Compactification of quasi-local algebras on the lattice
Jun Ikeda
TL;DR
This work constructs a functorial compactification of quasi-local algebras over countable metric spaces with group actions, connecting infinite-volume observables to periodic-boundary Tube algebra realizations in 1D and enabling a rigorous obstruction framework for implementing bulk TQFT symmetries on lattice models. By introducing local-generation/presentation and a parameterized (D, T)-compactified algebra, then extending to an asymptotic Λ-compactification, the authors show that fusion spin chain Tubes recover the full algebra in the infinite-volume limit and that bounded-spread automorphisms induce obstructions to braided autoequivalences of the bulk Drinfeld center. The framework yields concrete obstructions to Kramers-Wannier type dualities on symmetric subalgebras and offers a principled route to study symmetry implementation in topological lattice models, including abelian and nonabelian examples. Overall, the paper bridges AQFT, Tube algebra techniques, and lattice realizations of bulk topological orders, with potential implications for understanding symmetry actions and dualities in higher dimensions.
Abstract
We introduce a compactification construction for abstract quasi-local C*-algebras over countable metric spaces equipped with an isometric group action which is functorial with respect to bounded spread isomorphisms. In $1$D, the construction recovers Ocneanu's Tube algebra for fusion spin chains, and provides a canonical bridge between infinite-volume observables and observables with periodic boundary conditions. We exploit this connection to derive an obstruction for the implementability of such topological symmetries as Kramers-Wannier type dualities on symmetric subalgebras.