A well-known theorem by Connes and Skandalis provides sufficient conditions for finding the Kasparov product of two (bounded) Kasparov modules. In the unbounded picture of KK-theory, a theorem by Kucerovsky provides analogous conditions for the Kasparov product of two unbounded Kasparov modules.

While Kucerovsky’s theorem has been very influential for the construction of the unbounded Kasparov product, I will explain that it still has a few shortcomings. Namely, Kucerovsky’s positivity condition is global in nature, while the positivity condition of Connes-Skandalis is only local. Furthermore, while the Connes-Skandalis conditions depend only on the principal symbol of the operators, Kucerovsky’s conditions depend also on the subprincipal symbol.

In this talk, I will generalise Kucerovsky’s theorem such that the aforementioned shortcomings are removed. In particular, I will explain the construction of a ‘local representative’ for the KK-class of a cycle in unbounded KK-theory. A major advantage of this localisation procedure is that it furthermore allows to deal with Hilsum’s half-closed modules, which are cycles in unbounded KK-theory with symmetric but possibly non-selfadjoint operators.

This talk is based on https://arxiv.org/abs/2006.10616

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