Equivalence of categories of bivariant K-theory for C*-algebras over topological spaces via reflection functors

@misc{hashimoto2026equivalence,
  title = {{Equivalence of categories of bivariant K-theory for C*-algebras over topological spaces via reflection functors}},
  author = {Nanami Hashimoto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.22317},
  eprint = {2608.22317},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22317},
}

TY  - UNPB
TI  - Equivalence of categories of bivariant K-theory for C*-algebras over topological spaces via reflection functors
AU  - Nanami Hashimoto
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2608.22317
UR  - https://arxiv.org/abs/2608.22317
AB  - In this paper, for various pairs of topological spaces X and Y, we introduce an analogue of the Bernšte\uın-Gel'fand-Ponomarev reflection functors between the categories of C*-algebras over X and Y. We prove that these functors induce equivalences KK(X)_loc ≃ KK(Y)_loc between Meyer and Nest's localizing subcategories in Kirchberg's ideal-related KK-theory, and E(X) ≃ E(Y) between the entire categories in Dadarlat and Meyer's ideal-related E-theory. Both equivalences restrict to the respective bootstrap categories: ℬ(X) ≃ ℬ(Y) and ℬ_E(X) ≃ ℬ_E(Y). Combining these equivalences with a combinatorial argument in quiver representation theory, we obtain the same equivalences for any two finite T_0-spaces whose Hasse diagrams are orientations of the same tree. Moreover, we prove a rearrangement property of reflection functors, which yields an analogue of Coxeter functors. Finally, we apply reflection functors to give a conceptual alternative proof of Bentmann and Köhler's universal coefficient theorem for filtrated K-theory over finite T_0-spaces whose Hasse diagrams are orientations of Dynkin diagrams of type A.
ER  -