$q$-Independence of the $C^{\ast }$-algebra of "Continuous Functions" on a Quantum $2\times 2$-Matrix Ball

For $0<q<1,$ let $\mathrm{P}\mathrm{o}\mathrm{l}({\mathrm{M}\mathrm{a}\mathrm{t}}_2{)}_q$ be the $q$-analogue of regular functions on the open matrix ball of $2\times 2$-matrices introduced by L. Vaksman. We show that the universal enveloping $C^{\ast }$-algebras of $\mathrm{P}\mathrm{o}\mathrm{l}({\mathrm{M}\mathrm{a}\mathrm{t}}_2{)}_q,$ $0<q<1,$ are isomorphic for all values of $q\mathrm{.}$