An Atiyah-Segal homomorphism for approximate representations of infinite groups, and almost multiplicative ucp maps
Abstract
The Atiyah-Segal homomorphism relates the representation ring of a finite group to the K-theory ring of its classifying space. We develop a version of this for infinite groups, but based on a ring of approximate representations rather than representations. We then use known results on the Baum-Connes conjecture to show that our homomorphism is an isomorphism for many groups: this includes all torsion-free amenable groups, and all torsion-free fundamental groups of compact manifolds of dimension at most three. The main motivation is the desire to understand approximate representations using topological tools: as a sample application we give some examples of torsion phenomena in approximate representation theory. The methods of proof are C*-algebraic: the key idea is to relate our Atiyah-Segal type homomorphism to Kasparov's coassembly map. The main step realizes (Hausdorffized) KK-theory as approximately multiplicative ucp maps modulo stable approximate unitary equivalence for appropriate C*-algebras. This is in the spirit of the stable uniqueness theorems of Dadarlat-Eilers and Lin, and may be of purely C*-algebraic interest.
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BibTeX
@misc{willett2026atiyahsegal,
title = {{An Atiyah-Segal homomorphism for approximate representations of infinite groups, and almost multiplicative ucp maps}},
author = {Rufus Willett},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24252},
eprint = {2610.24252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24252},
}
RIS
TY - UNPB
TI - An Atiyah-Segal homomorphism for approximate representations of infinite groups, and almost multiplicative ucp maps
AU - Rufus Willett
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24252
UR - https://arxiv.org/abs/2610.24252
AB - The Atiyah-Segal homomorphism relates the representation ring of a finite group to the K-theory ring of its classifying space. We develop a version of this for infinite groups, but based on a ring of approximate representations rather than representations. We then use known results on the Baum-Connes conjecture to show that our homomorphism is an isomorphism for many groups: this includes all torsion-free amenable groups, and all torsion-free fundamental groups of compact manifolds of dimension at most three. The main motivation is the desire to understand approximate representations using topological tools: as a sample application we give some examples of torsion phenomena in approximate representation theory. The methods of proof are C^*-algebraic: the key idea is to relate our Atiyah-Segal type homomorphism to Kasparov's coassembly map. The main step realizes (Hausdorffized) KK-theory as approximately multiplicative ucp maps modulo stable approximate unitary equivalence for appropriate C^*-algebras. This is in the spirit of the stable uniqueness theorems of Dadarlat-Eilers and Lin, and may be of purely C^*-algebraic interest.
ER -