On the Unbounded Picture of -Theory
In the founding paper on unbounded -theory, it was established by Baaj and Julg that the bounded transform, which associates a class in -theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbou...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210766 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the Unbounded Picture of -Theory. Jens Kaad. SIGMA 16 (2020), 082, 21 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862643579053146112 |
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| author | Kaad, Jens |
| author_facet | Kaad, Jens |
| citation_txt | On the Unbounded Picture of -Theory. Jens Kaad. SIGMA 16 (2020), 082, 21 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In the founding paper on unbounded -theory, it was established by Baaj and Julg that the bounded transform, which associates a class in -theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbounded Kasparov modules, and we thereby describe the kernel of the bounded transform. This allows us to introduce a notion of topological unbounded -theory, which becomes isomorphic to -theory via the bounded transform. The equivalence relation is formulated entirely at the level of unbounded Kasparov modules and consists of homotopies together with an extra degeneracy condition. Our degenerate unbounded Kasparov modules are called spectrally decomposable since they admit a decomposition into a part with positive spectrum and a part with negative spectrum.
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| first_indexed | 2026-03-15T08:37:20Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210766 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T08:37:20Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kaad, Jens 2025-12-17T14:31:19Z 2020 On the Unbounded Picture of -Theory. Jens Kaad. SIGMA 16 (2020), 082, 21 pages 1815-0659 2020 Mathematics Subject Classification: 19K35; 58B34 arXiv:1901.05161 https://nasplib.isofts.kiev.ua/handle/123456789/210766 https://doi.org/10.3842/SIGMA.2020.082 In the founding paper on unbounded -theory, it was established by Baaj and Julg that the bounded transform, which associates a class in -theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbounded Kasparov modules, and we thereby describe the kernel of the bounded transform. This allows us to introduce a notion of topological unbounded -theory, which becomes isomorphic to -theory via the bounded transform. The equivalence relation is formulated entirely at the level of unbounded Kasparov modules and consists of homotopies together with an extra degeneracy condition. Our degenerate unbounded Kasparov modules are called spectrally decomposable since they admit a decomposition into a part with positive spectrum and a part with negative spectrum. The starting point for this paper was a series of conversations with Bram Mesland during the thematic program on Bivariant -theory in Geometry and Physics at the Erwin Schrödinger Institute in Vienna in November 2018. I would like to thank the ESI for their hospitality and support, and the organizers of the thematic programme for this great opportunity to meet and discuss the unbounded approach to -theory and its applications in mathematical physics. As always, I am also grateful to my friends and collaborators, Magnus Goeng, Bram Mesland, and Adam Rennie, for many insightful conversations on unbounded -theory and its relationship to -theory. Finally, I would like to thank the anonymous referee for her/his comments regarding spectral decomposability and its relationship to the work of Skandalis. The author was partially supported by the DFF-Research Project 2 Automorphisms and Invariants of Operator Algebras, no. 7014-00145B. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Unbounded Picture of -Theory Article published earlier |
| spellingShingle | On the Unbounded Picture of -Theory Kaad, Jens |
| title | On the Unbounded Picture of -Theory |
| title_full | On the Unbounded Picture of -Theory |
| title_fullStr | On the Unbounded Picture of -Theory |
| title_full_unstemmed | On the Unbounded Picture of -Theory |
| title_short | On the Unbounded Picture of -Theory |
| title_sort | on the unbounded picture of -theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210766 |
| work_keys_str_mv | AT kaadjens ontheunboundedpictureoftheory |