Real Star-Algebra is Commutative

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

$\blacksquare$