Projections in 𝐿¹(𝐺): the unimodular case

Alaghmandan, Mahmood; Ghandehari, Mahya; Spronk, Nico; Taylor, Keith

doi:10.1090/proc/13142

Projections in $L^1(G)$: the unimodular case
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by Mahmood Alaghmandan, Mahya Ghandehari, Nico Spronk and Keith F. Taylor

Proc. Amer. Math. Soc. 144 (2016), 4929-4941

DOI: https://doi.org/10.1090/proc/13142

Published electronically: April 27, 2016

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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 $\textrm {SL}_2({\mathbb R})$ and all second countable almost connected nilpotent locally compact groups.

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