Spectral Triples for Hensel-Steinitz Algebras
Shelley Hebert, Slawomir Klimek, Matt McBride
TL;DR
Addresses explicit $K$-homology representatives for the HS(s) algebras. Using the six-term exact sequence and the identification $I_s \cong C({\mathbb Z}_s^\times)\otimes \mathcal K$, together with the bootstrap category and the UCT, the paper identifies $K^0(HS(s))\cong \operatorname{Hom}(C_{\{1\}}({\mathbb Z}_s^\times,{\mathbb Z})\oplus{\mathbb Z},{\mathbb Z})\cong \operatorname{Hom}(C_{\{1\}}({\mathbb Z}_s^\times,{\mathbb Z}),{\mathbb Z})\oplus{\mathbb Z}$ and constructs explicit generators. It then provides concrete odd/even Fredholm modules corresponding to the dual functionals $e_{(y)}$ and evaluation maps, with an index computation matching $\Phi(1_{(x)})$. Finally, it builds even spectral triples on a smooth-Lipschitz subalgebra $HS_L^\infty(s)$ representing the same $K^0$-classes via the index pairing, with a Dirac operator defined by a growth function $\Lambda(y,l)$ ensuring compact resolvent and bounded commutators.
Abstract
We explicitly construct Fredholm modules and spectral triples representing any element of $K$-homology groups of Hensel-Steinitz algebras.