Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/proc/13142
Published electronically: April 27, 2016
MathSciNet review: 3544540
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Gilbert Arsac, Sur l'espace de Banach engendré par les coefficients d'une représentation unitaire, Publ. Dép. Math. (Lyon) 13 (1976), no. 2, 1-101 (French). MR 0444833
  • [2] Bruce A. Barnes, The role of minimal idempotents in the representation theory of locally compact groups, Proc. Edinburgh Math. Soc. (2) 23 (1980), no. 2, 229-238. MR 597128, https://doi.org/10.1017/S0013091500003114
  • [3] I. N. Bernšteĭn, All reductive $ {\mathfrak{p}}$-adic groups are of type I, Funkcional. Anal. i Priložen. 8 (1974), no. 2, 3-6 (Russian). MR 0348045
  • [4] Paul J. Cohen, Homomorphisms and idempotents of group algebras, Bull. Amer. Math. Soc. 65 (1959), 120-122. MR 0151548
  • [5] Jacques Dixmier, $ C^*$-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. MR 0458185
  • [6] Pierre Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236 (French). MR 0228628
  • [7] Pierre Eymard and Marianne Terp, La transformation de Fourier et son inverse sur le groupe des $ ax+b$ d'un corps local, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976-1978), II, Lecture Notes in Math., vol. 739, Springer, Berlin, 1979, pp. 207-248 (French). MR 560840
  • [8] Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
  • [9] Frederick P. Greenleaf, Norm decreasing homomorphisms of group algebras, Pacific J. Math. 15 (1965), 1187-1219. MR 0194911
  • [10] Karlheinz Gröchenig, Eberhard Kaniuth, and Keith F. Taylor, Compact open sets in duals and projections in $ L^1$-algebras of certain semi-direct product groups, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 545-556. MR 1151331, https://doi.org/10.1017/S0305004100075629
  • [11] Henry Helson, Note on harmonic functions, Proc. Amer. Math. Soc. 4 (1953), 686-691. MR 0058027
  • [12] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups, integration theory, group representations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. MR 551496
  • [13] Eberhard Kaniuth and Keith F. Taylor, Projections in $ C^*$-algebras of nilpotent groups, Manuscripta Math. 65 (1989), no. 1, 93-111. MR 1006629, https://doi.org/10.1007/BF01168369
  • [14] Eberhard Kaniuth and Keith F. Taylor, Minimal projections in $ L^1$-algebras and open points in the dual spaces of semi-direct product groups, J. London Math. Soc. (2) 53 (1996), no. 1, 141-157. MR 1362692, https://doi.org/10.1112/jlms/53.1.141
  • [15] Eberhard Kaniuth and Keith F. Taylor, Compact open sets in dual spaces and projections in group algebras of $ [{\rm FC}]^-$ groups, Monatsh. Math. 165 (2012), no. 3-4, 335-352. MR 2891258, https://doi.org/10.1007/s00605-010-0251-7
  • [16] Eberhard Kaniuth and Keith F. Taylor, Induced representations of locally compact groups, Cambridge Tracts in Mathematics, vol. 197, Cambridge University Press, Cambridge, 2013. MR 3012851
  • [17] E. Kotzmann, V. Losert, and H. Rindler, Dense ideals of group algebras, Math. Ann. 246 (1979/80), no. 1, 1-14. MR 554127, https://doi.org/10.1007/BF01352021
  • [18] Ronald L. Lipsman, Group representations, Lecture Notes in Mathematics, Vol. 388, Springer-Verlag, Berlin-New York, 1974. A survey of some current topics. MR 0372116
  • [19] Theodore W. Palmer, Banach algebras and the general theory of $ *$-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79, Cambridge University Press, Cambridge, 2001. MR 1819503
  • [20] Walter Rudin, Idempotent measures on Abelian groups, Pacific J. Math. 9 (1959), 195-209. MR 0105593
  • [21] Ross Stokke, Homomorphisms of convolution algebras, J. Funct. Anal. 261 (2011), no. 12, 3665-3695. MR 2838038, https://doi.org/10.1016/j.jfa.2011.08.014
  • [22] Alain Valette, Minimal projections, integrable representations and property $ ({\rm T})$, Arch. Math. (Basel) 43 (1984), no. 5, 397-406. MR 773186, https://doi.org/10.1007/BF01193846

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 43A20, 43A22

Retrieve articles in all journals with MSC (2010): 43A20, 43A22


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

American Mathematical Society