Quasi-invariant states with uniformly bounded cocycles

TL;DR

The paper analyzes $G$-quasi-invariant states on von Neumann algebras under a group of automorphisms, focusing on uniformly bounded Radon–Nikodym cocycles. It proves that such states are associated with an invariant state via a bounded density $d$, with the cocycle given by $x_g=d\,g^{-1}(d^{-1})$, and provides a spatial unitary implementation of $G$ in the GNS framework. A Kovács–Szücs-type theorem yields a unique $G$-invariant conditional expectation onto the fixed-point algebra, with a complementary description in terms of a mean over $G$ in the strongly quasi-invariant case. Finally, for amenable $G$ acting ergodically on the center of a semifinite algebra, the work establishes a canonical $G$-invariant semifinite trace and characterizes the density transformation under the cocycle, reinforcing the structural role of bounded cocycles in noncommutative dynamical systems.

Abstract

We investigate the notion of quasi-invariant states introduced in [2, 3] from an analytic viewpoint.We give the structures of quasi-invariant states with uniformly bounded cocycles. As a consequence, we can apply a Theorem of Kovacs and Szucs to get a conditional expectation on fixed points and another of Stormer to get an invariant semifinite trace under extra assumptions.

Theorems & Definitions (11)

• Definition 2.1
• Lemma 2.2: SZ, 5.20. Lemma
• Remark 2.3
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 4.1
• Theorem 4.2
• Proposition 4.3
• Theorem 5.1