Journal of the
Korean Mathematical Society JKMS

We define the spherical non-commutative torus $\Bbb L_{\omega}$ as the crossed product obtained by an iteration of $l$ crossed products by actions of $\Bbb Z$, the first action on $C(S^{2n+1})$. Assume the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus $A_{\rho}$ with a matrix algebra $M_{m}(\Bbb C)$ $(m>1)$. We prove that $\Bbb L_{\omega} \otimes M_p(\Bbb C)$ is not isomorphic to $C(\operatorname{Prim}(\Bbb L_{\omega})) \otimes A_{\rho} \otimes M_{mp}(\Bbb C)$, and that the tensor product of $\Bbb L_{\omega}$ with a $UHF$-algebra $M_{p^{\infty}}$ of type $p^{\infty}$ is isomorphic to $C(\operatorname{Prim}(\Bbb L_{\omega})) \otimes A_{\rho} \otimes M_{m}(\Bbb C) \otimes M_{p^{\infty}}$ if and only if the set of prime factors of $m$ is a subset of the set of prime factors of $p$. Furthermore, it is shown that the tensor product of $\Bbb L_{\omega}$ with the $C^*$-algebra $\Cal K(\Cal H)$ of compact operators on a separable Hilbert space $\Cal H$ is not isomorphic to $C(\operatorname{Prim}(\Bbb L_{\omega})) \otimes A_{\rho} \otimes M_{m}(\Bbb C) \otimes \Cal K(\Cal H)$ if $\operatorname{Prim}(\Bbb L_{\omega})$ is homeomorphic to $L^k(n) \times \Bbb T^{l'}$ for $k$ and $l'$ non-negative integers $(k> 1)$, where $L^k(n)$ is the lens space.

Keywords: tensor product, crossed product, $K$-theory, homogeneous $C^*$-algebra, twisted group $C^*$-algebra, non-commutative torus, $UHF$-algebra, lens space, and Cuntz algebra

MSC numbers: Primary 46L05, 46L87, Secondary 55R15