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Nonunital spectral triples and metric completeness in unbounded KK-theory

Journal Article


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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.

Publication Date


  • 2016

Citation


  • Mesland, B. & Rennie, A. C. (2016). Nonunital spectral triples and metric completeness in unbounded KK-theory. Journal of Functional Analysis, 271 (9), 2460-2538.

Scopus Eid


  • 2-s2.0-84994128787

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=7170&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/6140

Number Of Pages


  • 78

Start Page


  • 2460

End Page


  • 2538

Volume


  • 271

Issue


  • 9

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.

Publication Date


  • 2016

Citation


  • Mesland, B. & Rennie, A. C. (2016). Nonunital spectral triples and metric completeness in unbounded KK-theory. Journal of Functional Analysis, 271 (9), 2460-2538.

Scopus Eid


  • 2-s2.0-84994128787

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=7170&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/6140

Number Of Pages


  • 78

Start Page


  • 2460

End Page


  • 2538

Volume


  • 271

Issue


  • 9