Compactification of quasi-local algebras on the lattice

TL;DR

This paper introduces a new compactification method for quasi-local C*-algebras on lattices, connecting infinite-volume and periodic observables, and explores implications for topological symmetries.

Contribution

It presents a functorial compactification construction for quasi-local algebras that generalizes Ocneanu's Tube algebra and links different boundary conditions.

Findings

Recovers Ocneanu's Tube algebra in 1D

Establishes a connection between infinite-volume and periodic observables

Derives obstructions for implementing certain topological symmetries

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.