Projections in 𝐿¹(𝐺): the unimodular case

Abstract

We consider the issue of describing all self-adjoint idempotents (projections) in L 1 ( G ) L^1(G) when G 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 G and the topology of the dual space of G G . We obtain an explicit description of any projection in L 1 ( G ) 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 ) L^1(G) for G G belonging to a class of groups that includes SL 2 ( R ) \textrm {SL}_2({\mathbb R}) and all second countable almost connected nilpotent locally compact groups.