The structure of quantum spheres (Q2731925)

In [Q. J. Math. 48, No. 192, 503-510 (1997; Zbl 0992.46060)], the author showed that the \(C^*\)-algebra \(C({\mathbb S}^{2n+1}_q)\) of the quantum sphere \({\mathbb S}^{2n+1}_q\), \(q>1\), is an extension of \(C({\mathbb S}^{2n-1}_q)\) by \(C({\mathbb T})\otimes{\mathcal K}\), where \(\mathbb T\) is the unit circle and \(\mathcal K\) is the \(C^*\)-algebra of compact operators. In the present paper, the author gives an explicit description of this extension. It is shown that \(C({\mathbb S}^{2n+1}_q)\) consists of continuous fields \(\{f_t\}_{t\in{\mathbb T}}\) of operators \(f_t\) in a \(C^*\)-algebra \(\mathcal A\), which contains \(\mathcal K\) with \({\mathcal A}/{\mathcal K}\cong C({\mathbb S}^{2n-1}_q)\), such that \(\rho_*(f_t)\) is a constant function of \(t\), where \(\rho:{\mathcal A}\to {\mathcal A}/{\mathcal K}\) is the quotient map.