Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative

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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 nicely normed alternative algebra if and only if $A$ is a nicely normed associative algebra.


Proof

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

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


In order to streamline notation, let $\oplus$ and $\oplus'$ both be denoted by product notation:

\(\ds a \oplus b\) \(=:\) \(\ds a b\)
\(\ds x \oplus' y\) \(=:\) \(\ds x y\)

The context will make it clear which is meant.


Let $A$ be a nicely normed associative algebra.

Then:

\(\ds \paren {\tuple {a, b} \tuple {a, b} } \tuple {c, d}\) \(=\) \(\ds \tuple {a a - b b^*, a^* b + a b} \tuple {c, d}\)
\(\ds \) \(=\) \(\ds \tuple {\paren {a a - b b^*} c - d \paren {a^* b + a b}^*, \paren {a a - b b^*}^* d + c \paren {a^* b + a b} }\)
\(\ds \) \(=\) \(\ds \tuple {\paren {a a - b b^*} c - d \paren {b^* a + b^* a^*}, \paren {a^* a^* - b b^*} d + c \paren {a^* b + a b} }\) Definition of Conjugation on Algebra
\(\ds \) \(=\) \(\ds \tuple {a a c - b b^* c - d b^* a - d b^* a^*, a^* a^* d - b b^* d + c a^* b + c a b}\) $A$ is associative
\(\ds \) \(=\) \(\ds \tuple {a a c - b b^* c - d b^* \paren {a + a^*}, a^* a^* d - b b^* d + c \paren {a^* + a} b}\)
\(\ds \) \(=\) \(\ds \tuple {a a c - \norm b^2 c - d b^* \paren {2 \map \Re a}, a^* a^* d - \norm b^2 d + c \paren {2 \map \Re a} b}\) $A$ is nicely normed

Similarly:

\(\ds \tuple {a, b} \paren {\tuple {a, b} \tuple {c, d} }\) \(=\) \(\ds \tuple {a, b} \tuple {a c - d b^*, a^* d + c b}\)
\(\ds \) \(=\) \(\ds \tuple {a \paren {a c - d b^*} - \paren {a^* d + c b} b^*, a^* \paren {a^* d - c b} + \paren {a c + d b^*} b}\)
\(\ds \) \(=\) \(\ds \tuple {a a c - a d b^* - a^* d b^* - c b b^*, a^* a^* d + a^* c b + a c b - d b^* b}\) $A$ is associative
\(\ds \) \(=\) \(\ds \tuple {a a c - \paren {a + a^*} d b^* - c \norm b^2, a^* a^* d + \paren {a^* + a} c b - d b^* b}\)
\(\ds \) \(=\) \(\ds \tuple {a a c - \paren {2 \map \Re a} d b^* - c \norm b^2, a^* a^* d + \paren {2 \map \Re a} c b - d \norm b^2}\) $A$ is nicely normed


Thus it can be seen that:

$\paren {\tuple {a, b} \tuple {a, b} } \tuple {c, d} = \tuple {a, b} \paren {\tuple {a, b} \tuple {c, d} }$


Similarly it can be shown that:

$\paren {\tuple {c, d} \tuple {a, b} } \tuple {a, b} = \tuple {c, d} \paren {\tuple {a, b} \tuple {a, b} }$

and so $A'$ is seen to be an alternative algebra.


It follows from reversing the chain of equalities that if $A'$ is a nicely normed and alternative algebra then $A$ has to be a nicely normed associative algebra.

$\Box$


Then from Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed, we have that $A'$ is a nicely normed algebra if and only if $A$ is also a nicely normed algebra.

Hence the result.

$\blacksquare$


Sources

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