Definition:Nicely Normed Star-Algebra

Definition

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

Then $A$ is a nicely normed $*$-algebra iff:

$\forall a \in A: a + a^* \in \R$

$\forall a \in A, a \ne 0: 0 < a \oplus a^* = a^* \oplus a \in \R$

Real Part

Let $a \in A$ be an element of a nicely normed $*$-algebra.

Then the real part of $a$ is given by:

$\Re \left({a}\right) = \dfrac {a + a^*} 2$

Imaginary Part

Let $a \in A$ be an element of a nicely normed $*$-algebra.

Then the imaginary part of $a$ is given by:

$\Im \left({a}\right) = \dfrac {a - a^*} 2$

Norm

Let $a \in A$ be an element of a nicely normed $*$-algebra.

Then we can define a norm on $a$ by:

$\left\Vert{a}\right\Vert^2 = a \oplus a^*$

Sources

• John C. Baez: The Octonions (2002): 2.2 The Cayley-Dickson Construction