Isomorphisms of quantum spheres

For $n\in \mathbb{N}$ and $q\in [0,1[$, the Vaksman-Soibelman quantum sphere $S_q^{2n+1}$ is described by an associative algebra $\mathcal{A}(S_q^{2n+1})$ deforming the algebra of polynomial functions on the 2n+1 dimensional unit sphere. Its C*-enveloping algebra is known to be independent of the deformation parameter q. In contrast to what happens in the C*-algebraic setting, we show here that, for all $q,q^{\mathrm{\prime }}$ in the above range, $\mathcal{A}(S_q^{2n+1})$ is isomorphic to $\mathcal{A}(S_{q^{\mathrm{\prime }}}^{2n+1})$ only if $q=q^{\mathrm{\prime }}$.