Group of automorphisms for strongly quasi invariant states

TL;DR

This paper investigates the properties of $G$-quasi invariant states under automorphism groups on $C^*$- and von Neumann algebras, extending theories from invariant states to broader quasi-invariant contexts.

Contribution

It develops new properties of $G$-strongly quasi invariant states and explores their relationships with modular automorphisms, invariant subalgebras, and ergodicity, broadening the understanding of automorphism group actions.

Findings

Extended theories from $G$-invariant to $G$-quasi invariant states.

Analyzed relationships with modular automorphism groups.

Provided examples illustrating the theoretical results.

Abstract

For a $*$-automorphism group $G$ on a $C^*$- or von Neumann algebra, we study the $G$-quasi invariant states and their properties. The $G$-quasi invariance or $G$-strongly quasi invariance are weaker than the $G$-invariance and have wide applications. We develop several properties for $G$-strongly quasi invariant states. Many of them are the extensions of the already developed theories for $G$-invariant states. Among others, we consider the relationship between the group $G$ and modular automorphism group, invariant subalgebras, ergodicity, modular theory, and abelian subalgebras. We provide with some examples to support the results.