Type of Local von Neumann Algebras in Abelian Quantum Double Models

We show that the local von Neumann algebra on convex areas of the frustration-free ground state of abelian quantum double models is of type II∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$II_\infty $$\end{document}.