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

Partha Sarathi Chakraborty () and Arup Kumar Pal ()
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Partha Sarathi Chakraborty: Indian Statistical Institute
Arup Kumar Pal: Indian Statistical Institute

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 3, 879-892

Abstract: 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.

Keywords: Quantum groups; Representations; Approximate equivalence; 20G42; 58B32; 46L67; 19K35 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-024-00633-0

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