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# Note on Kasparov Product of $C^*$-algebra Extensions

Published:2012-03-05
Printed: Dec 2013
• Changguo Wei,
School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China
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## Abstract

Using the Dadarlat isomorphism, we give a characterization for the Kasparov product of $C^*$-algebra extensions. A certain relation between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when $B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general.
 Keywords: extension, Kasparov product, $KK$-group
 MSC Classifications: 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]

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