Group of automorphisms for strongly quasi invariant states
Ameur Dhahri, Chul Ki Ko, Hyun Jae Yoo
TL;DR
This paper addresses the analysis of states that are quasi invariant under a group of $*$-automorphisms $G$ on a $C^*$- or von Neumann algebra, focusing on $G$-quasi invariant and $G$-strongly quasi invariant states and their connections to the modular automorphism group. The authors develop a cocycle framework with Radon–Nikodym derivatives $x_g$, relate invariance to fixed subalgebras $\mathcal{F}(G)$ and centralizers, and construct conditional expectations $E_G$ for amenable groups. They establish modular-theoretic relations in this seting, characterize $G$-strongly quasi invariant states via tracial representations, and provide several quantum and classical examples including spin systems and Gibbs measures to illustrate the theory. Finally, they introduce projection techniques onto the $G$-invariant subspace and analyze abelianness criteria for the invariant algebras under the action. The results extend the canonical theory of $G$-invariant states to the weaker quasi invariant regime and offer tools for studying symmetry, ergodicity, and modular structure in quantum dynamics.
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.