{\mathbb R} }^n$ (also called the *commutative sphere*) is given by: $$S_{{ {\mathbb R} }}^{n-1}=\{(x_1,\ldots,x_n)\in{ {\mathbb R} }^n\;|\;\sum_i x_i^2=1\}$$ The algebra of continuous functions over it may be written as a universal $C^*$-algebra: $$C(S_{{ {\mathbb R} }}^{n-1})=C^*(x_1,\ldots,x_n\;|\; x_i=x_i^*, \sum_i x_i^2=1, x_ix_j=x_jx_i \;\forall i,j)$$

A natural noncommutative analogue of the sphere is given by the *(maximaly) noncommutative sphere* as introduced by Banica and Goswami [@BanGos]: $$C(S_{{ {\mathbb R} },+}^{n-1}):=C^*(x_1,\ldots,x_n\;|\; x_i=x_i^*, \sum_i x_i^2=1)$$ In the philosophy of noncommutative compact spaces, we may speak of the noncommutative sphere $S_{{ {\mathbb R} },+}^{n-1}$ as a noncommutative compact space which is only defined via the algebra $C(S_{{ {\mathbb R} },+}^{n-1})$ of (noncommutative) functions over it. Our next definition is an interpolation between the above spheres governed by the matrix ${\varepsilon}$.

The *${\varepsilon}$-sphere* $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$ is defined via the universal $C^*$-algebra: $$C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1}):=C^*(x_1,\ldots,x_n\;|\; x_i=x_i^*, \sum_i x_i^2=1, x_ix_j=x_jx_i, \textnormal{ if } {\varepsilon}_{ij}=1)$$

Note, that the commutative sphere $S_{{ {\mathbb R} }}^{n-1}$ is an ${\varepsilon}$-sphere for the matrix ${\varepsilon}_{\textnormal{comm}}$ of Example \[ExEps\](a). Moreover, the noncommutative sphere $S_{{ {\mathbb R} },+}^{n-1}$ is an ${\varepsilon}$-sphere with respect to ${\varepsilon}_{\textnormal{free}}$ of Example \[ExEps\](b). The ${\varepsilon}$-sphere is noncommutative, if ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$, i.e. the $C^*$-algebra $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$ is 