In this paper we investigate the unbounded Kasparov product between a differentiable module and an unbounded cycle of a very general kind that includes all unbounded Kasparov modules and hence also all spectral triples. Our assumptions on the differentiable module are weak and we do in particular not require that it satisfies any kind of smooth projectivity conditions. The algebras that we work with are furthermore not required to possess a smooth approximate identity. The lack of an adequate projectivity condition on our differentiable module entails that the usual class of unbounded Kasparov modules is not flexible enough to accommodate the unbounded Kasparov product and it becomes necessary to twist the commutator condition by an automorphism.

We show that the unbounded Kasparov product makes sense in this twisted setting and that it recovers the usual interior Kasparov product after taking bounded transforms. Since our unbounded cycles are twisted (or modular) we are not able to apply the work of Kucerovsky for recognizing unbounded representatives for the bounded Kasparov product, instead we rely directly on the connection criterion developed by Connes and Skandalis. In fact, since we do not impose any twisted Lipschitz regularity conditions on our unbounded cycles, even the passage from an unbounded cycle to a bounded Kasparov module requires a substantial amount of extra care.

在本文中，我们研究了可微模块和一个非常普遍的无界循环之间的无界 Kasparov 乘积，该循环包括所有无界 Kasparov 模块，因此也包括所有谱三元组。我们对可微模块的假设很弱，我们特别不要求它满足任何类型的平滑投影条件。此外，我们使用的代数不需要具有平滑的近似恒等式。由于我们的可微模块缺乏足够的投影条件，因此通常的无界 Kasparov 模块类别不够灵活，无法容纳无界 Kasparov 乘积，因此有必要通过自同构来扭曲交换子条件。

我们表明无界 Kasparov 乘积在这种扭曲设置中是有意义的，并且它在进行有界变换后恢复了通常的内部 Kasparov 乘积。由于我们的无界环是扭曲的（或模块化的），我们无法应用 Kucerovsky 的工作来识别有界 Kasparov 乘积的无界代表，而是直接依赖于 Connes 和 Skandalis 开发的连接标准。事实上，由于我们没有对我们的无界环强加任何扭曲的 Lipschitz 正则条件，即使从无界环到有界 Kasparov 模块的过渡也需要大量额外的注意。