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 $\map \Im b$.



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)
\(\ds \) \(=\) \(\ds \norm a^2 \norm b^2\)

$\blacksquare$


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