n}z_i$$

The proof is straightforward.

\[VglSphaeren\] We have $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}\neq S_{{ {\mathbb R} },{\varepsilon}'}^{n-1}$ for ${\varepsilon}\neq{\varepsilon}'$ in the sense that there is no $^*$-isomorphism $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C(S_{{ {\mathbb R} },{\varepsilon}'}^{n-1})$ sending generators to generators. Moreover, $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$ is noncommutative as soon as ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$.

If ${\varepsilon}\neq {\varepsilon}'$, we may find indices $i$ and $j$ such that ${\varepsilon}_{ij}=1$ and ${\varepsilon}'_{ij}=0$ (possibly after swapping the names for ${\varepsilon}$ and ${\varepsilon}'$). Assume that there is a $^*$-homomorphism $\psi:C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C(S_{{ {\mathbb R} },{\varepsilon}'}^{n-1})$ mapping generators to generators. Composing it with $\sigma_{{\varepsilon}'}\circ{\varphi}_{{\varepsilon}'}$ of the above lemma yields a contradiction, since $x_i$ and $x_j$ commute in $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$, but their images under $\sigma_{{\varepsilon}'}\circ{\varphi}_{{\varepsilon}'}\circ\psi$ do not. Noncommutativity of $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$ for ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$ follows directly from applying $\sigma_{\varepsilon}\circ{\varphi}_{\varepsilon}$.

Let ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$ and ${\varepsilon}\neq{\varepsilon}_{\operatorname{free}}$. Seen as noncommutative compact spaces, we have: $$S_{{ {\mathbb R} }}^{n-1}\subsetneq S_{{ {\mathbb R} },{\varepsilon}}^{n-1}\subsetneq S_{{ {\mathbb R} },+}^{n-1}$$ This means, we have surjective but non-injective $*$-homomorphisms $$C(S_{{ {\mathbb R} }}^{n-1})\leftarrow C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1}) \leftarrow C(S_{{ {\mathbb R} },+}^{n-1})$$ sending generators to generators.

The study of noncommutative spheres has a long history and goes back to Podleś [@QSphere]; see also the work of 