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For $0<q<1,$ let $\mathrm{Pol}(\mathrm{Mat}_{2})_q$ be the $q$-analogue ofregular functions on the open matrix ball of $2\times 2$-matrices introduced byL. Vaksman. We show that the universal enveloping $C^{*}$-algebras of$\mathrm{Pol}(\mathrm{Mat}_{2})_{q},$ $0<q<1,$ are isomorphic for all values of$q.$

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