Multiplicative Inverse in Nicely Normed Star-Algebra

Theorem
Let $A = \struct {A_F, \oplus}$ be a nicely normed $*$-algebra whose conjugation is denoted $*$.

Let $a \in A$.

Then the multiplicative inverse of $a$ is given by:
 * $a^{-1} = \dfrac {a^*} {\norm a^2}$

where:
 * $a^*$ is the conjugate of $a$
 * $\norm a$ is the norm of $a$.

Proof
For the result to hold, we need to show that $a \oplus \dfrac {a^*} {\norm a^2} = 1 = \dfrac {a^*} {\norm a^2} \oplus a$.

Note that this construction works whether $\oplus$ is associative or not.