ast$

$\blacksquare$

Sources

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