Nonunital spectral triples and metric completeness in unbounded KK-theory

By B. Mesland, A. Rennie. 2015

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Abstract 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.

Subfields: Algebra and Number Theory, Astronomy and Astrophysics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d7da8550c604087dd06704dd37601aa52861d93