Definition:Nicely Normed Star-Algebra

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

Then $A$ is a nicely normed $*$-algebra :
 * $\forall a \in A: a + a^* \in \R$
 * $\forall a \in A, a \ne 0: 0 < a \oplus a^* = a^* \oplus a \in \R$

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

Then the real part of $a$ is given by:
 * $\map \Re a = \dfrac {a + a^*} 2$

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

Then the imaginary part of $a$ is given by:
 * $\map \Im a = \dfrac {a - a^*} 2$

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

Then we can define a norm on $a$ by:
 * $\norm a^2 = a \oplus a^*$