Multiplicative Inverse in Nicely Normed Star-Algebra

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

Let $a \in A$.

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

where:
 * $a^*$ is the conjugate of $a$
 * $\left \Vert {a}\right \Vert$ is the norm of $a$.

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

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