Nonunital spectral triples and metric completeness in unbounded KK-theory

Mesland, B.; Rennie, A. (2016)

Article / Letter to editor

We consider the general properties of bounded approximate units in non-self-adjoint operator algebras. Such algebras arise naturally from the differential structure of spectral triples and unbounded Kasparov modules. Our results allow us to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, self-adjointness and regularity of induced operators on tensor product C*-modules and the lifting of Kasparov products to the unbounded category. In particular, we prove novel existence results for quasicentral approximate units in non-self-adjoint operator algebras, allowing us to strengthen Kasparov's technical theorem and extend it to this realm. Finally, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product. (C) 2016 Elsevier Inc. All rights reserved.

• All authors: Mesland, B.; Rennie, A.
• Date: 2016-11-01
• Volume: 271
• Issue: 9
• Pages: 2460 - 2538
• DOI: doi:10.1016/j.jfa.2016.08.004