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Journal of Operator Theory

Volume 82, Issue 1, Summer 2019  pp. 115-145.

On a theorem of Kucerovsky for half-closed chains

Authors:  Jens Kaad (1), Walter D. van Suijlekom (2)
Author institution:(1) Department of Mathematics and Computer Science, Syddansk Universitet, Campusvej 55, 5230, Odense M, Denmark
(2) Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Summary: Kucerovsky's theorem provides a method for recognizing the interior Kasparov product of selfadjoint unbounded cycles. In this paper we present a partial extension of 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.

Keywords: unbounded Kasparov modules, half-closed chains, unbounded modular cycles, $KK$-theory, unbounded $KK$-theory, Kasparov product, unbounded Kasparov product

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