$C(SO_q(4)/SO_q(2))$ as a Groupoid $C^*$-algebra
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| Format: | Preprint |
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2026
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| _version_ | 1866910224362766336 |
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| author | Bhatt, Shreema Subhash Deshpande, Vinay Saurabh, Bipul |
| author_facet | Bhatt, Shreema Subhash Deshpande, Vinay Saurabh, Bipul |
| contents | In this paper, we prove that $C(SO_q(4)/SO_q(2))$ is isomorphic to the $C^*$-algebra of the tight groupoid $\mathcal{G}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit $C(SO_0(4)/SO_0(2))$. We show that all four orbits of the unit space $\mathcal{G}_{\mathrm{tight}}^{(0)}$ under the natural action of $\mathcal{G}_{\mathrm{tight}}$ are locally closed, and that the associated isotropy groups are isomorphic to $\mathbb{Z}$. Consequently, every irreducible representation of $C^*(\mathcal{G}_{\mathrm{tight}})$ is induced from an irreducible representation of $C^*(\mathbb{Z})$, which are parametrized by $\mathbb{T}$. In this way, we obtain four families of irreducible representations parametrized by $\mathbb{T}$, and we explicitly construct their equivalence with the corresponding Soibelman irreducible representations of $C(SO_q(4)/SO_q(2))$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_10047 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $C(SO_q(4)/SO_q(2))$ as a Groupoid $C^*$-algebra Bhatt, Shreema Subhash Deshpande, Vinay Saurabh, Bipul Operator Algebras 22A22, 46L05, 46L67 In this paper, we prove that $C(SO_q(4)/SO_q(2))$ is isomorphic to the $C^*$-algebra of the tight groupoid $\mathcal{G}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit $C(SO_0(4)/SO_0(2))$. We show that all four orbits of the unit space $\mathcal{G}_{\mathrm{tight}}^{(0)}$ under the natural action of $\mathcal{G}_{\mathrm{tight}}$ are locally closed, and that the associated isotropy groups are isomorphic to $\mathbb{Z}$. Consequently, every irreducible representation of $C^*(\mathcal{G}_{\mathrm{tight}})$ is induced from an irreducible representation of $C^*(\mathbb{Z})$, which are parametrized by $\mathbb{T}$. In this way, we obtain four families of irreducible representations parametrized by $\mathbb{T}$, and we explicitly construct their equivalence with the corresponding Soibelman irreducible representations of $C(SO_q(4)/SO_q(2))$. |
| title | $C(SO_q(4)/SO_q(2))$ as a Groupoid $C^*$-algebra |
| topic | Operator Algebras 22A22, 46L05, 46L67 |
| url | https://arxiv.org/abs/2604.10047 |