An approximate equivalence for the GNS representation of the Haar state of $$SU_{q}(2)$$

We use the crystallised \(C^*\)-algebra \(C(SU_{q}(2))\) 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 \(C^{*}\)-algebra \(C(SU_{q}(2))\) 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 \(\widehat{SU_q(2)}\) with coefficients in a \(C^*\)-algebra in the sense of Mishchenko.