Unbounded bivariant K-theory and correspondences in noncommutative geometry

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By introducing a notion of smooth connection for unbounded KK-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable C*-module. The theory of operator spaces provides the required tools. Finally, the above mentioned KK-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples.