J. Korean Math. Soc. 1997; 34(4): 859-869
Printed December 1, 1997
Copyright © The Korean Mathematical Society.
Chun-Gil Park
Chungnam National University
It is shown that for $A_{k,m}$ a $k$-homogeneous $C^*$-algebra over $S^{2n-1} \times S^1$ such that no non-trivial matrix algebra can be factored out of $A_{k,m}$ and $A_{k,m} \otimes M_l(\Bbb C)$ has a non-trivial bundle structure for any positive integer $l$, we construct an $A_{k,m}$-$C(S^{2n-1} \times S^1) \otimes M_k(\Bbb C)$-equivalence bimodule to show that every $k$-homogeneous $C^*$-algebra over $S^{2n-1} \times S^1$ is strongly Morita equivalent to $C(S^{2n-1} \times S^1)$. Moreover, we prove that the tensor product of the $k$-homogeneous $C^*$-algebra $A_{k,m}$ with a $UHF$-algebra of type $p^{\infty}$ has the trivial bundle structure if and only if the set of prime factors of $k$ is a subset of the set of prime factors of $p$.
Keywords: $UHF$-algebra, homogeneous $C^*$-algebra, strong Morita equivalence, equivalence bimodule, tensor product, and $K$-theory
MSC numbers: Primary 46L87, 46L05; Secondary 55R15
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