# On the uniqueness of invariant states

## TL;DR

This paper investigates conditions for the uniqueness of invariant states in symplectic twisted group algebras associated with abelian groups, with applications to quantum Chern-Simons theory.

## Contribution

It introduces criteria for the uniqueness of invariant states in symplectic twisted group *-algebras and explores their implications in quantum abelian Chern-Simons theory.

## Key findings

- Criteria for the uniqueness of invariant states established.
- Application to natural states in quantum abelian Chern-Simons theory.
- Insights into ergodic actions of symplectic automorphism groups.

## Abstract

Given an abelian group G endowed with a T-pre-symplectic form, we assign to it a symplectic twisted group *-algebra W_G and then we provide criteria for the uniqueness of states invariant under the ergodic action of the symplectic group of automorphism. As an application, we discuss the notion of natural states in quantum abelian Chern-Simons theory.