Nicely Normed Alternative Algebra is Normed Division Algebra
From ProofWiki
Theorem
$A = \left({A_F, \oplus}\right)$ 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 $\Im (a)$ and $\Im (b)$.
Prove the above statement.
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So as $A$ is an alternative algebra, it follows that $\oplus$ is associative for $a, b, a^*, b^*$.
So:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left \Vert{a b}\right \Vert^2\) | \(=\) | \(\displaystyle \left({a \oplus b}\right) \oplus \left({a \oplus b}\right)^*\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of norm in nicely normed $*$-algebra | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle a \oplus b \oplus \left({b^* \oplus a^*}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of conjugate | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle a \oplus \left({b \oplus b^*}\right) \oplus a^*\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Associativity of $\oplus$ (from above) | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \left \Vert{a}\right \Vert^2 \left \Vert{b}\right \Vert^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
$\blacksquare$
