# Multiplicative Inverse in Nicely Normed 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 $a \oplus \dfrac {a^*} {\norm a^2} = 1 = \dfrac {a^*} {\norm a^2} \oplus a$.

 $\ds$  $\ds a \oplus \dfrac {a^*} {\norm a^2}$ $\ds$ $=$ $\ds a \oplus a^* \cdot \dfrac 1 {\norm a^2}$ $\ds$ $=$ $\ds \norm a^2 \cdot \dfrac 1 {\norm a^2}$ Definition of Nicely Normed Star-Algebra $\ds$ $=$ $\ds 1$ $\ds$ $=$ $\ds \dfrac 1 {\norm a^2} \cdot \norm a^2$ $\ds$ $=$ $\ds \dfrac 1 {\norm a^2} \cdot a^* \oplus a$ Definition of Nicely Normed Star-Algebra $\ds$ $=$ $\ds \dfrac {a^*} {\norm a^2}\oplus a$

$\blacksquare$

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