Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties:
$\forall \lambda \in H, a,b \in A$
1.$\;\lambda(ab)=(\lambda a)b$
$\; \parallel ab\parallel \leq \parallel a \parallel \parallel b \parallel$
$\;(ab)^{*}=b^{*}a^{*}$
4.$\;\;\parallel ab\parallel \leq \parallel a \parallel \parallel b \parallel$
- $\;\; \parallel aa^{*} \parallel= \parallel a \parallel^{2}$
There is a natural definition of spectrum of an element $a\in A$ (as a subset of $H$). There is also a natural definition of morphism and isomorphisms between two $H^{*}$ algebras.
Example: For a compact Hausdorff space $X$ put $A=H(X)=$ The space of all continuous $f:X \to H$ with obvious structures
Questions:
- Is it true to say that the spectrum is always non empty and compact?
- Is it true to say that two compact space $X$ and $Y$ are homeomorphic if and only if $H(X) \simeq H(Y)$?
The motivation for this question is that we search for an alternative proof for the Borsuk Ulam theorem for $f;S^{4} \to \mathbb{R}^{4} \simeq H$ via consideration of a $\mathbb{Z}/2\mathbb{Z}$ graded structure for $H(S^{4})$. see the following related post