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.$\;\lambda(ab)=(\lambda a)b$
$\; \parallel ab\parallel \leq \parallel a \parallel \parallel b \parallel,\;\;\; \parallel \lambda a\parallel=\parallel \lambda \parallel \parallel a\parallel$
$\;(ab)^{*}=b^{*}a^{*}$
4.$\;\;\parallel ab\parallel \leq \parallel a \parallel \parallel b \parallel$
$\;\; \parallel aa^{*} \parallel= \parallel a \parallel^{2}$
(If A is unital) $\lambda 1 \lambda' 1= \lambda \lambda' 1$
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
Banach algebraic proof of the Borsuk Ulam theorem
Edit: According to the answer of Andre Henriques we ask:
- Assume that $A$ is a real $C^{*}$ algebra which contains the Quaternions $H$. Under what algebraic conditions, $A$ is in the form $H(X)$=The space of continuous functions $f:X \to H$ for some compact Hausdorff $X$?
