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We establish the continuous homotopy invariance of bivariant local cyclichomology on the category of all \sigma-C^*-algebras. The argument reliesvitally on an isomorphism between the smooth and continuous cylinderconstructions using a technical criterion due to Meyer. As a consequence wecompute the local cyclic homology of the infinite sphere.

Continuous homotopy invariance of bivariant local cyclic homology for \sigma-C^*-algebras