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

TL;DR

This paper establishes an approximate equivalence between the GNS representation of the Haar state on quantum SU(2) and a direct integral of its irreducible representations, using crystallized C*-algebra at q=0.

Contribution

It introduces a novel approach connecting the GNS representation with irreducible representations via an approximate equivalence using crystallized C*-algebra at q=0.

Findings

Constructs a unitary for approximate equivalence

Derives a KK class via quasihomomorphisms

Provides a Fredholm representation of the dual quantum group

Abstract

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.