Spectral geometry of saturated Fell bundles
Abstract
Let ℬ=(Bg)g∈ G be a saturated Fell bundle over a countable discrete group, equipped with a spectral triple. We give explicit sufficient conditions for lifting this triple to the reduced cross-sectional C\Star-algebra Cr*(ℬ). Using finite projective realizations of the homogeneous Morita equivalence Be-bimodules, we transport the base Dirac operator to the individual fibres and assemble the resulting operators into a fibrewise Dirac operator. We introduce a spectral-regularity condition ensuring bounded commutators and give a sufficient criterion in terms of uniform derivative bounds on smooth factor-system data. Combining the fibrewise operator with a length operator associated with a proper translation-bounded matrix-valued length function yields the lifted spectral triple. When Be is separable, the lifted triple, with a suitable grading, represents the Kasparov product of the vertical class with the class of the unit-fibre triple. We also establish a summability criterion and sufficient conditions for stability under central cocycle twisting. A range of examples illustrates the scope of the construction, including twisted crossed products, free compact Abelian actions, and Fell bundles with nonfree homogeneous fibres.
Related
- Actions of Fell bundles, II: tensor products and Fourier-Stieltjes algebras
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BibTeX
@misc{wagner2026spectral,
title = {{Spectral geometry of saturated Fell bundles}},
author = {Stefan Wagner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15395},
eprint = {2610.15395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15395},
}
RIS
TY - UNPB
TI - Spectral geometry of saturated Fell bundles
AU - Stefan Wagner
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15395
UR - https://arxiv.org/abs/2610.15395
AB - Let ℬ=(B_g)_g∈ G be a saturated Fell bundle over a countable discrete group, equipped with a spectral triple. We give explicit sufficient conditions for lifting this triple to the reduced cross-sectional C\Star-algebra C_r^*(ℬ). Using finite projective realizations of the homogeneous Morita equivalence B_e-bimodules, we transport the base Dirac operator to the individual fibres and assemble the resulting operators into a fibrewise Dirac operator. We introduce a spectral-regularity condition ensuring bounded commutators and give a sufficient criterion in terms of uniform derivative bounds on smooth factor-system data. Combining the fibrewise operator with a length operator associated with a proper translation-bounded matrix-valued length function yields the lifted spectral triple. When B_e is separable, the lifted triple, with a suitable grading, represents the Kasparov product of the vertical class with the class of the unit-fibre triple. We also establish a summability criterion and sufficient conditions for stability under central cocycle twisting. A range of examples illustrates the scope of the construction, including twisted crossed products, free compact Abelian actions, and Fell bundles with nonfree homogeneous fibres.
ER -