Projections in $L^1(G)$; the unimodular case

TL;DR

This paper characterizes all self-adjoint idempotents in the group algebra $L^1(G)$ for unimodular groups, providing explicit descriptions for certain classes including $SL(2,R)$ and nilpotent groups.

Contribution

It offers a complete description of projections in $L^1(G)$ for a broad class of unimodular groups, extending known results to new group types.

Findings

Explicit description of projections in $L^1(G)$ for certain groups

Characterization of projections in the coefficient space of finite sums of irreducible representations

Applicable to groups including $SL(2,R)$ and almost connected nilpotent groups

Abstract

We consider the issue of describing all self-adjoint idempotents (projections) in $L^1(G)$ when $G$ is a unimodular locally compact group. The approach is to take advantage of known facts concerning subspaces of the Fourier-Stieltjes and Fourier algebras of $G$ and the topology of the dual space of $G$. We obtain an explicit description of any projection in $L^1(G)$ which happens to also lie in the coefficient space of a finite direct sum of irreducible representations. This leads to a complete description of all projections in $L^1(G)$ for $G$ belonging to a class of groups that includes $SL(2,R)$ and all almost connected nilpotent locally compact groups.