Trace class groups

TL;DR

This paper investigates trace class groups, characterizing their properties, criteria for semi-direct products to be trace class, and linking trace class representations to distribution spaces.

Contribution

It establishes that trace class groups are type I, provides criteria for semi-direct products to be trace class, and connects trace class representations with distribution space realizations.

Findings

Trace class groups are type I.

Criteria for semi-direct products to be trace class.

Trace class representations relate to distribution spaces.

Abstract

A representation $\pi$ of a locally compact group $G$ is called \e{trace class}, if for every test function $f$ the induced operator $\pi(f)$ is a trace class operator. The group $G$ is called \e{trace class}, if every $\pi\in G$ is trace class. We show that trace class groups are type I and give a criterion for semi-direct products to be trace class and show that a representation $\pi$ is trace class if and only if $\pi\otimes\pi'$ can be realized in the space of distributions.