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.$\;\lambda(ab)=(\lambda a)b$ 2. $\; \parallel ab\parallel \leq \parallel a \parallel \parallel b \parallel,\;\;\; \parallel \lambda a\parallel=\parallel \lambda \parallel \parallel a\parallel$ 3. $\;(ab)^{*}=b^{*}a^{*}$ 4.$\;\;\parallel ab\parallel \leq \parallel a \parallel \parallel b \parallel$ 5. $\;\; \parallel aa^{*} \parallel= \parallel a \parallel^{2}$ 6. (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:** > 1. Is it true to say that the spectrum is always non empty and compact? > >2. 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 https://mathoverflow.net/questions/218679/banach-algebraic-proof-of-the-borsuk-ulam-theorem **Edit:** According to the answer of Andre Henriques we ask: >3. 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 $X$?