On connected Lie groups and the Approximation Property

TL;DR

This paper provides a simplified proof characterizing connected Lie groups with the Approximation Property, extends the result to almost connected groups using property (T*), and discusses limitations beyond this class.

Contribution

It offers a shorter proof for the characterization and extends the result to almost connected groups, avoiding the need for property (T*).

Findings

Complete characterization of connected Lie groups with the Approximation Property.

Extension of the characterization to almost connected groups using property (T*).

Discussion on the limitations of extending beyond almost connected groups.

Abstract

Recently, a complete characterization of connected Lie groups with the Approximation Property was given. The proof used of the newly introduced property (T*). We present here a short proof of the same result avoiding the use of property (T*). Using property (T*), however, the characterization is extended to every almost connected group. We end with some remarks about the impossibility of going beyond the almost connected case.