Nicely Normed Alternative Algebra is Normed Division Algebra
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Theorem
$A = \struct {A_F, \oplus}$ be a nicely normed $*$-algebra which is also an alternative algebra.
Then $A$ is a normed division algebra.
Proof
Let $a, b \in A$.
Then all of $a, b, a^*, b^*$ can be generated by $\map \Im a$ and $\map \Im b$.
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So as $A$ is an alternative algebra, it follows that $\oplus$ is associative for $a, b, a^*, b^*$.
So:
\(\ds \norm {a b}^2\) | \(=\) | \(\ds \paren {a \oplus b} \oplus \paren {a \oplus b}^*\) | Definition of Norm in Nicely Normed $*$-Algebra | |||||||||||
\(\ds \) | \(=\) | \(\ds a \oplus b \oplus \paren {b^* \oplus a^*}\) | Definition of Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds a \oplus \paren {b \oplus b^*} \oplus a^*\) | Associativity of $\oplus$ (from above) | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm a^2 \norm b^2\) |
$\blacksquare$