THE TENSOR PRODUCTS OF SPHERICAL NON-COMMUTATIVE TORI WITH CUNTZ ALGEBRAS

The spherical non-commutative $\mathbb{S}_{\omega}$ were defined in [2,3]. Assume that no non-trivial matrix algebra can be factored out of the $\mathbb{S}_{\omega}$, and that the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus with a matrix algebra $M_k(\mathbb{C})$. It is shown that the tensor product of the spherical non-commutative torus $\mathbb{S}_{\omega}$ with the even Cuntz algebra $\mathcal{O}_{2d}$ has a trivial bundle structure if and only if k and 2d - 1 are relatively prime, and that the tensor product of the spherical non-commutative torus $S_{\omega}$ with the generalized Cuntz algebra $\mathcal{O}_{\infty}$ has a non-trivial bundle structure when k > 1.