An approximate equivalence for the GNS representation of the Haar state of $S{U}_q(2)$

We use the crystallised $C^{\ast }$-algebra $C(S{U}_q(2))$ at $q=0$ to obtain a unitary that gives an approximate equivalence involving the GNS representation on the $L^2$ space of the Haar state of the quantum $SU(2)$ group and the direct integral of all the infinite dimensional irreducible representations of the $C^{\ast }$-algebra $C(S{U}_q(2))$ for nonzero values of the parameter $q$. This approximate equivalence gives a $KK$ class via the Cuntz picture in terms of quasihomomorphisms as well as a Fredholm representation of the dual quantum group $\widehat{S{U}_q(2)}$ with coefficients in a $C^{\ast }$-algebra in the sense of Mishchenko.