Analytic Kasparov product for non-separable C^*-algebras

Abstract

We develop an analytic framework for the explicit construction of Kasparov products on uncountably generated Hilbert modules over non-separable C^*-algebras by extending the classical Kasparov bimodule picture. In particular, this yields a well-defined functor from the category of general C^*-algebras whose restriction to separable C^*-algebras coincides with classical KK-theory and which satisfies several properties expected of non-separable KK-theory. A key idea of the extension is to replace a single Fredholm operator on a Kasparov bimodule with a net and to encode the directed system of separable C^*-subalgebras. We also introduce the notion of freely complementable Hilbert modules and develop cropping techniques to extend Kasparov's technical theorem beyond the countably generated setting. It naturally gives rise to a sufficiently broad class of C^*-algebras that provides a non-commutative analogue of paracompactness and includes all σ-unital algebras.

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BibTeX

@misc{choi2026analytic,
  title = {{Analytic Kasparov product for non-separable C$^*$-algebras}},
  author = {Ikhan Choi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40351},
  eprint = {2610.40351},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40351},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Analytic Kasparov product for non-separable C^*-algebras
AU  - Ikhan Choi
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40351
UR  - https://arxiv.org/abs/2610.40351
AB  - We develop an analytic framework for the explicit construction of Kasparov products on uncountably generated Hilbert modules over non-separable C^*-algebras by extending the classical Kasparov bimodule picture. In particular, this yields a well-defined functor from the category of general C^*-algebras whose restriction to separable C^*-algebras coincides with classical KK-theory and which satisfies several properties expected of non-separable KK-theory. A key idea of the extension is to replace a single Fredholm operator on a Kasparov bimodule with a net and to encode the directed system of separable C^*-subalgebras. We also introduce the notion of freely complementable Hilbert modules and develop cropping techniques to extend Kasparov's technical theorem beyond the countably generated setting. It naturally gives rise to a sufficiently broad class of C^*-algebras that provides a non-commutative analogue of paracompactness and includes all σ-unital algebras.
ER  -

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