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

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  • Partha Sarathi Chakraborty

    (Indian Statistical Institute)

  • Arup Kumar Pal

    (Indian Statistical Institute)

Abstract

We use the crystallised $$C^*$$ C ∗ -algebra $$C(SU_{q}(2))$$ C ( S U q ( 2 ) ) at $$q=0$$ q = 0 to obtain a unitary that gives an approximate equivalence involving the GNS representation on the $$L^{2}$$ 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 $$C^{*}$$ C ∗ -algebra $$C(SU_{q}(2))$$ 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 $$\widehat{SU_q(2)}$$ S U q ( 2 ) ^ with coefficients in a $$C^*$$ C ∗ -algebra in the sense of Mishchenko.

Suggested Citation

  • Partha Sarathi Chakraborty & Arup Kumar Pal, 2024. "An approximate equivalence for the GNS representation of the Haar state of $$SU_{q}(2)$$ S U q ( 2 )," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(3), pages 879-892, September.
    • Handle: RePEc:spr:indpam:v:55:y:2024:i:3:d:10.1007_s13226-024-00633-0
    DOI: 10.1007/s13226-024-00633-0
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