Continuity of States on Non-Unital Differential Algebras in Loop Quantum Cosmology

TL;DR

This paper corrects a previous proof of the uniqueness of a state in loop quantum cosmology and extends the algebraic framework to facilitate future research in higher-dimensional models.

Contribution

It fixes an error in the proof of state uniqueness in loop quantum cosmology and broadens the algebraic approach for potential higher-dimensional applications.

Findings

Corrected the proof of state uniqueness in homogeneous isotropic loop quantum cosmology.

Extended the operator algebraic framework for future higher-dimensional studies.

Maintained the core uniqueness result despite initial proof issues.

Abstract

In a recent paper, Engle, Hanusch and Thiemann showed that there is a unique state on the reduced holonomy-flux $\ast$-algebra of homogeneous isotropic loop quantum cosmology, that is invariant under residual diffeomorphims. This result has been claimed to be true both for the Ashtekar-Bojowald-Lewandowski framework and for that introduced by the present author. Unfortunately, the uniqueness proof relies on an incorrect argument which spoils the second case. In our short note, we are going to patch this issue, this way keeping the nice uniqueness result in both cases. Moreover, we will even extend the underlying operator algebraic statements as this might help later for studying higher-dimensional models.