Cayley-Dickson Construction from Real Star-Algebra is Commutative

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Theorem

Let $A = \struct {A_F, \oplus}$ be a $*$-algebra.

Let $A' = \struct {A_F, \oplus'}$ be constructed from $A$ using the Cayley-Dickson construction.


Then $A$ is a real star-algebra if and only if $A'$ is a commutative algebra.


Proof

Let the conjugation operator on $A$ be $*$.

Let $\tuple {a, b}, \tuple {c, d} \in A'$.


Let $A$ be a real star-algebra.

\(\ds \tuple {a, b} \oplus' \tuple {c, d}\) \(=\) \(\ds \tuple {a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}\)
\(\ds \) \(=\) \(\ds \tuple {a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}\)
\(\ds \) \(=\) \(\ds \tuple {a \oplus c - d \oplus b, a \oplus d + c \oplus b}\) Definition of Real Star-Algebra: $a = a^*$ and $b = b^*$
\(\ds \) \(=\) \(\ds \tuple {c \oplus a - b \oplus d, c \oplus b + a \oplus d}\) Real Star-Algebra is Commutative, Real Addition is Commutative
\(\ds \) \(=\) \(\ds \tuple {c \oplus a - b \oplus d^*, c^* \oplus b + a \oplus d}\) Definition of Real Star-Algebra: $d = d^*$ and $c = c^*$
\(\ds \) \(=\) \(\ds \tuple {c, d} \oplus' \tuple {a, b}\)

So $A'$ is a commutative algebra.

$\Box$


Let $A'$ be a commutative algebra.

By picking apart the above equations, it is clear that for $A'$ to be a commutative algebra it is necessary for $A$ to be both real and commutative.

Hence the result.

$\blacksquare$


Sources

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