Isomorphisms of quantum spheres

For n and q∈ [0,1[, the Vaksman-Soibelman quantum sphere S²ⁿ⁺¹q is described by an associative algebra A(S²ⁿ⁺¹q) 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 $q,q'$ in the above range, A(S²ⁿ⁺¹q) is isomorphic to A(S²ⁿ⁺¹q') only if $q=q'$.