We prove a generalization of the Kasparov technical theorem. It is known that KK-theory cycles can be defined using unbounded operators ([3]), even in the equivariant case ([15]). We apply this generalized Kasparov technical theorem to a problem involving the Kasparov product of cycles defined by unbounded operators. In some earlier work ([15] and [16]) we showed that the Kasparov product can be defined directly in terms of unbounded cycles, provided that the cycles satisfy certain conditions. In this paper we show that these conditions are necessary as well ad sufficient, after equivalence.
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