Real Star-Algebra is Commutative
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Theorem
Let $A = \struct {A_F, \oplus}$ be a real $*$-algebra whose conjugation is denoted as $*$.
Then:
- $\forall a, b \in A, a \oplus b = b \oplus a$
That is, real $*$-algebra is commutative.
Proof
\(\ds a \oplus b\) | \(=\) | \(\ds \paren {a \oplus b}^*\) | Definition of Real $*$-Algebra | |||||||||||
\(\ds \) | \(=\) | \(\ds b^* \oplus a^*\) | Definition of Conjugation on Algebra | |||||||||||
\(\ds \) | \(=\) | \(\ds b \oplus a\) | Definition of Real $*$-Algebra |
$\blacksquare$