$C_*$-algebras Associated with Lens Spaces
Commun. Korean Math. Soc. 1998 Vol. 13, No. 4, 759-764
Deok-Hoon Boo, Sei-Qwon Oh and Chun-Gil Park
Chungnam National University, Chungnam National University, Chungnam National University
Abstract : We define the rational lens algebra $\Bbb L_{\frac{m}{k}}(n)$ as the crossed product by an action of $\Bbb Z$ on $C(S^{2n+1})$. Assume the fibres are $M_k(\Bbb C)$. We prove that $\Bbb L_{\frac{m}{k}}(n) \otimes M_p(\Bbb C)$ is not isomorphic to $C(\operatorname{Prim}(\Bbb L_{\frac{m}{k}}(n)))\break \otimes M_{kp}(\Bbb C)$ if $k>1$, and that $\Bbb L_{\frac{m}{k}}(n) \otimes M_{p^{\infty}}$ is isomorphic to \break$C(\operatorname{Prim}(\Bbb L_{\frac{m}{k}}(n))) \otimes M_k(\Bbb C) \otimes M_{p^{\infty}}$ if and only if the set of prime factors of $k$ is a subset of the set of prime factors of $p$. It is moreover shown that if $k>1$ then $\Bbb L_{\frac{m}{k}}(n)$ is not stably isomorphic to $C(\operatorname{Prim}(\Bbb L_{\frac{m}{k}}(n))) \otimes M_k(\Bbb C)$.
Keywords : $K$-theory, $UHF$-algebra, crossed product, tensor product
MSC numbers : Primary 46L05, 46L87; Secondary 55R15
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