An approximate equivalence for the GNS representation of the Haar state of $SU_{q}(2)$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909248652312576 |
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| author | Chakraborty, Partha Sarathi Pal, Arup Kumar |
| author_facet | Chakraborty, Partha Sarathi Pal, Arup Kumar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_06908 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An approximate equivalence for the GNS representation of the Haar state of $SU_{q}(2)$ Chakraborty, Partha Sarathi Pal, Arup Kumar Operator Algebras Quantum Algebra 58B32, 46L67, 19K35 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. |
| title | An approximate equivalence for the GNS representation of the Haar state of $SU_{q}(2)$ |
| topic | Operator Algebras Quantum Algebra 58B32, 46L67, 19K35 |
| url | https://arxiv.org/abs/2203.06908 |