On a theorem of Kucerovsky for half-closed chains

Kucerovsky’s theorem provides a method for recognizing the interior Kasparov product of selfadjoint unbounded cycles. In this paper we extend Kucerovsky’s theorem to the non-selfadjoint setting by replacing unbounded Kasparov modules with Hilsum’s half-closed chains. On our way we show that any half-closed chain gives rise to a multitude of twisted selfadjoint unbounded cycles via a localization procedure. These unbounded modular cycles allow us to provide verifiable criteria avoiding any reference to domains of adjoints of symmetric unbounded operators.