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


Authors: Mahmood Alaghmandan, Mahya Ghandehari, Nico Spronk and Keith F. Taylor
Journal: Proc. Amer. Math. Soc. 144 (2016), 4929-4941
MSC (2010): Primary 43A20; Secondary 43A22
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 $ {\rm SL}_2({\mathbb{R}})$ and all second countable almost connected nilpotent locally compact groups.


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Additional Information

Mahmood Alaghmandan
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, S-412 96 Göteborg, Sweden
Email: mahala@chalmers.se

Mahya Ghandehari
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email: mahya@udel.edu

Nico Spronk
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: nspronk@uwaterloo.ca

Keith F. Taylor
Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Email: keith.taylor@dal.ca

DOI: https://doi.org/10.1090/proc/13142
Keywords: $L^1$-projection, locally compact group, unimodular, square-integrable representation
Received by editor(s): October 21, 2015
Received by editor(s) in revised form: January 22, 2016
Published electronically: April 27, 2016
Additional Notes: The first two authors were supported by Fields Institute postdoctoral fellowships and postdoctoral fellowships at the University of Waterloo. The second author was supported by an AARMS postdoctoral fellowship while at Dalhousie University. The third and fourth authors were supported by NSERC of Canada Discovery Grants.
Communicated by: Pamela Gorkin
Article copyright: © Copyright 2016 American Mathematical Society