Posted on 7th October 2019

A preprint arXiv:1907.03661 which has been on the arXiv for a while, but which is now submitted. The paper is partly an exposition of some techniques for dealing with Analytic Generators of one-parameter isometry groups: as people explore Type III arguments in von Neumann algebras, I want to revisit these old(ish) ideas and show that developing theory can be rather useful.

My main motivation was the study of locally compact quantum groups and in particular the fact that the antipode can be understood through studying the scaling group. One of the facets of topological quantum group theory is the interaction between the \( C^* \)-algebraic and von Neumann algebraic theories; in particular we have an inclusion \( \newcommand{\G}{\mathbb G}C_0(\G) \rightarrow L^\infty(\G) \) which intertwines the relevant scaling groups. I show that in this general setup, there is a Kaplansky Density type result for the analytic generators involved. The techniques of the proof also allow me to prove an "automatic normality" result, giving a description of \( L^1_\sharp(\G) \) the natural dense \( \ast \)-subalgebra of \( L^1(\G) \).