$C^*$-algebras over spheres with fibres noncommutative tori

Abstract

All $C^*$-algebras of sections of locally trivial $C^*$-algebra bundles over $\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}$ with fibres $M_c(A_{\omega})$ are constructed under the assumption that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras. It is shown that each $C^*$-algebra of sections of a locally trivial $C^*$-algebra bundle over $\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}$ with fibres $M_c(A_{\omega})$ is stably isomorphic to $C(\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}) \otimes M_c(A_{\omega})$.

Let $A_{cd}$ be a $cd$-homogeneous $C^*$-algebra over $\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1} \times \mathbb{T}^{r+2}$ of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torus $\mathbb{S}_{\rho}^{cd}$ is defined by twisting $C^* (\widehat{\mathbb{T}^{r+2}} \times \mathbb{Z}^{m-2})$ in $A_{cd} \otimes C^* (\mathbb{Z}^{m-2})$ by a totally skew multiplier $\rho$ on $\widehat{\mathbb{T}^{r+2}} \times \mathbb{Z}^{m-2}$. We prove that $\mathbb{S}_{\rho}^{cd} \otimes M_{p\infty}$ is isomorphic to $C(\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}) \otimes C^* (\widehat{\mathbb{T}^{r+2}} \times \mathbb{Z}^{m-2}, \rho) \otimes M_{cd} (\mathbb{C}) \otimes M_{p\infty}$ if and only if the set of prime factors of $cd$ is a subset of the set of those of $p$.